Perverse Sheaves and the Geometric Satake Equivalence

The idea behind “perverse sheaves” originally had its roots in the work of Mark Goresky and Robert MacPherson on “intersection homology”, but has since taken a life of its own after the foundational work of Alexander Beilinson, Joseph Bernstein, and Pierre Deligne and has found many applications in mathematics. In this post, we will describe what perverse sheaves are, and state an important result in representation theory called the geometric Satake equivalence, which makes use of this language.

A perverse sheaf is a certain object of the “derived category of sheaves with constructible cohomology”, satisfying certain conditions. This is quite a lot of new words, but we shall be defining them in this post, starting with “constructible”.

Let X be an algebraic variety with a stratification, i.e. a decomposition

\displaystyle X=\coprod_{\lambda\in\Lambda}X_{\lambda}

of X into a finite disjoint union of connected, locally closed, smooth subsets X_{\lambda} called strata, such that the closure of any stratum is a union of strata.

A sheaf \mathcal{F} on X is constructible if its restriction \mathcal{F}\vert_{X_{\lambda}} to any stratum X_{\lambda} is locally constant (for every point x of X_{\lambda} there is some open set V containing x on which the restriction \mathcal{F}\vert_{V} to U is a constant sheaf). A locally constant sheaf which is finitely generated (its stalks are finitely generated modules over some ring of coefficients) is also called a local system. Local systems are quite important in arithmetic geometry – for instance, local sheaves on X correspond to representations of the etale fundamental group \pi_{1}(X). The character sheaves discussed at the end of The Global Langlands Correspondence for Function Fields over a Finite Field are also examples of local systems (in fact, perverse sheaves, which we shall define later in this post, can be viewed as a generalization of local systems and are also important in the geometric Langlands program).

Now let us describe roughly what a derived category is. Given an abelian category (for example the category of abelian groups, or sheaves of abelian groups on some space X) A, we can think of the derived category D(A) as the category whose objects are the cochain complexes in A, but whose morphisms are not quite the morphisms of cochain complexes in A, but instead something “looser” that only reflects information about their cohomology.

Let us explain what we mean by this. Two morphisms between cochain complexes in A may be “chain homotopic”, in which case they induce the same morphisms of the corresponding cohomology groups. Therefore, as an intermediate step in constructing the derived category D(A), we first create a category K(A) where the objects are the cochain complexes in A, but where the morphisms are the equivalence classes of morphisms of cochain complexes in A where the equivalence relation is that of chain homotopy. The category K(A) is called the homotopy category of cochain complexes (in A).

Finally, a morphism of chain complexes in A is called a quasi-isomorphism if it induces an isomorphism of the corresponding cohomology groups. Therefore, since we want the morphisms of D(A) to reflect the information about the cohomology, we want the quasi-isomorphisms of chain complexes in A to actually become isomorphisms in the category D(A). So as our final step, to obtain D(A) from K(A), we “formally invert” the quasi-isomorphisms.

We do not yet have everything we need to define what a perverse sheaf is, but we have mentioned previously that they are an object of the derived category of sheaves on an algebraic variety X with constructible cohomology. We denote this latter category D_{c}^{b}(X) (this is used if there is some stratification of X for which we have this category; if the stratification \Lambda is specified, we say \Lambda-constructible instead of constructible, and we denote the corresponding category by D_{\Lambda}^{b}(X)).

Let us say a few things about the category D_{c}^{b}(X). Having “constructible cohomology” means that the cohomology sheaves of D_{c}^{b}(X) are complexes of sheaves, we can take their cohomology, and this cohomology is valued in sheaves (these sheaves are what we call cohomology sheaves) which are constructible, i.e. on each stratum X_{\lambda} they are local systems. The category D_{c}^{b}(X) is also equipped with a very useful extra structure (which we will also later need to define perverse sheaves) called the six-functor formalism.

