The Arithmetic Site and the Scaling Site


In The Riemann Hypothesis for Curves over Finite Fields, we gave a rough outline of Andre Weil’s strategy to prove the analogue of the famous Riemann hypothesis for curves over finite fields. A rather natural question to ask would be, does this strategy give us any suggestions on how to take on the original Riemann hypothesis? We mentioned briefly in The Field with One Element that some mathematicians hope to find in the theory of the so-called “field with one element” something that will allow them to apply the ideas of Weil’s proof to the original Riemann hypothesis, by viewing the scheme \text{Spec}(\mathbb{Z})  as some kind of “curve” over the “field with one element”.

In this post we will consider something along similar lines, examining a kind of “space” to which we can apply an analogue of Weil’s strategy. This approach is due to the mathematicians Alain Connes and Caterina Consani, and makes use of the concepts of sites and toposes (see More Category Theory: The Grothendieck Topos and Even More Category Theory: The Elementary Topos). This is perhaps appropriate, since sites or toposes are often referred to as “generalized spaces”.

We recall from The Riemann Hypothesis for Curves over Finite Fields some aspects of Weil’s strategy. The object in consideration is a curve C over a finite field \mathbb{F}_{q}. In order to write down the zeta function for C, we need to count the number of points over \mathbb{F}_{q^{n}}, for every n from 1 to infinity. We can do this by counting the fixed points of powers of the Frobenius morphism. Explicitly this means taking intersection numbers of the diagonal and the divisor formed by integral linear combinations of powers of the Frobenius morphism on \bar{C}\times_{\bar{\mathbb{F}}_{q}}\bar{C}, where \bar{\mathbb{F}}_{q} is an algebraic closure of \mathbb{F}_{q} (it is the direct limit of the directed system formed by all the \mathbb{F}_{q^{n}}) and \bar{C}=C\otimes_{\mathbb{F}_{q}}\bar{\mathbb{F}}_{q}. The number of points of \bar{\mathbb{F}}_{q} will be the same as the number of points of C over \mathbb{F}_{q^{n}}. Throughout this post we should keep these steps of Weil’s strategy in mind.

In order to transfer this strategy of Weil to the original Riemann hypothesis, Connes and Consani construct the arithmetic site, meant to be the analogue of C, and the scaling site, meant to be the analogue of \bar{C}. The intuition behind these constructions is that the points of the scaling site, which is the same as the points of the arithmetic site “over \mathbb{R}_{+}^{\text{max}}“, is the same as the points of the “adele class space\mathbb{Q}^{\times}\backslash\mathbb{A}_{\mathbb{Q}}/\hat{\mathbb{Z}}^{\times}, which originally came up in earlier work of Connes where he constructed a quantum-mechanical system which gives Riemann’s prime-counting function (whose study provided the historical origin of the Riemann hypothesis), in the form of Weil’s “explicit formula”, as a quantum-mechanical trace formula! In essence this work restates the Riemann hypothesis in terms of mathematical language more commonly associated to physics, and is part of Connes’ pioneering work in noncommutative geometry, a new area of mathematics also closely related to physics, in particular quantum mechanics and quantum field theory. In the definition of the adele class space, \mathbb{A}_{\mathbb{Q}} refers to the ring of adeles of \mathbb{Q} (see Adeles and Ideles), while \hat{\mathbb{Z}} refers to \prod_{p}\mathbb{Z}_{p}, where \mathbb{Z}_{p} are the p-adic integers, which can be defined as the inverse limit of the inverse system formed by \mathbb{Z}/p^{n}\mathbb{Z}.

The Arithmetic Site

We now proceed to discuss the arithmetic site. It is described as the pair (\widehat{\mathbb{N}^{\times}},\mathbb{Z}_{\text{max}}), where \widehat{\mathbb{N}^{\times}} a Grothendieck topos, which, as we may recall from More Category Theory: The Grothendieck Topos, is defined as a category equivalent to the category \text{Sh}(\mathbf{C},J) of sheaves on a site (\mathbf{C},J). In the case of \widehat{\mathbb{N}^{\times}}, \mathbf{C} is the category with only one object, and whose morphisms correspond to the multiplicative monoid of nonzero natural numbers \mathbb{N}^{\times} (we also use \mathbb{N}^{\times} to denote this category, and \mathbb{N}_{0}^{\times} to denote the category with one object and whose morphisms correspond to \mathbb{N}^{\times}\cup\{0\}), while J is the indiscrete, or chaotic, Grothendieck topology, where all presheaves are also sheaves.

As part of the definition of the arithmetic site, we must also specify a structure sheaf. In this case is provided by \mathbb{Z}_{\text{max}}, the semiring (a semiring is like a ring, but is only a monoid, and not a group, under the “addition” operation – a semiring is also sometimes called a “rig“, because it is a ring without the “n” – the negative elements, and the most common example is the natural numbers \mathbb{N} with the usual addition and multiplication) whose elements are just the integers, together with -\infty, but where the “addition” is provided by the “maximum” operation, and the “multiplication” is provided by the ordinary addition! With the arithmetic site thus defined, we denote it by \mathcal{A}.

We digress for a while to discuss the semiring \mathbb{Z}_{\text{max}}, as well as the closely related semirings \mathbb{R}_{\text{max}} (defined similarly to \mathbb{Z}_{\text{max}}, but with the real numbers instead of the integers), \mathbb{R}_{+}^{\text{max}} (whose elements are the positive real numbers, with the addition given by the maximum operation, and the multiplication given by the ordinary multiplication), and the so-called Boolean semifield \mathbb{B} (whose elements are 0 and 1, with the addition again given by the maximum operation, and the multiplication again given by the ordinary multiplication). These semirings have origins in the area of mathematics known as tropical geometry, so named because one of its pioneers, Imre Simon, comes from Brazil, which is a tropical country. However, another source of inspiration is the work of the mathematical physicist Viktor Pavlovich Maslov in “semiclassical” quantum mechanics, where certain approximations could be made as the quantum mechanical systems being studied approached the classical limit. Maslov considered a “conjugated” addition

\displaystyle \lim_{\epsilon\to 0}(x^{\frac{1}{\epsilon}}+y^{\frac{1}{\epsilon}})^{\epsilon}

and this just happened to be the same as \text{max}(x,y).

Going back to the arithmetic site, we now discuss its points. Recall from Even More Category Theory: The Elementary Topos that a point of a topos (we discussed elementary toposes in that post, but this also applies to Grothendieck toposes) is defined by a geometric morphism from the topos \mathfrak{P} of sheaves of sets on the singleton set (the set with a single element) to the topos. This refers to a pair of adjoint functors such that the left-adjoint is left-exact (preserves finite limits). Therefore, for the arithmetic site, a point p is given by such a pair p^{*} and p_{*} such that p^{*}:\widehat{\mathbb{N}^{\times}}\rightarrow\textbf{Sets} is left-exact. The point p is also uniquely determined by the covariant functor \mathscr{P}=p^{*}\circ\epsilon:\mathbb{N}^{\times}\rightarrow\textbf{Sets} where \epsilon:\mathbb{N}^{\times}\rightarrow\widehat{\mathbb{N}^{\times}} is the Yoneda embedding.

There is an equivalence of categories between the category of points of the arithmetic site and the category of totally ordered groups which are isomorphic to the nontrivial subgroups of (\mathbb{Q},\mathbb{Q}_{+}) and injective morphisms of ordered groups. For such an ordered group \textbf{H} we therefore have a point \mathscr{P}_{\textbf{H}}. This gives us a correspondence with \mathbb{Q}_{+}^{\times}\backslash\mathbb{A}_{\mathbb{Q}}^{f}/\hat{\mathbb{Z}}^{\times} (where \mathbb{A}_{\mathbb{Q}}^{f} refers to the ring of finite adeles of \mathbb{Q}, which is defined similarly to the ring of adeles of \mathbb{Q} except that the infinite prime is not considered) because any such ordered group \textbf{H} is of the form \textbf{H}_{a}, the ordered group of all rational numbers q such that aq\in\hat{\mathbb{Z}}, for some unique a\in \mathbb{A}_{\mathbb{Q}}^{f}/\hat{\mathbb{Z}}. We can also now describe the stalks of the structure sheaf \mathbb{Z}_{\text{max}} at the point \mathscr{P}_{\textbf{H}}; it is isomorphic to the semiring H_{\text{max}}, with elements given by the set (\textbf{H}\cup\{-\infty\}), addition given by the maximum operation, and multiplication given by the ordinary addition.

The arithmetic site is analogous to the curve C over the finite field \mathbb{F}_{q}. As for the finite field \mathbb{F}_{q}, its analogue is given by the Boolean semifield \mathbb{B} mentioned earlier, which has “characteristic 1“, reminiscent of the field with one element (see The Field with One Element). Next we want to find the analogues of the algebraic closure \bar{\mathbb{F}}_{q}, as well as the Frobenius morphism. The former is given by the semiring \mathbb{R}_{+}^{\text{max}}, which contains \mathbb{B}, while the latter is given by multiplicative group of the positive real numbers \mathbb{R}_{+}^{\times}, as it is isomorphic to the group of automorphisms of \mathbb{R}_{+}^{\text{max}} that keep \mathbb{B} fixed.

But while we do know that the points of the arithmetic topos are given by geometric morphisms p:\mathfrak{P}\rightarrow \widehat{\mathbb{N}^{\times}} and determined by contravariant functors \mathscr{P}_{\textbf{H}}:\mathbb{N}^{\times}\rightarrow\textbf{Sets}, what do we mean by its “points over \mathbb{R}_{+}^{\text{max}}“? A point of the arithmetic site “over \mathbb{R}_{+}^{\text{max}}” refers to the pair (\mathscr{P}_{\textbf{H}},f_{\mathscr{P}}^{\#}), where \mathscr{P}_{\textbf{H}}:\mathbb{N}^{\times}\rightarrow\textbf{Sets} as earlier, and f_{\mathscr{P}_{\textbf{H}}}^{\#}:H_{\text{max}}\rightarrow\mathbb{R}_{+}^{\text{max}} (we recall that H_{\text{max}} are the stalks of the structure sheaf \mathbb{Z}_{\text{max}}). The points of the arithmetic site over \mathbb{R}_{+}^{\text{max}} include its points “over \mathbb{B}“, which are what we discussed earlier, and mentioned to be in correspondence with \mathbb{Q}_{+}^{\times}\backslash\mathbb{A}_{\mathbb{Q}}^{f}/\hat{\mathbb{Z}}^{\times}. But in addition, there are also other points of the arithmetic site over \mathbb{R}_{+}^{\text{max}} which are in correspondence with \mathbb{Q}_{+}^{\times}\backslash((\mathbb{A}_{\mathbb{Q}}^{f}/\hat{\mathbb{Z}}^{\times})\times\mathbb{R}_{+}^{\times}), just as \mathbb{R}_{+}^{\text{max}} contains all of \mathbb{B} but also other elements. Altogether, the points of the arithmetic site over \mathbb{R}_{+}^{\text{max}} correspond to the disjoint union of \mathbb{Q}_{+}^{\times}\backslash\mathbb{A}_{\mathbb{Q}}^{f}/\hat{\mathbb{Z}}^{\times} and \mathbb{Q}_{+}^{\times}\backslash((\mathbb{A}_{\mathbb{Q}}^{f}/\hat{\mathbb{Z}}^{\times})\times\mathbb{R}_{+}^{\times}), which is \mathbb{Q}^{\times}\backslash\mathbb{A}_{\mathbb{Q}}/\hat{\mathbb{Z}}^{\times}, the adele class space as mentioned earlier.

There is a geometric morphism \Theta:\text{Spec}(\mathbb{Z})\rightarrow \widehat{\mathbb{N}_{0}^{\times}} (here \widehat{\mathbb{N}_{0}^{\times}} is defined similarly to \widehat{\mathbb{N}^{\times}}, but with \mathbb{N}_{0}^{\times} in place of \mathbb{N}^{\times}) uniquely determined by

\displaystyle \Theta^{*}:\mathbb{N}_{0}^{\times}\rightarrow \text{Sh}(\text{Spec}(\mathbb{Z}))

which sends the single object of \mathbb{N}_{0}^{\times} to the sheaf \mathcal{S} on \text{Spec}(\mathbb{Z}), which we now describe. Let H_{p} denote the set of all rational numbers q such that a_{p}q is an element of \hat{Z}, where a_{p} is the adele with a 0 for the p-th component and 1 for all other components. Then the sheaf \mathcal{S} can be described in terms of its stalks \mathcal{S}_{\mathscr{P}}, which are given by H_{p}^{+}, the positive part of H_{p}, and \mathcal{S}_{0}, given by \{0\}. The sections \Gamma(U,\mathcal{S}) are given by the maps \xi:U\rightarrow \coprod_{p}H_{p}^{+} such that \xi_{p}\neq 0 for finitely many p\in U.

The Scaling Site

Now that we have defined the arithmetic site, which is the analogue of C, and the points of the arithmetic site over \mathbb{R}_{+}^{\text{max}}, which is the analogue of the points of C over the algebraic closure \bar{\mathbb{F}}_{q}, we now proceed to define the scaling site, which is the analogue of \bar{C}=C\otimes_{\mathbb{F}_{q}}\bar{\mathbb{F}}_{q}. The points of the scaling site are the same as the points of the arithmetic site over \mathbb{R}_{+}^{\text{max}}, which is analogous to the points of \bar{C} being the same as the points of C over \bar{\mathbb{F}}_{q}. But the importance of the scaling site lies in the fact that we can construct the analogue of a sheaf of rational functions on it, and a Riemann-Roch theorem, which, as we may recall from The Riemann Hypothesis for Curves over Finite Fields, it is also an important part of Weil’s proof.

The scaling site is once again given by a pair ([0,\infty)\rtimes\mathbb{N}^{\times},\mathcal{O}), where [0,\infty)\rtimes\mathbb{N}^{\times} is a Grothendieck topos and \mathcal{O} is a structure sheaf, but both are quite sophisticated constructions compared to the arithmetic site. To describe the Grothendieck topos [0,\infty)\rtimes\mathbb{N}^{\times} we recall that it must be a category equivalent to the category \text{Sh}(\mathbf{C},J) of sheaves on some site (\mathbf{C},J). Here \mathbf{C} is the category whose objects are given by bounded open intervals \Omega\subset [0,\infty), including the empty interval \null, and whose morphisms are given by

\displaystyle \text{Hom}(\Omega,\Omega')=\{n\in\mathbb{N}^{\times}|n\Omega\subset\Omega'\}

and in the special case that \Omega is the empty interval \null, we have

\displaystyle \text{Hom}(\Omega,\Omega')=\{*\}.

The Grothendieck topology J here is defined by the collection K(\Omega) of all ordinary covers of \Omega for any object \Omega of the category \mathbf{C}:

\displaystyle \{\Omega_{i}\}_{i\in I}=\{\Omega_{i}\subset\Omega|\cup_{i}\Omega_{i}=\Omega\}

Now we have to describe the structure sheaf \mathcal{O}. We start by considering \mathbb{Z}_{\text{max}}, the structure sheaf of the arithmetic site. By “extension of scalars” from \mathbb{B} to \mathbb{R}_{+}^{\text{max}} we obtain the reduced semiring \mathbb{Z}_{\text{max}}\hat{\otimes}_{\mathbb{B}}\mathbb{R}_{+}^{\text{max}}. This is not yet the structure sheaf \mathcal{O}, because the underlying category and Grothendieck topology for the scaling site is more complicated than the arithmetic site, and unlike the case for the arithmetic site, for the scaling site not every presheaf is a sheaf. So we must first “localize” \mathbb{Z}_{\text{max}}\hat{\otimes}_{\mathbb{B}}\mathbb{R}_{+}^{\text{max}}, and this gives us the structure sheaf \mathcal{O}.

Let us describe \mathbb{Z}_{\text{max}}\hat{\otimes}_{\mathbb{B}}\mathbb{R}_{+}^{\text{max}} in more detail. Let H be a rank 1 subgroup of \mathbb{R}. Then an element of H_{\text{max}}\hat{\otimes}_{\mathbb{B}}\mathbb{R}_{+}^{\text{max}} is given by a Newton polygon N\subset\mathbb{R}^{2}, which is the convex hull of the union of finitely many quadrants (x_{j},y_{j}-Q), where Q=H\times\mathbb{R}_{+} and (x_{j},y_{j})\in H\times R (a set is a convex set if it contains the line segment connecting any two of its points; the convex hull of a set is the smallest convex set that contains it). The Newton polygon N is uniquely determined by the function

\displaystyle \ell_{N}(\lambda)=\text{max}(\lambda x_{j}+y_{j})

for \lambda\in\mathbb{R}_{+}. This correspondence gives us an isomorphism between H\hat{\otimes}_{\mathbb{B}}\mathbb{R}_{+}^{\text{max}} and \mathcal{R}(H), the semiring of convex, piecewise affine, continuous functions on [0,\infty) with slopes in H\subset\mathbb{R} and finitely many singularities, with the pointwise operations (function is a convex function if the points on and above its graph form a convex set).

Therefore, we can describe the sections \Gamma(\Omega,\mathcal{O}) of the structure sheaf \mathcal{O}, for any bounded open interval \Omega, as the set of all convex, piecewise affine, continuous functions from \Omega to \mathbb{R}_{\text{max}} with slopes in \mathbb{Z}. We can also likewise describe the stalks of the structure sheaf \mathcal{O} – for a point \mathfrak{p}_{H}:[0,\infty)\rtimes\mathbb{N}^{\times}\rightarrow\textbf{Sets} associated to a rank 1 subgroup H\subset\mathbb{R}, the stalk \mathcal{O}_{\mathfrak{p}_{H}} is given by the semiring \mathcal{R}_{H} of germs of \mathbb{R}_{+}^{\text{max}}-valued, convex, piecewise affine, continuous functions with slope in H. We also have points \mathfrak{p}_{H}^{0}:[0,\infty)\rtimes\mathbb{N}^{\times}\rightarrow\textbf{Sets} with “support \{0\}“, corresponding to the points of the arithmetic site over \mathbb{B}. For such a point, the stalk \mathcal{O}_{\mathfrak{p}_{H}^{0}} is given by the semiring (H\times\mathbb{R})_{\text{max}} associated to the totally ordered group H\times\mathbb{R}.

Now that we have decribed the Grothendieck topos [0,\infty)\rtimes\mathbb{N}^{\times} and the structure sheaf \mathcal{O}, we describe the scaling site as being given by the pair ([0,\infty)\rtimes\mathbb{N}^{\times},\mathcal{O}), and we denote it by \hat{\mathcal{A}}.

Our next task, now that we have described the arithmetic site and the scaling site, is to find the analogue of the Riemann-Roch theorem. We start by noting that we have a sheaf of semifields \mathcal{K}, defined by letting \mathcal{K}(\Omega) be the semifield of fractions of \mathcal{O}(\Omega). For an element f_{H} in the stalk \mathcal{K}_{\mathfrak{p}_{H}} of \mathcal{K}, we define its order as

\displaystyle \text{Order}_{H}(f):=h_{+}-h_{-}


\displaystyle h_{\pm}:=\lim_{\epsilon\to 0_{\pm}}(f((1+\epsilon)H)-f(H))/\epsilon

for \epsilon\in\mathbb{R}_{+}.

We let C_{p} be the set of all points \mathfrak{p}_{H}:[0,\infty)\rtimes\mathbb{N}^{\times}\rightarrow\textbf{Sets} of the scaling site \hat{\mathcal{A}} such that H is isomorphic to H_{p}. The C_{p} are the analogues of the orbits of Frobenius. There is a topological isomorphism \eta_{p}:\mathbb{R}_{+}^{\times}/p^{\mathbb{Z}}\rightarrow C_{p}. It is worth noting that the expression \mathbb{R}_{+}^{\times}/p^{\mathbb{Z}} is reminiscent of the Tate uniformization of an elliptic curve (which generalizes the idea that an elliptic curve over the complex numbers forms a lattice in the complex plane to other complete fields besides the complex numbers –  see The Moduli Space of Elliptic Curves).

We have a pullback sheaf \eta_{p}^{*}(\mathcal{O}|_{C_{p}}), which we denote suggestively by \mathcal{O}_{p}. It is the sheaf on \mathbb{R}_{+}^{\times}/p^{\mathbb{Z}} whose sections are convex, piecewise affine, continuous functions with slopes in H_{p}. We can consider the sheaf of quotients \mathcal{K}_{p} of \mathcal{O}_{p} and its global sections f:\mathbb{R}_{+}^{\times}\rightarrow\mathbb{R}, which are piecewise affine, continuous functions with slopes in H_{p} such that f(p\lambda)=f(\lambda) for all \lambda\in\mathbb{R}_{+}^{\times}. Defining

\displaystyle \text{Order}_{\lambda}(f):=\text{Order}_{\lambda H_{p}}(f\circ\eta_{p}^{-1})

we have the following property for any f\in H^{0}(\mathbb{R}_{+}^{\times}/p^{\mathbb{Z}},\mathcal{K}_{p}) (recall that the zeroth cohomology group H^{0}(\mathbb{R}_{+}^{\times}/p^{\mathbb{Z}},\mathcal{K}_{p}) is defined as the space of global sections of \mathcal{K}_{p}):

\displaystyle \sum_{\lambda\in\mathbb{R}_{+}^{\times}/p^{\mathbb{Z}}}\text{Order}_{\lambda}(f)=0

We now want to define the analogue of divisors on C_{p} (see Divisors and the Picard Group). A divisor D on C_{p} is a section C_{p}\rightarrow H, mapping \mathfrak{p}_{H}\in C_{p} to D(H)\in H, of the bundle of pairs (H,h), where H\subset\mathbb{R} is isomorphic to H_{p}, and h\in H. We define the degree of a divisor D as follows:

\displaystyle \text{deg}(D)=\sum_{\mathfrak{p}\in C_{p}}D(H)

Given a point \mathfrak{p}_{H}\in C_{p} such that H=\lambda H_{p} for some \lambda\in\mathbb{R}_{+}^{*}, we have a map \lambda^{-1}:H\rightarrow H_{p}. This gives us a canonical mapping

\displaystyle \chi: H\rightarrow H_{p}/(p-1)H_{p}\simeq\mathbb{Z}/(p-1)\mathbb{Z}

Given a divisor D on C_{p}, we define

\displaystyle \chi(D):=\sum_{\frak{p}_{H}\in C_{p}}\chi(D(H))\in\mathbb{Z}/(p-1)\mathbb{Z}

We have \text{deg}(D)=0 and \chi(D)=0 if and only if D=(f), for f\in H^{0}(\mathbb{R}_{+}^{\times}/p^{\mathbb{Z}}\mathcal{K}_{p}) i.e. D is a principal divisor.

We define the group J(C_{p}) as the quotient \text{Div}^{0}(C_{p})/\mathcal{P} of the group \text{Div}^{0}(C_{p}) of divisors of degree 0 on C_{p} by the group \mathcal{P} of principal divisors on C_{p}. The group J(C_{p}) is isomorphic to \mathbb{Z}/(p-1)\mathbb{Z}, while the group \text{Div}(C_{p})/\mathcal{P} of divisors on C_{p} modulo the principal divisors is isomorphic to \mathbb{R}\times(\mathbb{Z}/(p-1)\mathbb{Z}).

In order to state the analogue of Riemann-Roch theorem we need to define the following module over \mathbb{R}_{+}^{\text{max}}:

\displaystyle H^{0}(D):=\{f\in\mathcal{K}_{p}|D+(f)\geq 0\}

Given f\in H^{0}(C_{p},\mathcal{K}_{p}), we define

\displaystyle \|f\|_{p}:=\text{max}\{h(\lambda)|_{p}/\lambda,\lambda\in C_{p}\}

where h(\lambda) is the slope of f at \lambda. Then we have the following increasing filtration on H^{0}:

\displaystyle H^{0}(D)^{\rho}:=\{f\in H^{0}(D)|\|f\|_{p}\leq\rho\}

This allows us to define the following notion of dimension for H^{0}(D) (here \text{dim}_{\text{top}} refers to what is known as the topological dimension or Lebesgue covering dimension, a notion of dimension defined in terms of refinements of open covers):

\displaystyle \text{Dim}_{\mathbb{R}}(H^{0}(D))=\lim_{n\to\infty}p^{-n}\text{dim}_{\text{top}}(H^{0}(D)^{p^{n}})

The analogue of the Riemann-Roch theorem is now given by the following:

\displaystyle \text{Dim}_{\mathbb{R}}(H^{0}(D))+\text{Dim}_{\mathbb{R}}(H^{0}(-D))=\text{deg}(D)


This concludes our discussion of the arithmetic site and the scaling site, but I would like to discuss one more related topic also being explored by Connes and Consani – the use of \mathbb{S}-algebras, which is closely related to the \Gamma-sets we have already introduced in The Field with One Element. Both of these concepts have their origins in homotopy theory.

We recall from the short discussion at the end of The Riemann Hypothesis for Curves over Finite Fields that the Weil conjectures, which are Weil’s generalization of the Riemann hypothesis for curves over finite fields to varieties of higher dimension, were proven by making use of cohomology (in particular etale cohomology) to find the fixed points of the powers of the Frobenius morphism (the formula that gives us the fixed points of a map using cohomology is called the Lefschetz fixed point formula). Now, concepts such as monoids, semirings, and many others (including the mathematician Nikolai Durov’s approach to the field with one element, which he also uses to develop a new version of Arakelov geometry) are all subsumed under the concept of \mathbb{S}-algebras, and doing so allows us to make use of a cohomology theory called topological cyclic cohomology. Connes and Consani hope that topological cyclic cohomology will help prove the original Riemann hypothesis the way that etale cohomology helped prove the Weil conjectures. Let us discuss briefly the work of Connes and Consani on this topic.

We recall from The Field with One Element the definition of a \Gamma-set (there also referred to as a \Gamma-space). A \Gamma-set is defined to be a covariant functor from the category \Gamma^{\text{op}}, whose objects are pointed finite sets and whose morphisms are basepoint-preserving maps of finite sets, to the category \textbf{Sets}_{*} of pointed sets. An \mathbb{S}-algebra is defined to be a \Gamma-set \mathscr{A}:\Gamma^{\text{op}}\rightarrow \textbf{Sets}_{*} together with an associative multiplication \mu:\mathscr{A}\wedge \mathscr{A}\rightarrow\mathscr{A} and a unit 1:\mathbb{S}\rightarrow\mathscr{A}, where \mathbb{S}:\Gamma^{\text{op}}\rightarrow\textbf{Sets}_{*} is the inclusion functor (also known as the sphere spectrum). An \mathbb{S}-algebra is a monoid in the symmetric monoidal category of \Gamma-sets with the wedge product and the sphere spectrum.

Any monoid M defines an \mathbb{S}-algebra \mathbb{S}M via the following definition:

\displaystyle \mathbb{S}M(X):=M\wedge X

for any pointed finite set X. Here M\wedge X is the smash product of M and X as pointed sets, with the basepoint for M given by its zero element element. The maps are given by \text{Id}_{M}\times f, for f:X\rightarrow Y.

Similarly, any semiring R defines an \mathbb{S}-algebra HR via the following definition:

\displaystyle HR(X):=X^{R/*}

for any pointed finite set X. Here X^{R/*} refers to the set of basepoint preserving maps from R to X. The maps HR(f) are given by HR(f)(\phi)(y):=\sum_{x\in f^{-1}(y)}\phi(x) for f:X\rightarrow Y, x\in X, and y\in Y. The multiplication HR(X)\wedge HR(Y)\rightarrow HR(X\wedge Y) is given by \phi\psi(x,y)=\phi(x)\psi(y) for any x\in X\setminus * and y\in Y\setminus *. The unit 1_{X}:X\rightarrow HR(X) is given by 1_{X}(x)=\delta_{x} for all x in X, where \delta_{x}(y)=1 if x=y, and 0 otherwise.

Therefore we can see that the notion of \mathbb{S}-algebra subsumes the notions of monoids and semirings, and other notions such as that of “hyperrings“, which we leave to the references for the moment. Instead, we will discuss how \mathbb{S}-algebras are related to the approach of Durov to the field with one element and Arakelov geometry. As we mentioned in Arakelov Geometry, the main idea of the theory is to consider the “infinite prime” along with the other points of \text{Spec}(\mathbb{Z}). We therefore define \overline{\text{Spec}(\mathbb{Z})} as \text{Spec}(\mathbb{Z})\cup \{\infty\}. Let \mathcal{O}_{\text{Spec}(\mathbb{Z})} be the structure sheaf of \text{Spec}(\mathbb{Z}). We want to extend this to a structure sheaf on \overline{\text{Spec}(\mathbb{Z})}, and to accomplish this we will use the functor H from semirings to \mathbb{S}-algebras defined earlier. For any open set U containing \infty, we define

\displaystyle \mathcal{O}_{\overline{\text{Spec}(\mathbb{Z})}}(U):=\|H\mathcal{O}_{\text{Spec}(\mathbb{Z})}(U\cup\text{Spec}(\mathbb{Z}))\|_{1}.

The notation \|\|_{1} is defined for the \mathbb{S}-algebra HR associated to the semiring R as follows:

\displaystyle \|HR(X)\|_{1}:=\{\phi\in HR(X)|\sum_{X\*}\|\phi(x)\|\leq 1\}

where \|\| in this particular case comes from the usual absolute value on \mathbb{Q}. This becomes available to us because the sheaf \mathcal{O}_{\overline{\text{Spec}(\mathbb{Z})}} is a subsheaf of the constant sheaf \mathbb{Q}.

Given an Arakelov divisor on \overline{\text{Spec}(\mathbb{Z})} (in this context an Arakelov divisor is given by a pair (D_{\text{finite}},D_{\infty}), where D_{\text{finite}} is an ordinary divisor on \text{Spec}(\mathbb{Z}) and D_{\infty} is a real number) we can define the following sheaf of \mathcal{O}_{\overline{\text{Spec}(\mathbb{Z})}}-modules over \overline{\text{Spec}(\mathbb{Z})}:

\displaystyle \mathcal{O}_{\overline{\text{Spec}(\mathbb{Z})}}(D)(U):=\|H\mathcal{O}(D_{\text{finite}})(U\cup\text{Spec}(\mathbb{Z}))\|_{e^{a}}

where a is the real number “coefficient” of D_{\infty}, and \|\|_{\lambda} means, for an R-module E (here the \mathbb{S}-algebra HE is constructed the same as HR, except there is no multiplication or unit) with seminorm \|\|^{E} such that \|a\xi\|^{E}\leq\|a\|\|\xi\|^{E} for a\in R and \xi\in E,

\displaystyle \|HE(X)\|_{\lambda}:=\{\phi\in HE(X)|^{E}\sum_{X\*}\|\phi(x)\|^{E}\leq \lambda\}

With such sheaves of \mathbb{S}-algebras on \overline{\text{Spec}(\mathbb{Z})} now constructed, the tools of topological cyclic cohomology can be applied to it. The theory of topological cyclic cohomology is left to the references for now, but will hopefully be discussed in future posts on this blog.