These six functors are R\mathrm{Hom}, \otimes^{\mathbb{L}}, Rf_{*}, Rf^{*}, Rf_{!}, and Rf^{!}, the first four being the derived functors corresponding to the usual operations of Hom, tensor product, pushforward, and pullback, respectively, and the last two are the derived “shriek” functors (see also The Hom and Tensor Functors and Direct Images and Inverse Images of Sheaves). The functor \otimes^{\mathbb{L}} makes D_{c}^{b}(X) into a symmetric monoidal category, and R\mathrm{Hom} is its right adjoint. The functor Rf_{*} is right adjoint to Rf^{*}, and similarly Rf_{!} is right adjoint to Rf^{!}. In the case that f is proper, Rf_{!} is the same as Rf_{*}, and in the case that f is etale, Rf^{!} is the same as Rf^{*}. We note that it is quite common in the literature to omit the R from the notation, and to let the reader infer that the functor is “derived” from the context (i.e. it is a functor between derived categories).

A derived category is but a specific instance of the even more abstract concept of a triangulated category, which we have defined already, together with the related concepts of a t-structure and the heart of a t-structure, in The Theory of Motives.

In fact we will need the concept of a t-structure to define perverse sheaves. Let us now define this t-structure on the derived category of constructible sheaves. Let X=\coprod_{\lambda\in\Lambda} X_{\lambda} be an algebraic variety with its stratification, and for every stratum X_{\lambda} let d_{\lambda} denote its dimension. We write D_{\mathrm{const}}^{b} for the subcategory of D_{\Lambda}^{b} whose cohomology sheaves are locally constant, and for any object \mathfrak{F} of some derived category we write \mathcal{H}^{i}(\mathfrak{F}) for its i-th cohomology sheaf. We define

\displaystyle ^{p}D_{\lambda}^{\leq 0}=\lbrace\mathfrak{F}\in D_{\mathrm{const}}^{b}(X_{\lambda}):\mathcal{H}^{i}(\mathfrak{F})=0\ \mathrm{for}\ i> d_{\lambda}\rbrace

\displaystyle ^{p}D_{\lambda}^{\geq 0}=\lbrace\mathfrak{F}\in D_{\mathrm{const}}^{b}(X_{\lambda}):\mathcal{H}^{i}(\mathfrak{F})=0\ \mathrm{for}\ i< -d_{\lambda}\rbrace

Now let i_{\lambda}:X_{\lambda}\to X be the inclusion of a stratum X_{\lambda} into X. We further define

\displaystyle ^{p}D^{\leq 0}=\lbrace\mathfrak{F}\in D_{\Lambda}^{b}:Ri_{\lambda}^{*}\frak{F}\in ^{p}D_{\lambda}^{\leq 0}\ \mathrm{for}\ \mathrm{all}\ \lambda\in\Lambda\rbrace

\displaystyle ^{p}D^{\geq 0}=\lbrace\mathfrak{F}\in D_{\Lambda}:Ri_{\lambda}^{!}\frak{F}\in ^{p}D_{\lambda}^{\geq 0}\ \mathrm{for}\ \mathrm{all}\ \lambda\in\Lambda\rbrace

This defines a t-structure, and we define the category of perverse sheaves on X, denoted \mathrm{Perv}(X), as the heart of this t-structure.

With the definition of perverse sheaves in hand we can now state the geometric version of the Satake correspondence (see also The Unramified Local Langlands Correspondence and the Satake Isomorphism). Let k be either \mathbb{C} or \mathbb{F}_{q}, and let K=k((t)), and let \mathcal{O}=k[[t]]. Let G be a reductive group. The loop group LG is defined to be the scheme whose k-points are G(K) and the positive loop group L^{+}G is defined to be the scheme whose k-points are G(\mathcal{O}). The affine Grassmannian is then defined to be the quotient LG/L^{+}(G).

The geometric Satake equivalence states that there is equivalence between the category of perverse sheaves \mathrm{Perv}(\mathrm{Gr}_{G}) on the affine Grassmannian \mathrm{Gr}_{G} and the category \mathrm{Rep}(^{L}G) of representations of the Langlands dual group ^{L}G of G. It was proven by Ivan Mirkovic and Kari Vilonen using the Tannakian formalism (see also The Theory of Motives) but we will not discuss the details of the proof further here, and leave it to the references or future posts.

As we have seen in The Global Langlands Correspondence for Function Fields over a Finite Field, the geometric Satake equivalence is important in being able to define the excursion operators in Vincent Lafforgue’s approach to the global Langlands correspondence for function fields over a finite field. It has (in possibly different variants) also found applications in other parts of arithmetic geometry, for example in certain approaches to the local Langlands correspondence, as well as the study of Shimura varieties. We shall discuss more in future posts on this blog.