The approach of Connes and Consani, whether making use of the arithmetic site and the scaling site to apply Weil’s strategy to the original Riemann hypothesis, or making use of \mathbb{S}-algebras and topological cyclic cohomology in analogy with the proof of the Weil conjectures, is still currently facing several technical obstacles. In the former case, an intersection theory and a Riemann-Roch theorem on the square of the scaling site is yet to be constructed. In the latter, there is the problem of appropriate coefficients for the cohomology theory. There are already several proposed strategies for dealing with these obstacles. Such efforts, aside from aiming to prove the Riemann hypothesis, widens the scope of the mathematics that we have today, and, perhaps more importantly, uncovers more and more the mysterious geometry underlying the familiar everyday concept of numbers.


On the Geometry of the Adele Class Space of Q by Caterina Consani

An Essay on the Riemann Hypothesis by Alain Connes

The Arithmetic Site by Alain Connes and Caterina Consani

Geometry of the Arithmetic Site by Alain Connes and Caterina Consani

The Scaling Site by Alain Connes and Caterina Consani

Geometry of the Scaling Site by Alain Connes and Caterina Consani

Absolute Algebra and Segal’s Gamma Sets by Alain Connes and Caterina Consani

New Approach to Arakelov Geometry by Nikolai Durov


Arakelov Geometry

In many posts on this blog, such as Basics of Arithmetic Geometry and Elliptic Curves, we have discussed how the geometry of shapes described by polynomial equations is closely related to number theory. This is especially true when it comes to the thousands-of-years-old subject of Diophantine equations, polynomial equations whose coefficients are whole numbers, and whose solutions of interest are also whole numbers (or, equivalently, rational numbers, since we can multiply or divide both sides of the polynomial equation by a whole number). We might therefore expect that the more modern and more sophisticated tools of algebraic geometry (which is a subject that started out as just the geometry of shapes described by polynomial equations) might be extremely useful in answering questions and problems in number theory.

One of the tools we can use for this purpose is the concept of an arithmetic scheme, which makes use of the ideas we discussed in Grothendieck’s Relative Point of View. An arithmetic variety is defined to be a a regular scheme that is projective and flat over the scheme \text{Spec}(\mathbb{Z}). An example of this is the scheme \text{Spec}(\mathbb{Z}[x]), which is two-dimensional, and hence also referred to as an arithmetic surface.

We recall that the points of an affine scheme \text{Spec}(R), for some ring R, are given by the prime ideals of R. Therefore the scheme \text{Spec}(\mathbb{Z}) has one point for every prime ideal – one “closed point” for every prime number p, and a “generic point” given by the prime ideal (0).

However, we also recall from Adeles and Ideles the concept of the “infinite primes” – which correspond to the archimedean valuations of a number field, just as the finite primes (primes in the classical sense) correspond to the nonarchimedean valuations. It is important to consider the infinite primes when dealing with questions and problems in number theory, and therefore we need to modify some aspects of algebraic geometry if we are to use it in helping us with number theory.

We now go back to arithmetic schemes, taking into consideration the infinite prime. Since we are dealing with the ordinary integers \mathbb{Z}, there is only one infinite prime, corresponding to the embedding of the rational numbers into the real numbers. More generally we can also consider an arithmetic variety over \text{Spec}(\mathcal{O_{K}}) instead of \text{Spec}(\mathbb{Z}), where \mathcal{O}_{K} is the ring of integers of a number field K. In this case we may have several infinite primes, corresponding to the embediings of K into the real and complex numbers. In this post, however, we will consider only \text{Spec}(\mathbb{Z}) and one infinite prime.

How do we describe an arithmetic scheme when the scheme \text{Spec}(\mathbb{Z}) has been “compactified” with the infinite prime? Let us look at the fibers of the arithmetic scheme. The fiber of an arithmetic scheme X at a finite prime p is given by the scheme defined by the same homogeneous polynomials as X, but with the coefficients taken modulo p, so that they are elements of the finite field \mathbb{F}_{p}. The fiber over the generic point (0) is given by taking the tensor product of the coordinate ring of X with the rational numbers. But how should we describe the fiber over the infinite prime?

It was the idea of the mathematician Suren Arakelov that the fiber over the infinite prime should be given by a complex variety – in the case of an arithmetic surface, which Arakelov originally considered, the fiber should be given by a Riemann surface.  The ultimate goal of all this machinery, at least when Arakelov was constructing it, was to prove the famous Mordell conjecture, which states that the number of rational solutions to a curve of genus greater than or equal to 2 was finite. These rational solutions correspond to sections of the arithmetic surface, and Arakelov’s strategy was to “bound” the number of these solutions by constructing a “height function” using intersection theory (see Algebraic Cycles and Intersection Theory) on the arithmetic surface. Arakelov unfortunately was not able to carry out his proof. He had only a very short career, being forced to retire after being diagnosed with schizophrenia. The Mordell conjecture was eventually proved by another mathematician, Gerd Faltings, who continues to develop Arakelov’s ideas.

Since we will be dealing with a complex variety, we must first discuss a little bit of differential geometry, in particular complex geometry (see An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry). Let X be a smooth projective complex equidimensional variety with complex dimension d. The space A^{n}(X) of differential forms (see Differential Forms) of degree n on X has the following decomposition:

\displaystyle A^{n}(X)=\bigoplus_{p+q=n}A^{p,q}(X)

We say that A^{p,q}(X) is the vector space of complex-valued differential forms of type (p,q). We have differential operators

\displaystyle \partial:A^{p,q}(X)\rightarrow A^{p+1,q}(X)

\displaystyle \bar{\partial}:A^{p,q}(X)\rightarrow A^{p,q+1}(X).

\displaystyle d=\partial+\bar{\partial}:A^{n}\rightarrow A^{n+1}.

We let D_{p,q}(X) be the dual to the vector space A^{p,q}(X), and we write D^{p,q}(X) to denote D_{d-p,d-q}(X). We refer to an element of D^{p,q} as a current of type (p,q). We have an inclusion map

\displaystyle A^{p,q}\rightarrow D^{p,q}

mapping a differential form \omega of type (p,q) to a current [\omega] of type (p,q), given by

\displaystyle [\omega](\alpha)=\int_{X}\omega\wedge\alpha

for all \alpha\in A^{d-p,d-q}(X).

The differential operators \partial, \bar{\partial}, d, and induce maps \partial', \bar{\partial}', and d' on D^{p,q}. We define the maps \partial, \bar{\partial}, and d on D^{p,q} by

\displaystyle \partial=(-1)^{n+1}\partial'

\displaystyle \bar{\partial}=(-1)^{n+1}\bar{\partial}'

\displaystyle d=(-1)^{n+1}d'

We also define

\displaystyle d^{c}=(4\pi i)^{-1}(\partial-\bar{\partial}).

For every irreducible analytic subvariety i:Y\hookrightarrow X of codimension p, we define the current \delta_{Y}\in D^{p,p} by

\displaystyle \delta_{Y}(\alpha):=\int_{Y^{ns}}i^{*}\alpha

for all \alpha\in A^{d-p,d-q}, where Y^{ns} is the nonsingular locus of Y.

A Green current g for a codimension p analytic subvariety Y is defined to be an element of D^{p-1,p-1}(X) such that

\displaystyle dd^{c}g+\delta_{Y}=[\omega]

for some \omega\in A^{p,p}(X).

Let \tilde{X} be the resolution of singularities of X. This means that there exists a proper map \pi: \tilde{X}\rightarrow X such that \tilde X is smooth, E:=\pi^{-1}(Y) is a divisor with normal crossings (this means that each irreducible component of E is nonsingular, and whenever they meet at a point their local equations  are linearly independent) whenever Y\subset X contains the singular locus of X, and \pi: \tilde{X}\setminus E\rightarrow X\setminus Y is an isomorphism.

A smooth form \alpha on X\setminus Y is said to be of logarithmic type along Y if there exists a projective map \pi:\tilde{X}\rightarrow X such that E:= \pi^{-1}(Y) is a divisor with normal crossings, \pi:\tilde{X}\setminus E\rightarrow X\setminus Y is smooth, and \alpha is the direct image by \pi of a form \beta on X\setminus E satisfying the following equation

\displaystyle \beta=\sum_{i=1}^{k}\alpha_{i}\text{log}|z_{i}|^{2}+\gamma

where z_{1}z_{2} ... z_{k}=0 is a local equation of E for every x in X, \alpha_{i} are \partial and \bar{\partial} closed smooth forms, and \gamma is a smooth form.

For every irreducible subvariety Y\subset X there exists a smooth form g_{Y} on X\setminus Y of logarithmic type along Y such that [g_{Y}] is a Green current for Y:

\displaystyle dd^{c}[g_{Y}]+\delta_{Y}=[\omega]

where w is smooth on X. We say that [g_{Y}] is a Green current of logarithmic type.

We now proceed to discuss this intersection theory on the arithmetic scheme. We consider a vector bundle E on the arithmetic scheme X, a holomorphic vector bundle (a complex vector bundle E_{\infty} such that the projection map is holomorphic) on the fibers X_{\infty} at the infinite prime, and a smooth hermitian metric (a sesquilinear form h with the property that h(u,v)=\overline{h(v,u)}) on E_{\infty} which is invariant under the complex conjugation on X_{\infty}. We refer to this collection of data as a hermitian vector bundle \bar{E} on X.

Given an arithmetic scheme X and a hermitian vector bundle \bar{E} on X, we can define associated “arithmetic”, or “Arakelov-theoretic” (i.e. taking into account the infinite prime) analogues of the algebraic cycles and Chow groups that we discussed in Algebraic Cycles and Intersection Theory.

An arithmetic cycle on X is a pair (Z,g) where Z is an algebraic cycle on X, i.e. a linear combination \displaystyle \sum_{i}n_{i}Z_{i} of closed irreducible subschemes Z_{i} of X, of some fixed codimension p, with integer coefficients n_{i}, and g is a Green current for Z, i.e. g satisfies the equation

\displaystyle dd^{c}g+\delta_{Z}=[\omega]


\displaystyle \delta_{Z}(\eta)=\sum_{i}n_{i}\int_{Z_{i}}\eta

for differential forms \omega and \eta of appropriate degree.

We define the arithmetic Chow group \widehat{CH}^{p}(X) as the group of arithmetic cycles \widehat{Z}^{p}(X) modulo the subgroup \widehat{R}^{p}(X) generated by the pairs (0,\partial u+\bar{\partial}v) and (\text{div}(f),-\text{log}(|f|^{2})), where u and v are currents of appropriate degree and f is some rational function on some irreducible closed subscheme of codimension p-1 in X .

Next we want to have an intersection product on Chow groups, i.e. a bilinear pairing

\displaystyle \widehat{CH}^{p}(X)\times\widehat{CH}^{q}(X)\rightarrow\widehat{CH}^{p+q}(X)

We now define this intersection product. Let [Y,g_{Y}]\in\widehat{CH}^{p}(X) and [Z,g_{Z}]\in\widehat{CH}^{q}. Assume that Y and Z are irreducible. Let Y_{\mathbb{Q}}=Y\otimes_{\text{Spec}(\mathbb{Z})}\text{Spec}(\mathbb{Q}), and Z_{\mathbb{Q}}=Z\otimes_{\text{Spec}(\mathbb{Z})}\text{Spec}(\mathbb{Q}). If Y_{\mathbb{Q}} and Z_{\mathbb{Q}} intersect properly, i.e. \text{codim}(Y_{\mathbb{Q}}\cap Z_{\mathbb{Q}})=p+q, then we have

\displaystyle [(Y,g_{Y})]\cdot [(Z,g_{Z})]:=[[Y]\cdot[Z],g_{Y}*g_{Z}]

where [Y]\cdot[Z] is just the usual intersection product of algebraic cycles, and g_{Y}*g_{Z} is the *-product of Green currents, defined for a Green current of logarithmic type g_{Y} and a Green current g_{Z}, where Y and Z are closed irreducible subsets of X with Z not contained in Y, as

\displaystyle g_{Y}*g_{Z}:=[\tilde{g}_{Y}]*g_{Z}\text{ mod }(\text{im}(\partial)+\text{im}(\bar{\partial}))


\displaystyle [g_{Y}]*g_{Z}:=[g_{Y}]\wedge\delta_{Z}+[\omega_{Y}]\wedge g_{Z}



for q:\tilde{Z}\rightarrow X is the resolution of singularities of Z composed with the inclusion of Z into X.

In the case that Y_{\mathbb{Q}} and \mathbb{Q} do not intersect properly, there is a rational function f_{y} on y\in X_{\mathbb{Q}}^{p-1} such that \displaystyle Y+\sum_{y}\text{div}(f_{y}) and Z intersect properly, and if g_{y} is another rational function such that \displaystyle Y+\sum_{y}\text{div}(f_{y})_{\mathbb{Q}} and Z_{\mathbb{Q}} intersect properly, the cycle

\displaystyle (\sum_{y}\widehat{\text{div}}(f_{y})-\sum_{y}\widehat{\text{div}}(g_{y}))\cdot(Z,g_{Z})

is in the subgroup \widehat{R}^{p}(X). Here the notation \widehat{\text{div}}(f_{y}) refers to the pair (\text{div}(f),-\text{log}(|f|^{2})).

This concludes our little introduction to arithmetic intersection theory. We now give a short discussion what else can be done with such a theory. The mathematicians Henri Gillet and Christophe Soule used this arithmetic intersection theory to construct arithmetic analogues of Chern classes, Chern characters, Todd classes, and the Grothendieck-Riemann-Roch theorem (see Chern Classes and Generalized Riemann-Roch Theorems). These constructions are not so straightforward – for instance, one has to deal with the fact that unlike the classical case, the arithmetic Chern character is not additive on exact sequences. This failure to be additive on exact sequences is measured by the Bott-Chern character. The Bott-Chern character plays a part in defining the arithmetic analogue of the Grothendieck group \widehat{K}_{0}(X).

In order to define the arithmetic analogue of the Grothendieck-Riemann-Roch theorem, one must then define the direct image map f_{*}:\widehat{K}_{0}(X)\rightarrow\widehat{K}_{0}(Y) for a proper flat map f:X\rightarrow Y of arithmetic varieties. This involves constructing a canonical line bundle \lambda(E) on Y, whose fiber at y is the determinant of cohomology of X_{y}=f^{-1}(y), i.e.

\displaystyle \lambda(E)_{y}=\bigotimes_{q\geq 0}(\text{det}(H^{q}(X_{y},E))^{(-1)^{q}}

as well as a metric h_{Q}, called the Quillen metric, on \lambda(E). With such a direct image map we can now give the statement of the arithmetic Grothendieck-Riemann-Roch theorem. It was originally stated by Gillet and Soule in terms of components of degree one in the arithmetic Chow group \widehat{CH}(Y)\otimes_{\mathbb{Z}}\mathbb{Q}:


where \widehat{\text{ch}} denotes the arithmetic Chern character, \widehat{\text{Td}} denotes the arithmetic Todd class, Tf is the relative tangent bundle of f, a is the map from

\displaystyle \tilde{A}(X)=\bigoplus_{p\geq 0}A^{p,p}(X)/(\text{im}(\partial)+\text{im}(\bar{\partial}))

to \widehat{CH}(X) sending the element \eta in \tilde{A}(X) to the class of (0,\eta) in \widehat{CH}(X), and

\displaystyle R(L)=\sum_{m\text{ odd, }\geq 1}(2\zeta'(-m)+\zeta(m)(1+\frac{1}{2}+...+\frac{1}{m}))\frac{c_{1}(L)^{m}}{m!}.

Later on Gillet and Soule formulated the arithmetic Grothendieck-Riemann-Roch theorem in higher degree as

\displaystyle \widehat{\text{ch}}(f_{*}(x))=f_{*}(\widehat{\text{Td(g)}}\cdot(1-a(R(Tf_{\mathbb{C}})))\cdot\widehat{\text{ch}}(x))

for x\in\widehat{K}_{0}(X).

Aside from the work of Gillet and Soule, there is also the work of the mathematician Amaury Thuillier making use of ideas from p-adic geometry, constructing a nonarchimedean potential theory on curves that allows the finite primes and the infinite primes to be treated on a more equal footing, at least for arithmetic surfaces. The work of Thuillier is part of ongoing efforts to construct an adelic geometry, which is hoped to be the next stage in the evolution of Arakelov geometry.


Arakelov Theory on Wikipedia

Arithmetic Intersection Theory by Henri Gillet and Christophe Soule

Theorie de l’Intersection et Theoreme de Riemann-Roch Arithmetiques by Jean-Benoit Bost

An Arithmetic Riemann-Roch Theorem in Higher Degrees by Henri Gillet and Christophe Soule

Theorie du Potentiel sur les Courbes en Geometrie Analytique Non Archimedienne et Applications a la Theorie d’Arakelov by Amaury Thuillier

Explicit Arakelov Geometry by Robin de Jong

Notes on Arakelov Theory by Alberto Camara

Lectures in Arakelov Theory by C. Soule, D. Abramovich, J.-F. Burnol, and J. Kramer

Introduction to Arakelov Theory by Serge Lang

The Field with One Element

Introduction: Analogies between Function Fields and Number Fields

As has been hinted at, and mentioned in passing, in several previous posts on this blog, there are important analogies between numbers and functions. The analogy can perhaps be made most explicit in the case of \mathbb{Z} (the ring of ordinary integers) and \mathbb{F}_{p}[t] (the ring of polynomials in one variable t over the finite field \mathbb{F}_{p}). We also often say that the analogy is between \mathbb{Q} (the field of rational numbers) and \mathbb{F}_{p}(t) (the field of rational functions in one variable t over the finite field \mathbb{F}_{p}), which are the respective fields of fractions of \mathbb{Z} and \mathbb{F}_{p}[t]. Recall also from Some Basics of Class Field Theory that \mathbb{Q} and \mathbb{F}_{p}(t) are examples of what we call global fields, together with their respective finite extensions.

Let us go back to \mathbb{Z} and \mathbb{F}_{p}[t] and compare their similarities. They are both principal ideal domains, which means that all their ideals can be generated by a single element. They both have groups of units (elements which have multiplicative inverses) which are finite. They both have an infinite number of prime ideals (generated by prime numbers in the case of \mathbb{Z}, and by monic irreducible polynomials in the case of \mathbb{F}_{p}[t]), and finally, they share the property that their residue fields over these prime ideals are finite.

But of course, despite all these analogies, a rather obvious question still remains unanswered. Regarding this question we quote the words of the mathematician Christophe Soule:

“The analogy between number fields and function fields finds a basic limitation with the lack of a ground field. One says that \text{Spec}(\mathbb{Z}) (with a point at infinity added, as is familiar in Arakelov geometry) is like a (complete) curve, but over which field?”

This question led to the development of the idea of the “field with one element”, also written \mathbb{F}_{1}, or sometimes \mathbb{F}_{\text{un}} (it’s a pun taken from “un”, the French word for “one”). Taken literally, there is no such thing  as a “field” with one element – the way we define a field, it must always have a “one” and a “zero”, and these two elements must be different. Instead, the idea of the “field with one element” is just a name for ideas that extend the analogy between function fields and number fields, as if this “field” really existed. The name itself has historical origins in the work of the mathematician Jacques Tits involving certain groups called Chevalley groups and Weil groups, where surprising results appear in the limit when the number of elements of the finite fields involved goes to one – but in most approaches now, the “field with one element” is not a field, and often has more than one element. The whole point is that these ideas may still work, even though the “field” itself may not even exist! As one might expect, in order to pursue these ideas one must think out of the box, and different mathematicians have approached this question in different ways.

In this post, we will look at four approaches to the field with one element, developed by the mathematicians Anton Deitmar, Christophe Soule, Bertrand Toen and Michel Vaquie, and James Borger. There are many more approaches besides these, but we will perhaps discuss them in future posts.

Note: Throughout this post it will be helpful to remind ourselves that since there exists a map from the integers \mathbb{Z} to any ring, we can think of rings as \mathbb{Z}-algebras. One of the ways the idea of the field with one element is approached is by exploring what \mathbb{F}_{1}-algebras mean, if ordinary rings are \mathbb{Z}-algebras.

The Approach of Deitmar

Deitmar defines the “category of rings over \mathbb{F}_{1}” (this is the term Deitmar uses, but we can also think of this as the category of \mathbb{F}_{1}-algebras) as simply the category of monoids. A monoid A is also written as \mathbb{F}_{A} to emphasize its nature as a “ring over \mathbb{F}_{1}“. The “field with one element” \mathbb{F}_{1} is simply defined to be the trivial monoid.

For an \mathbb{F}_{1}-ring \mathbb{F}_{A} we define the base extension (see Grothendieck’s Relative Point of View) to \mathbb{Z} by taking the “monoid ring” \mathbb{Z}[A]:

\displaystyle \mathbb{F}_{A}\otimes_{\mathbb{F}_{1}}\mathbb{Z}=\mathbb{Z}[A]

We may think of this monoid ring as a ring whose elements are formal sums of elements of the monoid A with integer coefficients, and with a multiplication provided by the multiplication on A, commuting with the scalar multiplication.

Meanwhile we also have the forgetful functor F which simply “forgets” the additive structure of a ring, leaving us with a monoid under its multiplication operation. The base extension functor -\otimes_{\mathbb{F}_{1}}\mathbb{Z} is left adjoint to the forgetful functor F, i.e. for every ring R and every \mathbb{F}_{A}/\mathbb{F}_{1} we have \text{Hom}_{\text{Rings}}(\mathbb{F}_{A}\otimes_{\mathbb{F}_{1}}\mathbb{Z}, R)\cong\text{Hom}_{\mathbb{F}_{1}}(\mathbb{F}_{A},F(R)) (see also Adjoint Functors and Monads).

The important concepts of “localization” and “ideals” in the theory of rings, important to construct the structure sheaf of a variety or a scheme, have analogues in the theory of monoids. The idea is that they only make use of the multiplicative structure of rings, so we can forget the additive structure and consider monoids instead. Hence, we can define varieties or schemes over \mathbb{F}_{1}. Many other constructions of algebraic geometry can be replicated with only monoids instead of rings, such as sheaves of modules over the structure sheaf. Deitmar then defines the zeta function of a scheme over \mathbb{F}_{1}, and hopes to connect this with known ideas about zeta functions (see for example our discussion in The Riemann Hypothesis for Curves over Finite Fields).

Deitmar’s idea of using monoids is one of the earlier approaches to the idea of the field with one element, and has become somewhat of a template for other approaches. One may be able to notice the influence of Deitmar’s work in the other approaches that we will discuss in this post.

The Approach of Soule

Soule’s question, as phrased in his paper On the Field with One Element, is as follows:

“Which varieties over \mathbb{Z} are obtained by base change from \mathbb{F}_{1} to \mathbb{Z}?”

Soule’s approach to answering this question then makes use of three concepts. The first one is a suggestion from the early days of the development of the idea of the field with one element, apparently due to the mathematicians Andre Weil and Kenkichi Iwasawa, that the finite field extensions of the field with one element should consist of the roots of unity, together with zero.

The second concept is an important point that we only touched on briefly from Algebraic Spaces and Stacks, namely, that we may identify the functor of points of a scheme with the scheme itself. Now the functor of points of a scheme is uniquely determined by its values on affine schemes, and the category of affine schemes is the opposite category to the category of rings; therefore, we now redefine a scheme simply as a covariant functor from the category of rings to the category of sets, which is representable.

The third concept is the idea of an evaluation of a function at a point. Soule implements this concept by including a \mathbb{C}-algebra as part of his definition of a variety over \mathbb{F}_{1}, together with a natural transformation that expresses this evaluation.

We now give the details of Soule’s construction, proceeding in four steps. Taking into account the first concept mentioned earlier,  we consider the following expression, the base extension of \mathbb{F}_{1^{n}} to \mathbb{Z} over \mathbb{F}_{1}:

\displaystyle \mathbb{F}_{1^{n}}\otimes_{\mathbb{F}_{1}}\mathbb{Z}=\mathbb{Z}[T]/(T^{n}-1)=\mathbb{Z}[\mu_{n}]

We shall also denote this ring by R_{n}. We can form a category whose objects are the finite tensor products of R_{n}, for n\geq 1, and we denote this category by \mathcal{R}.

An affine gadget over \mathbb{F}_{1} is a triple (\underline{X},\mathcal{A}_{X},e_{X}) where \underline{X} is a covariant functor from the category \mathcal{R} to the category of sets, \mathcal{A}_{X} is a \mathbb{C}-algebra, and e_{X} is a natural transformation from \underline{X} to \text{Hom}(\mathcal{A}_{X},\mathbb{C}[-]).

A morphism of affine gadgets consists of a natural transformation \underline{\phi}:\underline{X}\rightarrow\underline{Y} and a morphism of algebras \phi^{*}:\mathcal{A}_{X}\rightarrow\mathcal{A}_{Y} such that f(\underline{\phi}(P))=(\phi^{*}(f))(P). A morphism (\underline{\phi}, \phi^{*}) is also called an immersion if \underline{\phi} and \phi^{*} are both injective.

An affine variety over \mathbb{F}_{1} is an affine gadget X=(\underline{X},\mathcal{A}_{X},e_{X}) over \mathbb{F}_{1} such that

(i) for any object R of \mathcal{R}, the set \underline{X}(R) is finite, and

(ii) there exists an affine scheme X_{\mathbb{Z}}=X\otimes_{\mathbb{F}_{1}}\mathbb{Z} of finite type over \mathbb{Z} and immersion i:X\rightarrow \mathcal{G}(X_{\mathbb{Z}}) with the universal property that for any other affine scheme V of finite type over \mathbb{Z} and morphism \varphi:X\rightarrow\mathcal{G}(V), there exists a unique morphism \varphi_{\mathbb{Z}}:X_{\mathbb{Z}}\rightarrow\mathcal{G}(V) such that \varphi=\mathcal{G}(\varphi_{\mathbb{Z}})\circ i.

An object over \mathbb{F}_{1} is a triple (\underline{\underline{X}},\mathcal{A}_{X},e_{X}) where \underline{\underline{X}} is a contravariant functor from the category of affine gadgets over \mathbb{F}_{1}\mathcal{A}_{X} is once again a \mathbb{C}-algebra, and e_{X} is a natural transformation from \underline{\underline{X}} to \text{Hom}(\mathcal{A}_{X},\mathbb{C}[-]).

A morphism of objects is defined in the same way as a morphism of affine gadgets.

A variety over \mathbb{F}_{1} is an object X=(\underline{\underline{X}},\mathcal{A}_{X},e_{X}) over \mathbb{F}_{1} such that such that

(i) for any object R of \mathcal{R}, the set \underline{\underline{X}}(\text{Spec}(R)) is finite, and

(ii) there exists a scheme X_{\mathbb{Z}}=X\otimes_{\mathbb{F}_{1}}\mathbb{Z} of finite type over \mathbb{Z} and immersion i:X\rightarrow \mathcal{O}b(X_{\mathbb{Z}}) with the universal property that for any other scheme V of finite type over \mathbb{Z} and morphism \varphi:X\rightarrow\mathcal{O}b(V), there exists a unique morphism \varphi_{\mathbb{Z}}:X_{\mathbb{Z}}\rightarrow\mathcal{O}b(V) such that \varphi=\mathcal{O}b(\varphi_{\mathbb{Z}})\circ i.

Like Deitmar, Soule constructs the zeta function of a variety over \mathbb{F}_{1}, and furthermore explores connections with certain kinds of varieties called “toric varieties”, which are also of interest in other approaches to the field with one element, and the theory of motives (see The Theory of Motives).

The Approach of Toen and Vaquie

We recall from Grothendieck’s Relative Point of View that we call a scheme X a scheme “over” S, or an S-scheme, if there is a morphism of schemes from X to S, and if S is an affine scheme defined as \text{Spec}(R) for some ring R, we also refer to it as a scheme over R, or an R-scheme. We recall also every scheme is a scheme over \text{Spec}(\mathbb{Z}), or a \mathbb{Z}-scheme. The approach of Toen and Vaquie is to construct categories of schemes “under” \text{Spec}(\mathbb{Z}).

From Monoidal Categories and Monoids we know that rings are the monoid objects in the monoidal category of abelian groups, and abelian groups are \mathbb{Z}-modules.

More generally, for a symmetric monoidal category (\textbf{C}, \otimes, \mathbf{1}) that is complete, cocomplete, and closed (i.e. possesses internal Homs related to the monoidal structure \otimes, see again Monoidal Categories and Monoids), we have in \textbf{C} a notion of monoid, for such a monoid A a notion of an A-module, and for a morphism of monoids A\rightarrow B a notion of a base change functor -\otimes_{A}B from A-modules to B-modules.

Therefore, if we have a category \textbf{C} with a symmetric monoidal functor \textbf{C}\rightarrow \mathbb{Z}\text{-Mod}, we obtain a notion of a “scheme relative to \textbf{C}” and a base change functor to \mathbb{Z}-schemes. This gives us our sought-for notion of schemes under \text{Spec}(\mathbb{Z}).

In particular, there exists a notion of commutative monoids (associative and with unit) in \textbf{C}, and they form a category which we denote by \textbf{Comm}(\textbf{C}). We define the category of affine schemes related to \textbf{C} as \textbf{Aff}_{\textbf{C}}:= \textbf{Comm}(\textbf{C})^{\text{op}}.

These constructions satisfy certain properties needed to define a category of schemes relative to (\textbf{C},\otimes,\mathbf{1}), such as a notion of Zariski topology. A relative scheme is defined as a sheaf on the site \textbf{Aff}_{\textbf{C}} provided with the Zariski topology, and which has a covering by affine schemes. The category of schemes obtained is denoted \textbf{Sch}(\textbf{C}). It is a subcategory of the category of sheaves on \textbf{Aff}_{\textbf{C}} which is closed under the formation of fiber products and disjoint unions. It contains a full subcategory of affine schemes, given by the representable sheaves, and which is equivalent to the category \textbf{Comm}(\textbf{C})^{\text{op}}. The purely categorical nature of the construction makes the category \textbf{Sch}(\textbf{C}) functorial in \textbf{C}.