Perverse sheaf on Wikipedia

Constructible sheaf on Wikipedia

Derived category on Wikipedia

Satake isomorphism on Wikipedia

An illustrated guide to perverse sheaves by Geordie Williamson

Langlands correspondence and Bezrukavnikov’s equivalence by Geordie Williamson and Anna Romanov

Topics in automorphic forms (notes by Chao Li from a course by Jack Thorne)

Perverse sheaves and fundamental lemmas (notes by Chao Li from a course by Wei Zhang)

Perverse sheaves in representation theory (notes by Chao Li from a course by Carl Mautner)

Geometric Langlands duality and representations of algebraic groups over commutative rings by Ivan Mirkovic and Kari Vilonen

Adic Spaces and Perfectoid Spaces

At the end of Formal Schemes we hinted at the concept of adic spaces, which subsumes both formal schemes and rigid analytic spaces (see also Rigid Analytic Spaces). In this post we will define what these are, give some examples, and introduce and discuss briefly a very special type of adic spaces, the perfectoid spaces, which generalizes what we discussed in Perfectoid Fields.

We begin by discussing the rings that we will need to construct adic spaces. A topological ring (see also Formal Schemes) A is called a Huber ring if it contains an open subring A_{0} which is adic with respect to a finitely generated ideal of definition I contained in A_{0}. This means that the nonnegative powers of I form a basis of open neighborhoods of 0. The subring A_{0} is called a ring of definition for A.

Here are some examples of Huber rings:

  • Any ring A, equipped with the discrete topology, with the ring of definition A_{0}=A, and the ideal of definition I=(0).
  • The p-adic numbers A=\mathbb{Q}_{p}, with the p-adic topology, with the ring of definition A_{0}=\mathbb{Z}_{p}, and the ideal of definition I=(p).
  • The field of formal Laurent series A=k((x)) over some field k, with the metric topology given by the nonarchimedean valuation defined by the order of vanishing at 0, with the ring of definition A_{0}=k[[x]], and the ideal of definition I=(x).
  • Generalizing the previous two examples, any nonarchimedean field K is an example of a Huber ring, with ring of definition A_{0}=\lbrace x:\vert x\vert\leq 1\rbrace and ideal of definition I=(\varpi) for some \varpi satisfying 0<\vert\varpi\vert <1.

A subset S of a Huber ring, or more generally a topological ring, is called bounded if, for any open neighborhood U of 0, we can always find another open neighborhood V of 0 such that all the products of elements of V with elements of S are contained inside U. An element of a Huber ring is called power bounded if the set of all its nonnegative powers is bounded. For a Huber ring A we denote the set of power bounded elements by A^{\circ}. Any element of the ring of definition will always be power bounded.

With the definition of power bounded elements in hand we give two more examples of Huber rings:

  • Let K be a nonarchimedean field as in the previous example, and let \varpi again be an element such that 0<\vert\varpi\vert<1. Its set of power bounded elements is given by K^{\circ}=\lbrace x:\vert x\vert\leq 1\rbrace. Now let A=K^{\circ}[[T_{1},\ldots, T_{n}]] with the I-adic topology (see also Formal Schemes), where I is the ideal (\varpi, T_{1},\ldots,T_{n}). Then A is a Huber ring with ring of definition A_{0}=A and ideal of definition I.
  • Let K, K^{\circ}, and \varpi be as above. Consider the Tate algebra A= K\langle T_{1},\ldots,T_{n}\rangle (see also Rigid Analytic Spaces), a topological ring whose topology is generated by a basis of open neighborhoods of 0 given by \varpi^{n} A. Then A is a Huber ring with ring of definition given by A_{0}=K^{\circ}\langle T_{1},\ldots,T_{n}\rangle and ideal of definition given by (\varpi).

A subring A^{+} of a Huber ring A which is open, integrally closed, and power bounded is called a ring of integral elements. A Huber pair is a pair (A,A^{+}) consisting of a Huber ring A and a ring of integral elements A^{+} contained in A. Note that the set of power bounded elements, A^{\circ}, is itself an example of a ring of integral elements! In fact, in many examples that we will consider the relevant Huber pair will be of the form (A,A^{\circ}).