In their paper, Toen and Vacquie give six examples of their construction, one of which is just the ordinary category of schemes, while the other five are schemes “under \text{Spec}(\mathbb{Z})“.

First we let (C,\otimes,\mathbf{1})=(\mathbb{Z}\text{-Mod},\otimes,\mathbb{Z}), the symmetric monoidal category of abelian groups (for the tensor product). The category of schemes obtained \mathbb{Z}\text{-Sch} is equivalent to the category of schemes in the usual sense.

The second example will be (C,\otimes,\mathbf{1})=(\mathbb{N}\text{-Mod},\otimes,\mathbb{N}) the category of commutative monoids, or abelian semigroups, with the tensor product, which could also be called \mathbb{N}-modules. The category of schemes in this case will be denoted \mathbb{N}\text{-Sch}, and the subcategory of affine schemes is equivalent to the opposite category of commutative semirings.

The third example is (C,\otimes,\mathbf{1})=(\text{Ens},\times, *), the symmetric monoidal category of sets with the direct product. The category of relative schemes will be denoted \mathbb{F}_{1}\text{-Sch}, and we can think of them as schemes or varieties defined on the field with one element. By definition, the subcategory of affine \mathbb{F}_{1}-schemes is equivalent to the opposite category of commutative monoids.

We have the base change functors

-\otimes_{\mathbb{F}_{1}}\mathbb{N}:\mathbb{F}_{1}\text{-Sch}\rightarrow \mathbb{N}\text{-Sch}


-\otimes_{\mathbb{N}}\mathbb{Z}:\mathbb{N}\text{-Sch}\rightarrow \mathbb{Z}\text{-Sch}

We can compose these base change functors and represent it with the following diagram:


The final three examples of “schemes under \text{Spec}(\mathbb{Z})” given by Toen and Vaquie make use of ideas from “homotopical algebraic geometry“. Homotopical algebraic geometry is a very interesting subject that unfortunately we have not discussed much on this blog. Roughly, in homotopical algebraic geometry the role of rings in ordinary algebraic geometry is taken over by ring spectra – spectra (in the sense of Eilenberg-MacLane Spaces, Spectra, and Generalized Cohomology Theories) with a “smash product” operation. This allows us to make use of concepts from abstract homotopy theory. In this post we will only introduce some very basic concepts that we will need to discuss Toen and Vaquie’s examples, and leave the rest to the references.

We will need the concepts of \Gamma-spaces and simplicial sets. We define the category \Gamma^{0} to be the category whose objects are “pointed” finite sets (a finite set where one element is defined to be the “basepoint”) and whose morphisms are maps of finite sets that preserve the basepoint. We also define the category \Delta to be the category whose objects are finite ordered sets [n]=\{0<1<2...<n\} and whose morphisms are monotone (non-decreasing) maps of finite ordered sets. A \Gamma-space is then simply a covariant functor from the category \Gamma^{0} to the category of pointed sets, while a simplicial set is a covariant functor from the category \Delta to the category of sets. Simplicial sets are rather abstract constructions, but they are inspired by simplices and simplicial complexes in algebraic topology (see Simplices).

Let M be a \Gamma-space. If there is a monoid structure on \pi_{0}M(1_{+}) (see Homotopy Theory), then we say that M is a special \Gamma-space. If, in addition, this structure is also an abelian group structure, then we say that M is a very special \Gamma-space.

The category of \Gamma-spaces and the category of simplicial sets are both symmetric monoidal categories, which we need to define relative schemes. For the category of \Gamma-spaces, we have the smash product, defined by the requirement that any morphism F_{1}\wedge F_{2}\rightarrow G to any functor G from \Gamma^{0}\times \Gamma^{0} to the category of pointed sets be a natural transformation, i.e. there are maps of pointed sets from F_{1}\wedge F_{2}(X\wedge Y) to G(X\wedge Y), natural in X and Y (here X\wedge Y refers to the smash product of pointed sets obtained by taking the direct product and collapsing the wedge sum, see Eilenberg-MacLane Spaces, Spectra, and Generalized Cohomology Theories),  and the unit is the sphere spectrum \mathbb{S}, which is just the inclusion functor from the category of pointed finite sets to the category of pointed sets.

For the category of simplicial sets, we have the direct product, defined as the functor X\times Y which sends the finite ordered set [n] to the set X([n])\times Y([n]), for two simplicial sets X and Y, and the unit is the functor *, which sends any finite ordered set to the set with a single element.

We now go back to Toen and Vaquie’s final three examples of relative schemes. The first of these examples is when one has (C,\otimes,\mathbf{1}) = (\mathcal{GS},\wedge,\mathbb{S}), the category of very special \Gamma-spaces. We thus have a category of schemes relative to \mathcal{GS}, which we will denote \mathbb{S}\text{-Sch}, where the notation \mathbb{S} recalls the sphere spectrum.

The second example is (C,\otimes,\mathbf{1})=(\mathcal{MS},\wedge,\mathbb{S}_{+}), the category of special \Gamma-spaces. The category of relative schemes will be noted \mathbb{S}_{+}\text{-Sch}, and its affine objects are homotopical analogs of commutative semirings. The notation \mathbb{S}_{+} intuitively means the semiring in spectra of positive integers, and is a homotopical version of the semiring \mathbb{N}.

The third example is (C,\otimes,\mathbf{1})=(\text{SEns},\times,*), the category of simplicial sets with its direct product. The schemes that we obtain are homotopical versions of the \mathbb{F}_{1}-schemes, and will be called \mathbb{S}_{1}-schemes, where \mathbb{S}_{1} may be thought of as the “ring spectrum with one element”, in analogy with \mathbb{F}_{1}, the “field with one element”.

Similar to the earlier cases, we also have the base change functors

-\otimes_{\mathbb{S}_{1}}\mathbb{S}_{+}:\mathbb{S}_{1}\text{-Sch}\rightarrow \mathbb{S}_{+}\text{-Sch}


-\otimes_{\mathbb{S}_{+}}\mathbb{S}:\mathbb{S}_{+}\text{-Sch}\rightarrow \mathbb{S}\text{-Sch}

which we can also compose and represent it with the following diagram:


Moreover, we also have the following functors:

-\otimes_{\mathbb{S}_{1}}\mathbb{F}_{1}:\mathbb{S}_{1}\text{-Sch}\rightarrow \mathbb{F}_{1}\text{-Sch}

-\otimes_{\mathbb{S}_{+}}\mathbb{N}:\mathbb{S}_{+}\text{-Sch}\rightarrow \mathbb{N}\text{-Sch}


-\otimes_{\mathbb{S}}\mathbb{Z}:\mathbb{S}\text{-Sch}\rightarrow \mathbb{Z}\text{-Sch}

which relate the “homotopical” relative schemes to the ordinary relative schemes. And so, all these schemes, both the new schemes “under \text{Spec}(\mathbb{Z})” as well as the ordinary schemes over \text{Spec}(\mathbb{Z}), are related to each other.

The Approach of Borger

Borger’s approach makes use of the idea of adjoint triples (see Adjoint Functors and Monads). Before we discuss the field with one element in this approach, let us first discuss something more elementary. Consider a field K and and a field extension L of K, and let G=\text{Gal}(L/K). We have the following adjoint triple:

\displaystyle \text{Weil restrict}:L\textbf{-Alg}\rightarrow K\textbf{-Alg}

\displaystyle -\otimes_{K}L: K\textbf{-Alg}\rightarrow L\textbf{-Alg}

\displaystyle \text{forget base}:L\textbf{-Alg}\rightarrow K\textbf{-Alg}

Grothendieck’s abstract reformulation of Galois theory says that there is an equivalence of categories between the category of K-algebras and the category of L-algebras with an action of G. This means that we can also consider the above adjoint triple in the following sense:

\displaystyle A\rightarrow\otimes_{G}A:L\textbf{-Alg}\rightarrow L\textbf{-Alg}\text{ (with }G\text{-action)}

\displaystyle \text{fgt}: L\textbf{-Alg}\text{ (with }G\text{-action)}\rightarrow L\textbf{-Alg}

\displaystyle A\rightarrow\prod_{G}A:L\textbf{-Alg}\rightarrow L\textbf{-Alg}\text{ (with }G\text{-action)}

Let us now go back to the field with one element. We want to construct the following adjoint triple:

\displaystyle \text{Weil restrict}:\mathbb{Z}\textbf{-Alg}\rightarrow\mathbb{F}_{1}\textbf{-Alg}

\displaystyle -\otimes_{\mathbb{F}_{1}}\mathbb{Z}:\mathbb{F}_{1}\textbf{-Alg}\rightarrow\mathbb{Z}\textbf{-Alg}

\displaystyle \text{forget base}:\mathbb{Z}\textbf{-Alg}\rightarrow\mathbb{F}_{1}\textbf{-Alg}

Following the above example of the field K and the field extension L of K, we will approach the construction of this adjoint triple by considering instead the following adjoint triple:

\displaystyle \Lambda\odot-:\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}

\displaystyle \text{fgt}:\Lambda\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}

\displaystyle W(-):\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}

We must now discuss the meaning of the concepts involved in the last adjoint triple. In particular, the must define the category of \Lambda-rings, as well as the adjoint functors \Lambda\odot-, \text{fgt}, and W(-) that form the adjoint triple.

Let R be a ring and let p be a prime number. A Frobenius lift is a ring homomorphism \psi_{p}:R\rightarrow R such that F\circ q=q\circ\psi_{p} where q:R\rightarrow R/pR is the quotient map and F:R/pR\rightarrow R/pR is the Frobenius map which sends an element x to the element x^{p}.

Closely related to the idea of Frobenius lifts is the idea of p-derivations. If the terminology is reminiscent of differential calculus, this is because Borger’s approach is closely related to the mathematician Alexandru Buium’s theory of “arithmetic differential equations“. If numbers are like functions, then what Buium wants to figure out is what the analogue of a derivative of a function should be for numbers.


\displaystyle \psi_{p}(x)=x^{p}+p\delta_{p}(x).

Being a ring homomorphism means that \psi satisfies the following properties:

(1) \psi_{p}(x+y)=\psi_{p}(x)+\psi_{p}(y)

(2) \psi_{p}(xy)=\psi_{p}(x)\psi_{p}(y)

(3) \psi_{p}(1)=1

(4) \psi_{p}(0)=0

Recalling that \psi_{p}(x)=x^{p}+p\delta_{p}(x), this means that \delta_{p}(x) must satisfy the following properties corresponding to the above properties for \psi_{p}(x):

(1) \delta_{p}(x+y)=\delta_{p}(x)+\delta_{p}(y)-\frac{1}{p}\sum_{i=1}^{p-1}\binom{p}{i}x^{p-i}y^{i}

(2) \delta_{p}(xy)=x^{p}\delta_{p}(y)+y^{p}\delta_{p}(x)+p\delta_{p}(x)\delta_{p}(y)

(3) \delta_{p}(1)=0

(4) \delta_{p}(0)=0.


\displaystyle \Lambda_{p}\odot A=\mathbb{Z}[\delta_{p}^{\circ n}(a)|n\geqslant 0,a\in A]/\sim

where \sim is the equivalence relation given by the “Liebniz rule”, i.e.

\displaystyle \delta_{p}^{\circ 0}(x+y)=\delta_{p}^{\circ 0}(x)+\delta_{p}^{\circ 0}(y)

\displaystyle \delta_{p}^{\circ 0}(xy)=\delta_{p}^{\circ 0}(x)\delta_{p}^{\circ 0}(y)

\displaystyle \delta_{p}^{\circ 1}(x+y)=\delta_{p}^{\circ 1}(x)+\delta_{p}^{\circ 1}(y)-\frac{1}{p}\sum_{i=1}^{p-1}\binom{p}{i}x^{p-i}y^{i}

\displaystyle \delta_{p}^{\circ 1}(xy)=\delta_{p}^{\circ 1}(x)\delta_{p}^{\circ 1}(y)+p\delta_{p}^{\circ 1}(x)\delta_{p}^{\circ 1}(y)

and so on.

We now discuss the closely related (and an also important part of modern mathematical research)  notion of Witt vectors. We define the ring of Witt vectors of the ring A by

\displaystyle W_{p}(A)=A\times A\times...

with ring operations given by

\displaystyle (a_{0},a_{1},...)+(b_{0},b_{1},...)=(a_{0}+b_{0},a_{1}+b_{1}-\sum_{i=1}^{p-1}\frac{1}{p}\binom{p}{i}a_{0}^{i}b_{0}^{p-i},...)

\displaystyle (a_{0},a_{1},...)(b_{0},b_{1},...)=(a_{0}b_{0},a_{0}^{p}b_{1}+a_{1}b_{0}^{p}+pa_{1}b_{1},...)

\displaystyle 0=(0,0,...)

\displaystyle 1=(1,0,...)

The functors

\displaystyle \Lambda_{p}\odot-:\mathbf{Rings}\rightarrow\mathbf{\delta_{p}}\textbf{-}\mathbf{Rings}

\displaystyle \text{fgt}:\mathbf{\delta_{p}}\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}

\displaystyle W_{p}(-):\mathbf{Rings}\rightarrow\mathbf{\delta_{p}}\textbf{-}\mathbf{Rings}

form an adjoint triple.

A \Lambda_{p}-ring is defined to be the smallest \Lambda_{p}^{'}-ring that contains e, where a \Lambda_{p}^{'}-ring is in turn defined to be a p-torsion free ring together with a Frobenius lift. But it so happens that a \Lambda_{p}-ring is also the same thing as a \delta_{p}-ring, so we also have the following adjoint triple:

\displaystyle \Lambda_{p}\odot-:\mathbf{Rings}\rightarrow\Lambda_{p}\textbf{-}\mathbf{Rings}

\displaystyle \text{fgt}:\Lambda_{p}\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}

\displaystyle W_{p}(-):\mathbf{Rings}\rightarrow\Lambda_{p}\textbf{-}\mathbf{Rings}

Now that we know the basics of a “p-typical” \Lambda-ring, which is a ring together with a Frobenius morphism \psi_{p} for one fixed p, we can also consider a ring together with a Frobenius morphism \psi_{p} for every prime p, to form a “big” \Lambda-ring. We will then obtain the following adjoint triple:

\displaystyle \Lambda\odot-:\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}

\displaystyle \text{fgt}:\Lambda\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}

\displaystyle W(-):\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}

This is just the adjoint triple that we were looking for in the beginning of this section. In other words, we now have what we need to construct rings over \mathbb{F}_{1}, or \mathbb{F}_{1}-algebras and moreover, we have an adjoint triple that relates them to ordinary rings (or \mathbb{Z}-algebras).

We can then generalize these constructions from rings to schemes. The definition of a \Lambda-structure on general schemes is complicated and left to the references, but when the scheme X is flat over \mathbb{Z} (see The Hom and Tensor Functors), a \Lambda-structure on X is simply defined to be a commuting family of endomorphisms \psi_{p}, one for each prime p, such that they agree with the p-th power Frobenius map on the fibers X\times_{\text{Spec}(\mathbb{Z})}\mathbb{F}_{p}.