Now we introduce the adic spectrum of an Huber pair (A,A^{+}), denoted \mathrm{Spa}(A,A^{+}). They will form the basic building blocks of adic spaces, like affine schemes are to schemes or affinoid rigid analytic spaces are to rigid analytic spaces. We will proceed in the usual manner; first we define the underlying set, then we put a topology on it, and then construct a structure sheaf – except that in the case of adic spaces, what we will construct is merely a structure presheaf and may not always be a sheaf! Then we will define more general adic spaces to be something that locally looks like the adic spectrum of some Huber pair.

The underlying set of the adic spectrum \mathrm{Spa}(A,A^{+}) is the set of equivalence classes of continuous valuations \vert\cdot\vert on A such that \vert a\vert\leq 1 whenever a is in A^{+}. From now on we will change our notation and let x denote a continuous valuation, and we write f for an element of A, so that we can write \vert f(x)\vert instead of \vert a\vert, to drive home the idea that these (equivalence classes of) continuous valuations are the points of our space, on which elements of our ring A are functions.

The underlying topological space of \mathrm{Spa}(A,A^{+}) is then obtained from the above set by equipping it with the topology generated by the subsets of the form

\displaystyle \lbrace x: \vert f(x)\vert \leq \vert g(x)\vert \neq 0\rbrace

for all f,g\in A.

Let us now define the structure presheaf. First let us define rational subsets. Let T be a subset of A such that the set consisting of all products of elements of T with elements of A is an open subset of A. We define the rational subset

\displaystyle U\left(\frac{T}{s}\right):=\lbrace x:\vert t(x)\vert \leq\vert s(x)\vert\neq 0\rbrace

for all t\in T. If U is a rational subset of the Huber pair (A,A^{+}), then there is a Huber pair (\mathcal{O}_{ \mathrm{Spa}(A,A^{+}) }(U),\mathcal{O}_{ \mathrm{Spa}(A,A^{+}) }^{+}(U)) such that the map \mathrm{Spa}(A,A^{+})\to \mathrm{Spa}(\mathcal{O}_{X}(U),\mathcal{O}_{X}^{+}(U)) factors through U and this map is universal among such maps.

Now we define our structure presheaf by assigning to any open set W the Huber pair (\mathcal{O}_{ \mathrm{Spa}(A,A^{+} )}(W) , \mathcal{O}_{ \mathrm{Spa}(A,A^{+} )}^{+}(W)) where \mathcal{O}_{ \mathrm{Spa}(A,A^{+} )}(W):=\varprojlim \mathcal{O}(U) where the limit is taken over all inclusions of rational subsets U\subseteq W, and \mathcal{O}_{ \mathrm{Spa}(A,A^{+} )}^{+}(W) is similarly defined.

Again, the structure presheaf of \mathrm{Spa}(A,A^{+}) may not necessarily be a sheaf; in the case that it is, we say that the Huber pair (A,A^{+}) is sheafy. In this case we will also refer to \mathrm{Spa}(A,A^{+}) (the underlying topological space together with the structure sheaf) as an affinoid adic space. We can now define more generally an adic space as the data of a topological space X, a structure sheaf \mathcal{O}_{X}, and for each point x of X, an equivalence class of continuous valuations on the stalk \mathcal{O}_{X,x}, such that it admits a covering of U_{i}‘s giving rise to the data of a structure sheaf and a collection of valuations, all of which is isomorphic to that given by an affinoid adic space.

Recall that we said above that the set of power-bounded elements, A^{\circ}, is an example of a ring of integral elements. Therefore (A,A^{\circ}) is an example of a Huber pair. It is convention that, if our Huber pair is given by (A,A^{\circ}) we write \mathrm{Spa}(A) instead of \mathrm{Spa}(A,A^{\circ}). Let us now look at some examples of adic spaces.

Consider \mathrm{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p}) (which by the previous paragraph we may also simply write as \mathrm{Spa}(\mathbb{Q}_{p}), since \mathbb{Z}_{p} is the set of power-bounded elements of \mathbb{Q}_{p}). Then the underlying topological space of \mathrm{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p}) consists of one point, corresponding to the usual p-adic valuation on \mathbb{Q}_{p}.

Next we consider \mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p}) (which by the same idea as above we may write as \mathrm{Spa}(\mathbb{Z}_{p}). The underlying topological space of \mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p}) now consists of two points, one of which is open, and one of which is closed. The open (or “generic”) point corresponds once again to the usual p-adic valuation \mathbb{Q}_{p} restricted to \mathbb{Z}_{p}. The closed point is the valuation which sends any \mathbb{Z}_{p} which contains a power of p to 0, and sends everything else to 1.

More complicated is \mathrm{Spa}(\mathbb{Q}_{p}\langle T\rangle,\mathbb{Z}_{p}\langle T\rangle), also known as the adic closed unit disc. We can compare this with the closed unit disc discussed in Rigid Analytic Spaces. In that post we the underlying set of the closed unit disc was given by the set of maximal ideals of \mathbb{Q}_{p}\langle T\rangle. But every such maximal ideal gives rise to a continuous valuation on \mathbb{Q}_{p}\langle T\rangle. So every point of the rigid analytic closed unit disc gives rise to a point of the adic closed unit disc. But the adic closed unit disc has more points!

An example of a point of the adic closed unit disc is as follows. Let \Gamma be the ordered abelian group \mathbb{R}_{>0}\times \gamma^{\mathbb{Z}}, where \gamma is such that a<\gamma<1 for all real numbers a<1 in this order. Define a continuous valuation \vert\cdot\vert_{x^{-}} on \mathbb{Q}_{p}\langle T\rangle as follows:

\displaystyle \vert \sum_{n=0}a_{n}T^{n}\vert_{x^{-}}=\sup_{n\geq 0}\vert a_{n}\vert\gamma^{n}

This valuation defines a point x^{-} of the adic closed unit disc. This valuation sees T as being infinitesimally less than 1, i.e. \vert T(x^{-})\vert=\vert T\vert_{x^{-}}<1, but \vert T(x^{-})\vert>a for all a<1 in \mathbb{Q}_{p}. This point x^{-} serves a useful purpose. Recall in Rigid Analytic Spaces that we were unable to disconnect the closed unit disc into two open sets (the “interior” and the “boundary”) because of the Grothendieck topology. In this case we do not have a Grothendieck topology but an honest-to-goodness actual topology, but still we will not be able to disconnect the adic closed unit disc into the analogue of these open sets. This is because the disjoint union of the open sets \cup_{n\geq 1}\vert T^{n}(x)\vert<\vert p\vert and \vert T(x)\vert=1 will not miss the point x^{-}, so just these two will not cover the adic closed unit disc.

Finally let us consider \mathrm{Spa}(\mathbb{Z}_{p}[[ T]],\mathbb{Z}_{p}[[T]]). This is the adic open unit disc. This has a map to \mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p}), and the preimage of the generic point of \mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p}) is called the generic fiber (this generic fiber may also be thought of as the adic open unit disc over \mathbb{Q}_{p}, which makes it more comparable to the example of the adic closed unit disc earlier). The adic open unit disc has many interesting properties (for example it is useful to study in closer detail if one wants to study the fundamental curve of p-adic Hodge theory, also known as the Fargues-Fontaine curve) but we will leave this to future posts.

Let us now introduce a very special type of adic space. First we define a very special type of Huber ring. We say that a Huber ring A is Tate if it contains a topologically nilpotent unit (also called a pseudo-uniformizer). An element \varpi is topologically nilpotent if its sequence of powers \varpi, \varpi^{2},\ldots converges to 0. For example, the Huber ring \mathbb{Q}_{p} (as discussed above) is Tate, with pseudo-uniformizer given by p.

If, in addition to being Tate, the Huber ring A is complete, uniform (which means that A^{\circ} is bounded in A), and contains a pseudo-uniformizer \varpi such that \varpi^{p}\vert p in A^{\circ} and the p-th power map map A/\varpi\to A/\varpi^{p} is an isomorphism, then we say that A is perfectoid. As can be inferred from the name, this generalizes the perfectoid fields we introduced in Perfectoid Fields. We recall the important property of perfectoid fields (which we can now generalize to perfectoid rings) – if R is perfectoid, then the category of finite etale R-algebras is equivalent to the category of finite etale R^{\flat}-algebras, where R^{\flat} is the tilt of R. For fields, this manifests as an isomorphism of their absolute Galois groups, which generalizes the famous Fontaine-Wintenberger theorem.