One may notice that in Borger’s approach an \mathbb{F}_{1}-scheme has more structure than a \mathbb{Z}-scheme, whereas in Deitmar’s approach \mathbb{F}_{1}-schemes, being commutative monoids, have less structure than \mathbb{Z}-schemes. One may actually think of the \Lambda-structure as “descent data” to \mathbb{F}_{1}. In other words, the \Lambda-structure tells us how a scheme defined over \mathbb{Z} is defined over \mathbb{F}_{1}. There is actually a way to use a monoid M to construct a \Lambda-ring \mathbb{Z}[M], where \mathbb{Z}[M] is just the monoid ring as described earlier in the approach of Deitmar, and the Frobenius lifts are defined by \psi_{p}=m^{p} for m\in M. We therefore have some sort of connection between Deitmar’s approach (which is also easily seen to be closely related to Soule’s and Toen and Vaquie’s approach) with Borger’s approach.


We have mentioned only four approaches to the idea of the field with one element in this rather lengthy post. There are many others, and these approaches are often related to each other. In addition, there are other approaches to uncovering even more analogies between function fields and number fields that are not commonly classified as being part of this circle of ideas. To end this post, we just mention that many open problems in mathematics, such as the abc conjecture and the Riemann hypothesis, have function field analogues that have already been solved (we have already discussed the function field analogue of the Riemann hypothesis in The Riemann Hypothesis for Curves over Finite Fields) – perhaps an investigation of these analogies would lead to the solution of their number field analogues – or, in the other direction, perhaps work on these problems would help uncover more aspects of these mysterious and beautiful analogies.


Field with One Element on Wikipedia

Field with One Element on the nLab

Function Field Analogy on the nLab

Schemes over F1 by Anton Deitmar

Lectures on Algebraic Varieties over F1 by Christophe Soule

Les Varieties sur le Corps a un Element by Christophe Soule

On the Field with One Element by Christophe Soule

Under Spec Z by Bertrand Toen and Michel Vaquie

Lambda-Rings and the Field with One Element by James Borger

Witt Vectors, Lambda-Rings, and Arithmetic Jet Spaces by James Borger

Mapping F1-Land: An Overview of Geometries over the Field with One Element by Javier Lopez-Pena and Oliver Lorscheid

Geometry and the Absolute Point by Lieven Le Bruyn

This Week’s Finds in Mathematical Physics (Week 259) by John Baez

Algebraic Number Theory by Jurgen Neukirch

The Local Structure of Algebraic K-Theory by Bjorn Ian Dundas, Thomas G. Goodwillie, and Randy McCarthy


Some Basics of Class Field Theory

Class field theory is one of the crown jewels of modern algebraic number theory. It can be viewed as a generalization of the “reciprocity laws” discovered by Carl Friedrich Gauss and other number theorists of the 19th century. However, as we have not yet discussed reciprocity laws in this blog, we will leave that point of view to the references for now. Instead, in order to describe class field theory, we rely on the following quote from the mathematician Claude Chevalley:

“The object of class field theory is to show how the abelian extensions of an algebraic number field K can be determined by objects drawn from our knowledge of K itself; or, if one prefers to present things in dialectic terms, how a field contains within itself the elements of its own transcending.”

Our approach in this post, as with much of the modern literature, will start from the “local” point of view, and then we will put together all these local pieces in order to have a “global” theory. Recall that there is an analogy between geometry and number theory, and that primes play the role of points in this analogy. Therefore, as in geometry “local” means “zooming in” on a point, in number theory “local” means “zooming in” on a prime. Putting these pieces together is akin to what we have done in Adeles and Ideles. This will become more clear when we define local fields and global fields.

Let K be a (nonarchimedean) local field. This means that K is complete with respect to a discrete valuation (see Valuations and Completions and Adeles and Ideles) and that its residue field is finite. These are either the fields \mathbb{Q}_{p} (the p-adic numbers), \mathbb{F}_{p}((t)) (the field of formal power series over a finite field \mathbb{F}_{p}), or their finite extensions. Let L be a finite extension of K.

We define the norm homomorphism as

\displaystyle N_{L|K}(x)=\prod_{\sigma}\sigma x

for x\in L and \sigma\in \text{Gal}(L|K) (note that there are many notations for the action of \sigma on x; in the book Algebraic Number Theory by Jurgen Neukirch, the notation x^{\sigma} is used instead). We let N_{L|K}L^{\times} stand for the image of the norm homomorphism in K. Then local class field theory tells us that we have the following isomorphism:

\displaystyle K^{\times}/N_{L|K}L^{\times}\xrightarrow{\sim}\text{Gal}(L|K)^{\text{ab}}.

We see that everything in the left-hand side belongs to the field K. This is related to Chevalley’s quote earlier. However, there is even more to this, as we shall see later.

Understanding more about this isomorphism requires the theory of Galois cohomology (see Etale Cohomology of Fields and Galois Cohomology). Namely, the Galois cohomology group H^{2}(\text{Gal}(L|K),L^{\times}) is isomorphic as a group to the group homomorphisms from \text{Gal}(L|K)^{\text{ab}} to K^{\times}/N_{L|K}L^{\times}. It is cyclic of degree equal to the degree of L over K.

There is an injective map from H^{2}(\text{Gal}(L|K),L^{\times}) to the quotient \mathbb{Q}/\mathbb{Z}, and the element of H^{2}(\text{Gal}(L|K),L^{\times}) that gets mapped to 1/n, where n is the degree of L over K, is precisely the element that corresponds to the inverse of the isomorphism K^{\times}/N_{L|K}L^{\times}\xrightarrow{\sim}\text{Gal}(L|K)^{\text{ab}}.

Now let K be a global field, which means that it is a finite extension either of \mathbb{Q} (the rational numbers) or of \mathbb{F}_{p}(t) (the function field over a finite field \mathbb{F}_{p}). Let L be a finite extension of K. Let C_{K} and C_{L} denote the idele class groups (see Adeles and Ideles) of K and L respectively. As in the local case, we will need a norm homomorphism, but this time it will be for idele class groups.

We will define this norm homomorphism “componentwise”. Writing an idele as (z_{w}), we take the norm N_{L_{w}|K_{v}}(z_{w}), and take the product for all primes w above v. We do this for every prime v of K, and thus we obtain an element of the group of ideles of K, and then we take the quotient to obtain an element of the idele class group of K. We denote by N_{L|K}C_{L} the image of this norm homomorphism in C_{K}.

Then global class field theory tells us that we have the following isomorphism:

\displaystyle C_{K}/N_{L|K}C_{L}\sim\text{Gal}(L|K)^{\text{ab}}

Again, as in the local case, everything in the left-hand side belongs to C_{K}.

As we have said earlier, we can obtain this isomorphism by putting together the local pieces from local class field theory, i.e. homomorphisms from K_{v}^{\times} to \text{Gal}(L_{w}|K_{v})^{\text{ab}} which come from the isomorphisms from K_{v}^{\times}/N_{L|K}L_{w}^{\times}, as ideles have components which are local fields, and then taking the quotient to obtain the isomorphism for idele class groups, similar to what we have done for the norm homomorphism.

However, in order to obtain the desired isomorphism, the map (called the Artin map)


from the group of ideles I_{K} of K to the group \text{Gal}(L|K)^{\text{ab}}, which is obtained from putting together the local pieces (before taking the quotients) must satisfy three properties:

(i) It has to be continuous with respect to the topologies on I_{K} and \text{Gal}(L|K) (the topology on the group of ideles is discussed in Adeles and Ideles, while the topology on \text{Gal}(L|K) is the so-called Krull topology – the latter is part of the theory of profinite groups).

(ii) The image of K^{\times} (as embedded in its group of ideles I_{K}) is equal to the identity.

(iii) It is equal to the Frobenius morphism for elements in I_{K}^{S} (see again Adeles and Ideles for the explanation of this notation), where S consists of the archimedean primes and those primes which are ramified in L (see Splitting of Primes in Extensions).

It is a challenging task in itself to prove that the Artin map does indeed satisfy these properties, and for now we leave it to the references. Instead, we mention a few more properties of class field theory. In the local case, class field theory also classifies the subgroups of K^{\times} which are of the form N_{L|K}L^{\times}, which correspond to the open subgroups of finite index in K^{\times}. Since the finite abelian extension L of K also obviously corresponds to the subgroup N_{L|K}L^{\times}, we then obtain a classification of the finite abelian extensions of K. Similarly, in the global case, class field theory classifies the subgroups of C_{K} which are of the form N_{L|K}C_{L}, which correspond to the open subgroups of finite index in C_{K}. The field which corresponds to the such a subgroup is called its class field. In the case that L is the maximal unramified abelian extension of K, L is called the Hilbert class field of K, and there we have the result that the ideal class group (see Algebraic Numbers) of K is isomorphic to the Galois group \text{Gal}(L|K). With the ideas discussed in this last paragraph, the goal of class field theory as expressed in the quote of Chevalley is fulfilled; we are able to describe the abelian extensions of K from knowledge only of K itself.


Class Field Theory on Wikipedia

Artin Reciprocity Law on Wikipedia

Profinite Group on Wikipedia

Class Field Theory by J.S. Milne

Algebraic Number Theory by Jurgen Neukirch

Algebraic Number Theory by J. W. S. Cassels and A. Frohlich

A Panorama of Pure Mathematics by Jean Dieudonne

Primes of the Form x^{2}+ny^{2} by David A. Cox


Some Useful Links: Knots in Physics and Number Theory

In modern times, “knots” have been important objects of study in mathematics. These “knots” are akin to the ones we encounter in ordinary life, except that they don’t have loose ends. For a better idea of what I mean, consider the following picture of what is known as a “trefoil knot“:


More technically, a knot is defined as the embedding of a circle in 3-dimensional space. For more details on the theory of knots, the reader is referred to the following Wikipedia pages:

Knot on Wikipedia

Knot Theory on Wikipedia

One of the reasons why knots have become such a major part of modern mathematical research is because of the work of mathematical physicists such as Edward Witten, who has related them to the Feynman path integral in quantum mechanics (see Lagrangians and Hamiltonians).

Witten, who is very famous for his work on string theory (see An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry) and for being the first, and so far only, physicist to win the prestigious Fields medal, himself explains the relationship between knot theory and quantum mechanics in the following article:

Knots and Quantum Theory by Edward Witten

But knots have also appeared in other branches of mathematics. For example, in number theory, the result in etale cohomology known as Artin-Verdier duality states that the integers are similar to a 3-dimensional object in some sense. In particular, because it has a trivial etale fundamental group (which is kind of an algebraic analogue of the fundamental group discussed in Homotopy Theory and Covering Spaces), it is similar to a 3-sphere (recall the common but somewhat confusing notation that the ordinary sphere we encounter in everyday life is called the 2-sphere, while a circle is also called the 1-sphere).

Note: The fact that a closed 3-dimensional space with a trivial fundamental group is a 3-sphere is the content of a very famous conjecture known as the Poincare conjecture, proved by Grigori Perelman in the early 2000’s.  Perelman refused the million-dollar prize that was supposed to be his reward, as well as the Fields medal.

The prime numbers, because their associated finite fields have one cover for every integer, are like circles, and recalling the definition of knots mentioned above, are therefore like knots on this 3-sphere. This analogy, originally developed by David Mumford and Barry Mazur, is better explained in the following post by Lieven le Bruyn on his blog neverendingbooks:

What is the Knot Associated to a Prime on neverendingbooks

Finally, given what we have discussed, could it be that knot theory can “tie together” (pun intended) physics and number theory? This is the motivation behind the new subject called “arithmetic Chern-Simons theory” which is introduced in the following paper by Minhyong Kim:

Arithmetic Chern-Simons Theory I by Minhyong Kim

Of course, it must also be clarified that this is not the only way by which physics and number theory are related. It is merely another way, a new and not yet thoroughly explored one, by which the unity of mathematics manifests itself via its many different branches helping one another.


Splitting of Primes in Extensions

In Algebraic Numbers we discussed how ideals factorize in an algebraic number field (recall that we had to look at factorization of ideals since the elements in the ring of integers of more general algebraic number fields may no longer factorize uniquely). In this post, we develop some more terminology related to this theory, and we also discuss how in the case of a so-called “Galois extension” the Galois group (see Galois Groups) may express information related to the factorization of ideals in an algebraic number field.

Let \mathfrak{p} be a prime ideal of the ring of integers \mathcal{O}_{K} of an algebraic number field K (we will sometimes also refer to \mathfrak{p} as a prime ideal of K – this is common practice and hopefully will not cause any confusion). In an algebraic number field L which contains K (we also say that L is an extension of K, and write L|K), this prime ideal \mathfrak{p} decomposes into a product of prime ideals \mathfrak{P}_{1},\mathfrak{P}_{2}...\mathfrak{P}_{r} in \mathcal{O}_L, with respective exponents e_{1},e_{2}...e_{r}, i.e.

\displaystyle \mathfrak{p}=\mathfrak{P}_{1}^{e_{1}}\mathfrak{P}_{2}^{e_{2}}...\mathfrak{P}_{n}^{e_{r}}.

The exponents e_{1},e_{2}...e_{r} are called the ramification indices of the prime ideals \mathfrak{P}_{1},\mathfrak{P}_{2},...\mathfrak{P}_{r}. If e_{i}=1, and the residue field extension \kappa(\mathfrak{P}_{i})|\kappa(\mathfrak{p}) (see below) is separable, we say that \mathfrak{P}_{i} is unramified over K. If e_{1}=e_{2}=...e_{r}=1, we say that the prime \mathfrak{p} is unramified. If all primes of K are unramified in L, we say that the extension L|K is unramified.

In the rest of this post we will continue to assume the factorization of \mathfrak{p} as shown above. The residue fields \kappa(\mathfrak{P}_{i}) and \kappa(\mathfrak{p}) of \mathcal{O}_{L} and \mathcal{O}_{K} at the primes \mathfrak{P}_{i} and \mathfrak{p} are defined as the quotients \mathcal{O}_{L}/\mathfrak{P}_{i} and \mathcal{O}_{K}/\mathfrak{p}, and the inertia degrees f_{i} are defined as the degrees of the fields \kappa(\mathfrak{P}_{i}) with respect to the field \kappa(\mathfrak{p}) (i.e. the dimensions of the vector spaces \kappa(\mathfrak{P}_{i}) over the field of scalars \kappa(\mathfrak{p})), i.e.

\displaystyle f_{i}=[\kappa(\mathfrak{P}_{i}):\kappa(\mathfrak{p})].

The ramification indices e_{i}, the inertia degrees f_{i}, and the degree n=[L:K] of the field extension L with respect to K are related by the following “fundamental identity“:

\displaystyle \sum_{i=1}^{r}e_{i}f_{i}=n

In order to understand these concepts better, we can look at the following “extreme” cases:

If e_{i}=1 and f_{i}=1 for all i, then r=n, and we say that the prime \mathfrak{p} splits completely in L.

If r=1 and f_{1}=1, then e_{1}=n, and we say that the prime \mathfrak{p} ramifies completely in L.

If r=1 and e_{1}=1, then f_{1}=n, and we say that the prime \mathfrak{p}  is inert in L.

Consider for example, the field \mathbb{Q}(i) as a field extension of the field \mathbb{Q}. The ring of integers of \mathbb{Q}(i) is the ring of Gaussian integers \mathbb{Z}[i] (see The Fundamental Theorem of Arithmetic and Unique Factorization), while the ring of integers of \mathbb{Q} is the ring of ordinary integers \mathbb{Z}. The degree [\mathbb{Q}(i):\mathbb{Q}] is equal to 2. The prime ideal (5) of \mathbb{Z} splits completely as the product (2+i)(2-i) in \mathbb{Z}[i], the prime ideal (2) of \mathbb{Q} ramifies completely as (1+i)^{2} in \mathbb{Z}[i], while the prime ideal (3) of \mathbb{Z} is inert in \mathbb{Z}[i].