A perfectoid space is an adic space which can be covered by affinoid adic spaces \mathrm{Spa}(A,A^{+}), where A is perfectoid. If X is a perfectoid space, we can associate to it its tilt X^{\flat}, by taking the tilts of the affinoid adic spaces that cover X and gluing them together. In fact, for a fixed perfectoid space X, there is an equivalence of categories between perfectoid spaces over X, and perfectoid spaces over X^{\flat}. This is the geometric version of the equivalence of categories of finite etale algebras over a perfectoid ring and its tilt. In addition, although we will not do it in this post, one can define the etale sites of X and X^{\flat}, and these will also be equivalent.

To end this post, we mention some properties of perfectoid spaces that make it useful form some applications. It turns out that if X is a smooth rigid analytic space, it always has a pro-etale cover by affinoid perfectoid spaces. A pro-etale map U\to X may be thought of as a completed inverse limit \varprojlim_{i} U_{i}\to X, where each U_{i}\to X is an etale map. An example of a pro-etale cover is as follows. If we let \mathbb{Q}_{p}^{\mathrm{cycl}} be the perfectoid field given by the completion of \cup_{n}\mathbb{Q}_{p}(\mu_{p^{n}}) (this is somewhat similar to the example involved in the Fontaine-Wintenberger theorem in Perfectoid Fields), then \mathrm{Spa}(\mathbb{Q}_{p}^{\mathrm{cycl}}) is a pro-etale cover of \mathrm{Spa}(\mathbb{Q}_{p}). To see why this is pro-etale, note that a finite separable extension of fields is etale, and \mathbb{Q}_{p}^{\mathrm{cycl}} is the completion of the infinite union (direct limit) of such finite separable extensions \mathbb{Q}_{p}(\mu_{p^{n}}) of \mathbb{Q}_{p}, but looking at the adic spectrum means the arrows go the other way, which is why we think of it as an inverse limit.

Another property of perfectoid spaces is the following. If U is a perfectoid affinoid space over \mathbb{C}_{p}, then for all i>0 H^{i}(U_{\mathrm{et}},\mathcal{O}_{X}^{+}) (this is the cohomology of the sheaf of functions bounded by 1 on the etale site of X) is annihilated by the maximal ideal of \mathcal{O}_{\mathbb{C}_{p}}. We also say that H^{i}(U_{\mathrm{et}},\mathcal{O}_{X}^{+}) is almost zero.

Together, what these two properties tell us is that we can compute the cohomology of a smooth rigid analytic space via the Cech complex associated to its covering by perfectoid affinoid spaces. This has been applied in the work of Peter Scholze to the mod p cohomology of the rigid analytic space associated to a Siegel modular variety, in order to relate it to Siegel cusp forms (see also Siegel modular forms). In this case the covering by perfectoid affinoid spaces is provided by a Siegel modular variety at “infinite level”, which happens to have a map (called the period map) to a Grassmannian (the moduli space of subspaces of a fixed dimension of some fixed vector space), and there are certain properties that we can then make use of (for instance, the line bundle on the Siegel modular variety whose sections are cusp forms can be obtained via pullback from a certain line bundle on the Grassmannian) together with p-adic Hodge theoretic arguments to relate the mod p cohomology to Siegel cusp forms.

All this has the following stunning application. Recall that in we may obtain Galois representations from cusp forms (see for example Galois Representations Coming From Weight 2 Eigenforms). This can also be done for Siegel cusp forms more generally. These cusp forms live on a modular curve or Siegel modular variety, which are obtained as arithmetic manifolds, double quotients \Gamma\backslash G(\mathbb{R})/K of a real Lie group G (in this case the symplectic group) by a maximal compact open subgroup K and an arithmetic subgroup \Gamma. But they are also algebraic varieties, so can be studied using the methods of algebraic geometry (see also Shimura Varieties). For example, we can use etale cohomology to obtain Galois representations.

But not all arithmetic manifolds are also algebraic varieties! For instance we have Bianchi manifolds, which are double quotients \Gamma\backslash\mathrm{SL}_{2}(\mathbb{C})/\mathrm{SU}(2), where \Gamma can be, say, a congruence subgroup of \mathrm{SL}_{2}(\mathbb{Z}[i]) (or we can also replace \mathbb{Z}[i] with the ring of integers of some other imaginary quadratic field). The groups involve look complex, but the theory of algebraic groups and in particular the method of Weil restriction allows us to look at them as real Lie groups. This is not an algebraic variety (one way to see this is that \mathrm{SL}_{2}(\mathbb{C})/\mathrm{SU}(2) is hyperbolic 3-space, so a Bianchi manifold has 3 real dimensions and as such cannot be related to an algebraic variety the way a complex manifold can).

Still, it has been conjectured that the singular cohomology (in particular its torsion subgroups) of such arithmetic manifolds which are not algebraic varieties can still be related to Galois representations! And for certain cases this has been proved using the following strategy. First, these arithmetic manifolds can be found as an open subset of the boundary of an appropriate compactification of a Siegel modular variety. Then, methods from algebraic topology (namely, the excision long exact sequence) allow us to relate the cohomology of the arithmetic manifold to the cohomology of the Siegel modular variety.

On the other hand, by our earlier discussion, the covering of the (rigid analytic space associated to the) Siegel modular variety by affinoid perfectoid spaces given by the Siegel modular variety at infinite level, together with the period map of the latter to the Grassmannian, allows one to show that the mod p cohomology of Siegel modular varieties is related to Siegel cusp forms, and it is known how to obtain Galois representations from these. Putting all of these together, this allows us to obtain Galois representations from the cohomology of manifolds which are not algebraic varieties.

A deeper look at aspects of perfectoid spaces, as well as their generalizations and applications (including a more in-depth look at the application to the mod p cohomology of Siegel modular varieties discussed in the previous couple of paragraphs), will hopefully be discussed in future posts.


Perfectoid space on Wikipedia

Adic spaces by Jared Weinstein

Berkeley lectures on p-adic geometry by Peter Scholze and Jared Weinstein

Reciprocity laws and Galois representations: recent breakthroughs by Jared Weinstein

Lecture notes on perfectoid Shimura varieties by Ana Caraiani

On torsion in the cohomology of locally symmetric varieties by Peter Scholze

p-adic Hodge theory for rigid analytic varieties by Peter Scholze

Formal Schemes

In Galois Deformation Rings we discussed the dual numbers as well as the concept of deformations. The dual numbers are numbers with an additional “tangent direction” or a “derivative” – we can further take into account higher order derivatives to consider deformations, which leads us to the concept of deformations.

In this post, we will consider spaces related to deformations, called formal schemes. Let us begin with a motivating example. Consider a field k. From algebraic geometry, we know that the underlying topological space of the scheme \mathrm{Spec}(k) is just a single point. What about the dual numbers, which is the ring k[x]/(x^{2}). What is the underlying topological space of the scheme \mathrm{Spec}(k[x]/(x^{2}))? It turns out it is also just the point!

So as far as the underlying topological spaces go, there is no difference between \mathrm{Spec}(k) and \mathrm{Spec}(k[x]/(x^{2})) – they are both just points. This is because they both have one prime ideal. For k this is the ideal (0), which is also its only ideal that is not itself. For k[x]/(x^{2}), its one prime ideal is the ideal (x); note that the ideal (0) in k[x]/(x^{2}) is not prime, which is related to this ring not being an integral domain. However a scheme is more than just its underlying topological space, one also has the data of its structure sheaf, i.e. its ring of regular functions, and in this regard \mathrm{Spec}(k) and \mathrm{Spec}(k[x]/(x^{2})) are different.

We sometimes consider \mathrm{Spec}(k[x]/(x^{2})) as a “thickening” of \mathrm{Spec}(k) – they are both just the point, but the functions on \mathrm{Spec}(k[x]/(x^{2})) have derivatives, as if there were tangent directions on the point that is the underlying space of \mathrm{Spec}(k[x]/(x^{2})) on which one can move “infinitesimally”, unlike on the point that is the underlying space of \mathrm{Spec}(k).