We now bring in Galois groups. We assume that L is a Galois extension of K. This means that the order of G(L|K), the Galois group of L over K, is equal to the degree of L over K. In this case, it turns out that we will have

\displaystyle e_{1}=e_{2}=...=e_{r}


\displaystyle f_{1}=f_{2}=...=f_{r}.

The fundamental identity then becomes


This is but the first of many simplifications we obtain whenever we are dealing with Galois extensions.

Given a prime ideal \mathfrak{P} of \mathcal{O}_{K}, we define the decomposition group G_{\mathfrak{P}} as the subgroup of the Galois group G that fixes \mathfrak{P}, i.e.

\displaystyle G_{\mathfrak{P}}=\{\sigma\in G|\sigma\mathfrak{P=\mathfrak{P}}\}.

The elements of L that are fixed by the decomposition group G_{\mathfrak{P}} form what is called the decomposition field of K over \mathfrak{P}, denoted Z_{\mathfrak{P}}:

 \displaystyle Z_{\mathfrak{P}}=\{x\in L|\sigma x=x,\forall\sigma\in G_{\mathfrak{P}}\}

Every element \sigma of G_{\mathfrak{P}} automorphism \bar{\sigma} of \kappa(\mathfrak{P}) sending the element given by a\text{ mod }\mathfrak{P} to the element given by \sigma a\text{ mod }\mathfrak{P}. The residue field of the decomposition field Z_{\mathfrak{P}} with respect to \mathfrak{p} is the same as the residue field of the field K with respect to \mathfrak{p}, which is \kappa(\mathfrak{p}). Therefore we have a surjective homomorphism

\displaystyle G_{\mathfrak{P}}\rightarrow G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))

which sends the element \sigma of G_{\mathfrak{P}} to the element \bar{\sigma} of G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})). The kernel of this homorphism is called the inertia group of \mathfrak{P} over K. Once again, the elements of L fixed by the inertia group I_{\mathfrak{P}} form what we call the inertia field of K over \mathfrak{P}, denoted T_{\mathfrak{P}}:

 \displaystyle T_{\mathfrak{P}}=\{x\in K|\sigma x=x,\forall\sigma\in I_{\mathfrak{P}}\}

The groups G_{\mathfrak{P}}, I_{\mathfrak{P}}, G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})) are related by the following exact sequence:

\displaystyle 0\rightarrow I_{\mathfrak{P}}\rightarrow G_{\mathfrak{P}}\rightarrow G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))\rightarrow 0

Meanwhile, the relationship between the fields K, Z_{\mathfrak{P}}, T_{\mathfrak{P}}, and L can be summarized as follows:

\displaystyle K\subseteq Z_{\mathfrak{P}}\subseteq T_{\mathfrak{P}}\subseteq L

The ramification index, inertia degree, and the number of primes in K into which a prime \mathfrak{p} in L splits are given in terms of the degrees of the aforementioned fields as follows:

\displaystyle e=[L:T_{\mathfrak{P}}]

\displaystyle f=[T_{\mathfrak{P}}:Z_{\mathfrak{P}}]

\displaystyle r=[Z_{\mathfrak{P}}:K]

Let \mathfrak{P}_{Z}=\mathfrak{P}\cap Z_{\mathfrak{P}}, and \mathfrak{P}_{T}=\mathfrak{P}\cap T_{\mathfrak{P}}. We also refer to \mathfrak{P}_{Z} (resp. \mathfrak{P}_{T}) as the prime ideal of Z_{\mathfrak{P}} (resp. T_{\mathfrak{P}}) below \mathfrak{P}.

The ramification index of \mathfrak{P} over \mathfrak{P}_{T} is equal to e, and its inertia degree is equal to 1. Meanwhile, the ramification index of \mathfrak{P}_{T} over \mathfrak{P}_{Z} is equal to 1, and its inertia degree is equal to e. Finally, the ramification index and inertia degree of \mathfrak{P}_{Z} over \mathfrak{p} are both equal to 1.

We can therefore see that in the case of a Galois extension, the theory of the splitting of primes becomes simple and elegant. Before we end this post, there is one more concept that we will define. Let \mathfrak{P} be a prime that is unramified over K. Then G_{\mathfrak{P}} is isomorphic to G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})), it is cyclic, and it is generated by the unique automorphism

\displaystyle \varphi_{\mathfrak{P}}\equiv a^{q}\text{ mod }\mathfrak{P}    for all    \displaystyle a\in \mathcal{O}_{K}

where q=[\kappa(\mathfrak{P}):\kappa(\mathfrak{p})]. The automorphism \varphi_{\mathfrak{P}} is called the Frobenius automorphism, and it is a very important concept that shows up in many aspects of algebraic number theory.


Splitting of Prime Ideals in Galois Extensions on Wikipedia

A Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael Rosen

Number Fields by Daniel Marcus

Algebraic Theory of Numbers by Pierre Samuel

Algebraic Number Theory by Jurgen Neukirch


SEAMS School Manila 2017: Topics on Elliptic Curves

A few days ago, from July 17 to 25, I attended the SEAMS (Southeast Asian Mathematical Society) School held at the Institute of Mathematics, University of the Philippines Diliman, discussing topics on elliptic curves. The school was also partially supported by CIMPA (Centre International de Mathematiques Pures et Appliquees, or International Center for Pure and Applied Mathematics), and I believe also by the Roman Number Theory Association and the Number Theory Foundation. Here’s the official website for the event:

Southeast Asian Mathematical Society (SEAMS) School Manila 2017: Topics on Elliptic Curves

There were many participants from countries all over Southeast Asia, including Indonesia, Malaysia, Philippines, and Vietnam, as well as one participant from Austria and another from India. The lecturers came from Canada, France, Italy, and Philippines.

Jerome Dimabayao and Michel Waldschmidt started off the school, introducing the algebraic and analytic aspects of elliptic curves, respectively. We have tackled these subjects in this blog, in Elliptic Curves and The Moduli Space of Elliptic Curves, but the school discussed them in much more detail; for instance, we got a glimpse of how Karl Weierstrass might have come up with the function named after him, which relates the equation defining an elliptic curve to a lattice in the complex plane. This requires some complex analysis, which unfortunately we have not discussed that much in this blog yet.

Francesco Pappalardi then discussed some important theorems regarding rational points on elliptic curves, such as the Nagell-Lutz theorem and the famous Mordell-Weil theorem. Then, Julius Basilla discussed the counting of points of elliptic curves over finite fields, often making use of the Hasse-Weil inequality which we have discussed inThe Riemann Hypothesis for Curves over Finite Fields, and the applications of this theory to cryptography. Claude Levesque then introduced to us the fascinating theory of quadratic forms, which can be used to calculate the class number of a quadratic number field (see Algebraic Numbers), and the relation of this theory to elliptic curves.

Richell Celeste discussed the reduction of elliptic curves modulo primes, a subject which we have also discussed here in the post Reduction of Elliptic Curves Modulo Primes, and two famous problems related to elliptic curves, Fermat’s Last Theorem, which was solved by Andrew Wiles in 1995, and the still unsolved Birch and Swinnerton-Dyer conjecture regarding the rank of the group of rational points of elliptic curves. Fidel Nemenzo then discussed the classical problem of finding “congruent numbers“, rational numbers forming the sides of a right triangle whose area is given by an integer, and the rather surprising connection of this problem to elliptic curves.

On the last day of the school, Jerome Dimabayao discussed the fascinating connection between elliptic curves and Galois representations, which we have given a passing mention to at the end of the post Elliptic Curves. Finally, Jared Guissmo Asuncion gave a tutorial on the software PARI which we can use to make calculations related to elliptic curves.

Participants were also given the opportunity to present their research work or topics they were interested in. I gave a short presentation discussing certain aspects of algebraic geometry related to number theory, focusing on the spectrum of the integers, and a mention of related modern mathematical research, such as Arakelov theory, and the view of the integers as a curve (under the Zariski topology) and as a three-dimensional manifold (under the etale topology).

Aside from the lectures, we also had an excursion to the mountainous province of Rizal, which is a short distance away from Manila, but provides a nice getaway from the environment of the big city. We visited a couple of art museums (one of which was also a restaurant serving traditional Filipino cuisine), an underground cave system, and a waterfall. We used this time to relax and talk with each other, for instance about our cultures, and many other things. Of course we still talked about mathematics, and during this trip I learned about many interesting things from my fellow participants, such as the class field theory problem and the subject of real algebraic geometry .

I believe lecture notes will be put up on the school website at some point by some of the participants of the school. For now, some of the lecturers have put up useful references for their lectures.

SEAMS School Manila 2017 was actually the first summer school or conference of its kind that I attended in mathematics, and I enjoyed very much the time I spent there, not only in learning about elliptic curves but also making new friends among the mathematicians in attendance. At some point I also hope to make some posts on this blog regarding the interesting things I have learned at that school.


Adeles and Ideles

In Valuations and Completions we introduced the p-adic numbers \mathbb{Q}_{p}, which, like the real numbers, are the completion of the rational numbers under a certain kind of valuation. There is one such valuation for each prime number p, and another for the “infinite prime”, which is just the usual absolute value. Each valuation may be thought of as encoding number theoretic information related to the prime p, or to the “infinite prime”, for the case of the absolute value (more technically, the p-adic valuations are referred to as nonarchimedean valuations, while the absolute value is an example of an archimedean valuation).

We can consider valuations not only for the rational numbers, but for more general algebraic number fields as well. In its abstract form, given an algebraic number field K, a (multiplicative) valuation of K is simply any function |\ | from K to \mathbb{R} satisfying the following properties:

(i) |x|\geq 0, where x=0 if and only if x=0

(ii) |xy|=|x||y|

(iii) |x+y|\leq|x|+|y|

If this seems reminiscent of the discussion in Metric, Norm, and Inner Product, it is because a valuation does, in fact, define a metric on K, and by extension, a topology. Two valuations are equivalent if they define the same topology; another way to phrase this statement is that two valuations |\ |_{1} and |\ |_{2} are equivalent if |x|_{1}=|x|_{2}^{s} for some positive real number s, for all x\in K.  The valuation is nonarchimedean if |x+y|\leq\text{max}\{|x|,|y|\}; otherwise, it is archimedean.

Just as in the case of rational numbers, we also have an exponential valuation, defined as a function v from the field K to \mathbb{R}\cup \infty satisfying the following conditions:

(i) v(x)=\infty if and only if x=0

(ii) v(xy)=v(x)+v(y)

(iii) v(x+y)\geq\text{min}\{v(x),v(y)\}

Two exponential valuations v_{1} and v_{2} are equivalent if v_{1}(x)=sv_{2}(x) for some real number s, for all x\in K.

The idea of valuations allows us to make certain concepts in algebraic number theory (see Algebraic Numbers) more abstract. We define a place v of an algebraic number field K as an equivalence class of valuations of K. We write K_{v} to denote the completion of K under the place v; these are the generalizations of the p-adic numbers and real numbers to algebraic number fields other than \mathbb{Q}. The nonarchimedean places are also called the finite places, while the archimedean places are also called the infinite places. To express whether a place v is a finite place or an infinite place, we write v|\infty or v\nmid\infty respectively.

The infinite places are of two kinds; the ones for which K_{v} is isomorphic to \mathbb{R} are called the real places, while the ones for which K_{v} is isomorphic to \mathbb{C} are called the complex places. The number of real places and complex places of K, denoted by r_{1} and r_{2} respectively, satisfy the equation r_{1}+2r_{2}=n, where n is the degree of K over \mathbb{Q}, i.e. n=[K:\mathbb{Q}].

By the way, in some of the literature, such as in the book Algebraic Number Theory by Jurgen Neukirch, “places” are also referred to as “primes“. This is intentional – one may actually think of our definition of places as being like a more abstract replacement of the definition of primes. This is quite advantageous in driving home the concept of primes as equivalence classes of valuations; however, to avoid confusion, we will stick to using the term “places” here, along with its corresponding notation.

When v is a nonarchimedean valuation, we let \mathfrak{o}_{v} denote the set of all elements x of K_{v} for which |x|_{v}\leq 1. It is an example of a ring with special properties called a valuation ring. This means that, for any x in K, either x or x^{-1} must be in \mathfrak{o}_{v}. We let \mathfrak{o}_{v}^{*} denote the set of all elements of \mathfrak{o}_{v} for which |x|_{v}=1, and we let \mathfrak{p}_{v} denote the set of all elements of \mathfrak{o}_{v} for which |x|_{v}< 1. It is the unique maximal ideal of \mathfrak{o}_{v}.

Now we proceed to consider the modern point of view in algebraic number theory, which is to consider all these equivalence classes of valuations together. This will lead us to the language of adeles and ideles.

An adele \alpha of K is a family (\alpha_{v}) of elements \alpha_{v} of K_{v} where \alpha_{v}\in K_{v}, and \alpha_{v}\in\mathfrak{o}_{v} for all but finitely many v. We can define addition and multiplication componentwise on adeles, and the resulting ring of adeles is then denoted \mathbb{A}_{K}. The group of units of the ring of adeles is called the group of ideles, denoted I_{K}. For a finite set of primes S that includes the infinite primes, we let

\displaystyle \mathbb{A}_{K}^{S}=\prod_{v\in S}K_{v}\times\prod_{v\notin S}\mathfrak{o}_{v}


\displaystyle I_{K}^{S}=\prod_{v\in S}K_{v}^{*}\times\prod_{v\notin S}\mathfrak{o}_{v}^{*}.

We denote the set of infinite primes by S_{\infty}. Then \mathfrak{o}_{K}, the ring of integers of the number field K, is given by K\cap\mathbb{A}_{K}^{S_{\infty}}, while \mathfrak{o}_{K}^{*}, the group of units of \mathfrak{o}_{K}, is given by K^{*}\cap I_{K}^{S_{\infty}}.

Any element of K is also an element of \mathbb{A}_{K}, and any element of K^{*} (the group of units of K) is also an element of I_{K}. The elements of I_{K} which are also elements of K^{*} are called the principal ideles. This should not be confused with the concept of principal ideals; however the terminology is perhaps suggestive on purpose. In fact, ideles and fractional ideals are related. Any fractional ideal \mathfrak{a} can be expressed in the form

\displaystyle \mathfrak{a}=\prod_{\mathfrak{p}}\mathfrak{p}^{\nu_{\mathfrak{p}}}.

Therefore, we have a mapping

\displaystyle \alpha\mapsto (\alpha)=\prod_{\mathfrak{p}}\mathfrak{p}^{v_{\mathfrak{p}}(\alpha_v)}

from the group of ideles to the group of fractional ideals. This mapping is surjective, and its kernel is I_{K}^{S_{\infty}}.

The quotient group I_{K}/K^{*} is called the idele class group of K, and is denoted by C_{K}. Again, this is not to be confused with the ideal class group we discussed in Algebraic Numbers, although the two are related; in the language of ideles, the ideal class group is defined as I_{K}/I_{K}^{S_{\infty}}K^{*}, and is denoted by Cl_{K}. There is a surjective homomorphism C_{K}\mapsto Cl_{K} induced by the surjective homomorphism from the group of ideles to the group of fractional ideals that we have described in the preceding paragraph.

An important aspect of the concept of adeles and ideles is that they can be equipped with topologies (see Basics of Topology and Continuous Functions). For the adeles, this topology is generated by the neighborhoods of 0 in \mathbb{A}_{K}^{S_{\infty}} under the product topology. For the ideles, this topology is defined by the condition that the mapping \alpha\mapsto (\alpha,\alpha^{-1}) from I_{K} into \mathbb{A}_{K}\times\mathbb{A}_{K} be a homeomorphism onto its image. Both topologies are locally compact, which means that every element has a neighborhood which is compact, i.e. every open cover of that neighborhood has a finite subcover. For the group of ideles, its topology is compatible with its group structure, which makes it into a locally compact topological group.

In this post, we have therefore seen how the theory of valuations can allow us to consider a more abstract viewpoint for algebraic number theory, and how considering all the valuations together to form adeles and ideles allows us to rephrase the usual concepts related to algebraic number fields, such as the ring of integers, its group of units, and the ideal class group, in a new form. In addition, the topologies on the adeles and ideles can be used to obtain new results; for instance, because the group of ideles is a locally compact topological (abelian) group, we can use the methods of harmonic analysis (see Some Basics of Fourier Analysis) to study it. This is the content of the famous thesis of the mathematician John Tate. Another direction where the concept of adeles and ideles can take us is class field theory, which relates the idele class group to the other important group in algebraic number theory, the Galois group (see Galois Groups). The language of adeles and ideles can also be applied not only to algebraic number fields but also to function fields of curves over finite fields. Together these fields are also known as global fields.


Adele Ring on Wikipedia

Tate’s Thesis on Wikipedia

Class Field Theory on Wikipedia

Algebraic Number Theory by Jurgen Neukirch

Algebraic Number Theory by J. W. S. Cassels and A. Frohlich

A Panorama of Pure Mathematics by Jean Dieudonne


Some Useful Links on the Hodge Conjecture, Kahler Manifolds, and Complex Algebraic Geometry

I’m going to be fairly busy in the coming days, so instead of the usual long post, I’m going to post here some links to interesting stuff I’ve found online (related to the subjects stated on the title of this post).

In the previous post, An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry, we discussed Calabi-Yau manifolds (which are special cases of Kahler manifolds) and how their interesting properties, namely their Riemannian, symplectic, and complex aspects figure into the branch of mathematics called mirror symmetry, which is inspired by the famous, and sometimes controversial, proposal for a theory of quantum gravity (and more ambitiously a candidate for the so-called “Theory of Everything”), string theory.

We also mentioned briefly a famous open problem concerning Kahler manifolds called the Hodge conjecture (which was also mentioned in Algebraic Cycles and Intersection Theory). The links I’m going to provide in this post will be related to this conjecture.

As with the post An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry, aside from introducing the subject itself, another of the primary intentions will be to motivate and explore aspects of algebraic geometry such as complex algebraic geometry, and their relation to other branches of mathematics.

Here is the page on the Hodge conjecture, found on the website of the Clay Mathematics Institute:

Hodge Conjecture on Clay Mathematics Institute

We have mentioned before that the Hodge conjecture is one of seven “Millenium Problems” for which the Clay Mathematics Institute is offering a million dollar prize. The page linked to above contains the official problem statement as stated by Pierre Deligne, and a link to a lecture by Dan Freed, which is intended for a general audience and quite understandable. The lecture by Freed is also available on Youtube:

Dan Freed on the Hodge Conjecture at the Clay Mathematics Institute on Youtube

Unfortunately the video of that lecture has messed up audio (although the lecture remains understandable – it’s just that the audio comes out of only one side of the speakers or headphones). Here is another set of videos by David Metzler on Youtube, which explains the Hodge conjecture (along with the other Millennium Problems) to a general audience:

Millennium Problem Talks on Youtube

The Hodge conjecture is also related to certain aspects of number theory. In particular, we have the Tate conjecture, which is another conjecture similar to the Hodge conjecture, but more related to Galois groups (see Galois Groups). Alex Youcis discusses it on the following post on his blog Hard Arithmetic:

The Tate Conjecture over Finite Fields on Hard Arithmetic

On the same blog there is also a discussion of a version of the Hodge conjecture called the p-adic Hodge conjecture on the following post:

An Invitation to p-adic Hodge Theory; or How I Learned to Stop Worrying and Love Fontaine on Hard Arithmetic

The first part of the post linked to above discusses the Hodge conjecture in its classical form, while the second part introduces p-adic numbers and related concepts, some aspects of which were discussed on this blog in Valuations and Completions.

A more technical discussion of the Hodge conjecture, Kahler manifolds, and complex algebraic geometry can be found in the following lecture of Claire Voisin, which is part of the Proceedings of the 2010 International Congress of Mathematicians in Hyderabad, India:

On the Cohomology of Algebraic Varieties by Claire Voisin

More about these subjects will hopefully be discussed on this blog at sometime in the future.


Reduction of Elliptic Curves Modulo Primes

We have discussed elliptic curves over the rational numbers, the real numbers, and the complex numbers in Elliptic Curves. In this post, we discuss elliptic curves over finite fields of the form \mathbb{F}_{p}, where p is a prime, obtained by “reducing” an elliptic curve over the integers modulo p (see Modular Arithmetic and Quotient Sets).

We recall that in Elliptic Curves we gave the definition of an elliptic curve as a polynomial equation that we may write as

\displaystyle y^{2}=x^{3}-ax+b

with a and b satisfying the condition that

\displaystyle 4a^{3}+27b^{2}\neq 0.

Still, we claimed that we will not be able to write the equation of the elliptic curve when the coefficients of the elliptic curve are of characteristic equal to 2 or 3, as is the case for the finite fields \mathbb{F}_{2} or \mathbb{F}_{3}, therefore we will give more general forms for the equation of the elliptic curve later, along with the appropriate conditions. To help us with the latter, we will first look at the case of curves over the real numbers, where we can still make use of the equations above, and see what happens when the conditions on a and b are not satisfied.

Let both a and b both be equal to 0, in which case the condition is not satisfied. Then our curve (which is not an elliptic curve) is given by the equation

\displaystyle y^{2}=x^{3}

whose graph in the xy plane is given by the following figure (plotted using the WolframAlpha software):


Next let a=-3 and b=2. Once again the condition is not satisfied. Our curve is given by

\displaystyle y^{2}=x^{3}-3x+2

and whose graph is given by the following figure (again plotted using WolframAlpha):


Note also that in both cases, the right hand side of the equations of the curves are polynomials in x with a double or triple root; for y^{2}=x^{3}, the right hand side, x^{3}, has a triple root at x=0, while for y^{2}=x^{3}-3x+2, the right hand side, x^{3}-3x+2, factors into y^{2}=(x-1)^{2}(x+2) and therefore has a double root at x=1.

The two curves, y^{2}=x^{3} and y^{2}=x^{3}-3x+2, are examples of singular curves. It is therefore a requirement for a curve to be an elliptic curve, that it must be nonsingular.

We now introduce the general form of an elliptic curve, applicable even when the coefficients belong to fields of characteristic 2 or 3, along with the general condition for it to be nonsingular. We note that the elliptic curve has a “point at infinity“; in order to make this idea explicit, we make use of the notion of projective space (see Projective Geometry) and write our equation in homogeneous coordinates X, Y, and Z:

\displaystyle Y^{2}Z+a_{1}XYZ+a_{3}YZ^{2}=X^{3}+a_{2}XZ^{2}+a_{4}X^{2}Z+a_{6}Z^{3}

This equation is called the long Weierstrass equation. We may also say that it is in long Weierstrass form.

We can now define what it means for a curve to be singular. Let

\displaystyle F=Y^{2}Z+a_{1}XYZ+a_{3}YZ^{2}-X^{3}-a_{2}XZ^{2}-a_{4}X^{2}Z-a_{6}Z^{3}

Then a singular point on this curve F is a point with coordinates a, b, and c such that

\displaystyle \frac{\partial F}{\partial X}(a,b,c)=\frac{\partial F}{\partial Y}(a,b,c)=\frac{\partial F}{\partial Z}(a,b,c)=0

It might be difficult to think of calculus when we are considering, for example, curves over finite fields, where there are a finite number of points on the curve, so we might instead just think of the partial derivatives of the curve as being obtained “algebraically” using the “power rule” of basic calculus,

\displaystyle \frac{d(x^{n})}{dx}=nx^{n-1}

and applying it, along with the usual rules for partial derivatives and constant factors, to every term of the curve. Such is the power of algebraic geometry; it allows us to “import” techniques from calculus and other areas of mathematics which we would not ordinarily think of as being applicable to cases such as curves over finite fields.

If a curve has no singular points, then it is called a nonsingular curve. We may also say that the curve is smooth. In order for a curve written in long Weierstrass form to be an elliptic curve, we require that it be a nonsingular curve as well.

If the coefficients of the curve are not of characteristic equal to 2, we can make a projective transformation of variables to write its equation in a simpler form, known as the short Weierstrass equation, or short Weierstrass form:


In this case the condition for the curve to be nonsingular can be written in the following form:

\displaystyle -4a_{2}^{3}a_{6}+a_{2}^{2}a_{4}^{2}+18a_{4}a_{2}a_{6}-4a_{4}^{3}-27a_{6}^{2}=0

The quantity

\displaystyle D=-4a_{2}^{3}a_{6}+a_{2}^{2}a_{4}^{2}+18a_{4}a_{2}a_{6}-4a_{4}^{3}-27a_{6}^{2}

is called the discriminant of the curve.

We note now, of course, that the usual expressions for the elliptic curve, in what we call affine coordinates x and y, can be recovered from our expression in terms of homogeneous coordinates X, Y, and Z simply by setting x=\frac{X}{Z} and y=\frac{Y}{Z}. The case Z=0 of course corresponds to the “point at infinity”.

We now consider an elliptic curve whose equation has coefficients which are rational numbers. We can make a projective transformation of variables to rewrite the equation into one which has integers as coefficients. Then we can reduce the coefficients modulo a prime p and investigate the points of the elliptic curve considered as having coordinates in the finite field \mathbb{F}_{p}.

It may happen that when we reduce an elliptic curve modulo p, the resulting curve over the finite field \mathbb{F}_{p} is no longer nonsingular. In this case we say that it has bad reduction at p. Consider, for example, the following elliptic curve (written in affine coordinates):

\displaystyle y^{2}=x^{3}-4x^{2}+16

Let us reduce this modulo the prime p=11. Then, since -4\equiv 7 \text{mod }11 and 16\equiv 5 \text{mod }11, we obtain the curve

\displaystyle y^{2}=x^{3}+7x^{2}+5

over \mathbb{F}_{11}. The right hand side actually factors into (x+1)^{2}(x+5) over \mathbb{F}_{11}, which means that it has a double root at x=10 (which is equivalent to x=-1 modulo 11), and has discriminant equal to zero over \mathbb{F}_{11}, hence, this curve over \mathbb{F}_{11} is singular, and the elliptic curve given by y^{2}=x^{3}+7x^{2}+5 has bad reduction at p=11. It also has bad reduction at p=2; in fact, we mentioned earlier that we cannot even write an elliptic curve in the form y^{2}=x^{3}+a_{2}x^{2}+a_{4}x+a_{6} when the field of coefficients have characteristic equal to 2. This is because such a curve will always be singular over such a field. The curve y^{2}=x^{3}+7x^{2}+5 remains nonsingular over all other primes, however; we also say that the curve has good reduction over all primes p except for p=2 and p=11.

In the case that an elliptic curve has bad reduction at p, we say that it has additive reduction if there is only one tangent line at the singular point (we also say that the singular point is a cusp), for example in the case of the curve y^{2}=x^{3}, and we say that it has multiplicative reduction if there are two distinct tangent lines at the singular point (in this case we say that the singular point is a node), for example in the case of the curve y^{2}=x^{3}-3x+2. If the slope of these tangent lines are given by elements of the same field as the coefficients of the curve (in our case rational numbers), we say that it has split multiplicative reduction, otherwise, we say that it has nonsplit multiplicative reduction. We note that since we are working with finite fields, what we describe as “tangent lines” are objects that we must define “algebraically”, as we have done earlier when describing the notion of a curve being singular.

As we have already seen in The Riemann Hypothesis for Curves over Finite Fields, whenever we have a curve over some finite field \mathbb{F}_{q} (where q=p^{n} for some natural number n), our curve will also have a finite number of points, and these points will have coordinates in \mathbb{F}_{q}. We denote the number of these points by N_{q}. In our case, we are interested in the case n=1, so that q=p. When our elliptic curve has good reduction over p, we define a quantity a_{p}, sometimes called the p-defect, or also known as the trace of Frobenius, as

\displaystyle a_{p}=p+1-N_{p}.

We can now define the Hasse-Weil L-function of an elliptic curve E as follows:

\displaystyle L_{E}(s)=\prod_{p}L_{p}(s)

where p runs over all prime numbers, and

\displaystyle L_{p}(s)=\frac{1}{(1-a_{p}p^{-s}+p^{1-2s})}    if E has good reduction at p

\displaystyle L_{p}(s)=\frac{1}{(1-p^{-s})}    if E has split multiplicative reduction at p

\displaystyle L_{p}(s)=\frac{1}{(1+p^{-s})}    if E has nonsplit multiplicative reduction at p

\displaystyle L_{p}(s)=1    if E has additive reduction at p.

The Hasse-Weil L-function encodes number-theoretic information related to the elliptic curve, and much of modern mathematical research involves this function. For example, the Birch and Swinnerton-Dyer conjecture says that the rank of the group formed by the rational points of the elliptic curve (see Elliptic Curves), also known as the Mordell-Weil group, is equal to the order of the zero of the Hasse-Weil L-function at s=1, i.e. we have the following Taylor series expansion of the Hasse-Weil L-function at s=1:

\displaystyle L_{E}(s)=c(s-1)^{r}+\text{higher order terms}

where c is a constant and r is the rank of the elliptic curve.

Meanwhile, the Shimura-Taniyama-Weil conjecture, now also known as the modularity conjecture, central to Andrew Wiles’s proof of Fermat’s Last Theorem, states that the Hasse-Weil L-function can be expressed as the following series:

\displaystyle L_{E}(s)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}

and the coefficients a_{n} are also the coefficients of the Fourier series expansion of some modular form f(E,\tau) (see The Moduli Space of Elliptic Curves):

\displaystyle f(E,\tau)=\sum_{n=1}^{\infty}a_{n}e^{2\pi i \tau}.

For more on the modularity theorem and Wiles’s proof of Fermat’s Last Theorem, the reader is encouraged to read the award-winning article A Marvelous Proof by Fernando Q. Gouvea, which is freely and legally available online. A link to this article (hosted on the website of the Mathematical Association of America) is provided among the list of references below.


Elliptic Curve on Wikipedia

Hasse-Weil Zeta Function on Wikipedia

Birch and Swinnerton-Dyer Conjecture on Wikipedia

Modularity Theorem on Wikipedia

Wiles’s Proof of Fermat’s Last Theorem on Wikipedia

The Birch and Swinnerton-Dyer Conjecture by Andrew Wiles

A Marvelous Proof by Fernando Q. Gouvea

A Friendly Introduction to Number Theory by Joseph H. Silverman

The Arithmetic of Elliptic Curves by Joseph H. Silverman

Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman

Invitation to the Mathematics of Fermat-Wiles by Yves Hellegouarch

A First Course in Modular Forms by Fred Diamond and Jerry Shurman