Just like in our discussion in Galois Deformation Rings, we may want to consider not only the “first-order derivatives” which appear in the dual numbers but also “higher-order derivatives”; we may even want to consider all of them together. This amounts to taking the inverse limit \varprojlim_{n}k[x]/(x^{n}), which is the formal power series ring k[[x]]. However, if we take \mathrm{Spec}(k[[x]]), we will see that it actually has two points, a “generic point” (corresponding to the ideal (0), which is prime because k[[x]] is an integral domain) and a “special point” (corresponding to the ideal (x), which is also prime and furthermore the lone maximal ideal), unlike \mathrm{Spec}(k) or \mathrm{Spec}(k[x]/(x^{2})) (or more generally \mathrm{Spec}(k[x])/(x^{n})), for any n, justified by similar reasons to the preceding argument).

This is where formal schemes come in – a formal scheme can express the “thickening” of some other scheme, with all the “higher-order derivatives”, where the underlying topological space is the same as that of the original scheme but the structure sheaf might be different, to reflect this “thickening”.

A topological ring is a ring A equipped with a topology such that the usual ring operations are continuous with respect to this topology. In this post we will mostly consider the I-adic topology, for some ideal I called the ideal of definition. This topology is generated by a basis consisting of sets of the form a+I^{n}, for a in A.

An example of a topological ring with the I-adic topology is the formal power series ring k[[x]] which we have discussed, together with the ideal of definition (x). Another example is the p-adic integers \mathbb{Z}_{p}, together with the ideal of definition (p). We note that all these examples are complete with respect to the I-adic topology.

More generally for higher dimension one can take, say, an affine variety cut out by some polynomial equation, say y^{2}=x^{3}, and consider the ring k[x,y]/(y^{2}-x^{3}). Note that \mathrm{Spec}(k[x,y]/(y^{2}-x^{3})) is an affine variety. Now we can form a topological ring complete with respect to the I-adic topology, by taking the completion with respect to the ideal I=(y^{2}-x^{3}), i.e. the inverse limit of the diagram

\displaystyle k[x,y]/(y^{2}-x^{3})\leftarrow  k[x,y]/(y^{2}-x^{3})^{2} \leftarrow  k[x,y]/(y^{2}-x^{3})^{3} \leftarrow\ldots

The formation of this ring is an important step in describing the “thickening” of the affine variety \mathrm{Spec}(k[x,y]/(y^{2}-x^{3})), but as said above, it cannot just be done by taking the “spectrum”. Therefore we introduce the idea of a formal spectrum.

Let A be a Noetherian topological ring and let I be an ideal of definition. In addition, let A be complete with respect to the I-adic topology. We define the formal spectrum of A, denoted \mathrm{Spf}(A), to be the pair (X,\mathcal{O}_{X}), where X is the underlying topological space of \mathrm{Spec}(A/I), and the structure sheaf \mathcal{O}_{X} is defined by setting \mathcal{O}_{X}(D_{f}) to be the I-adic completion of A[1/f], for D_{f} the distinguished open set corresponding to f. Applied to the examples earlier, this gives us what we want – a sort of “thickening” of some affine scheme, with an underlying topological space the same as that of the original scheme but with a structure sheaf of functions with “higher-order derivatives”.

More generally, to include the “non-affine” case a formal scheme is a topologically ringed space, i.e. a pair (X,\mathcal{O}_{X}) where X is a topological space and \mathcal{O}_{X} is a sheaf of topological rings, such that for any point x of X there is an open neighborhood of x which is isomorphic as a topologically ringed space to \mathrm{Spf}(A) for some Noetherian topological ring A.

Aside from being useful in deformation theory, formal schemes are also related to rigid analytic spaces (see also Rigid Analytic Spaces). For certain types of formal schemes (“locally formally of finite type over \mathrm{Spf}(\mathbb{Z}_{p})“) one can assign (functorially) a rigid analytic space. For example, this functor will assign to the formal scheme \mathrm{Spf}(\mathbb{Z}_{p}[[x]]) the open unit disc (the interior of the closed unit disc in Rigid Analytic Spaces). This functor is called the generic fiber functor, which is an odd name, because \mathrm{Spf}(\mathbb{Z}_{p}[[x]]) has no “generic points”! However, there is a way to make this name make more sense using the language of adic spaces, which also subsumes the theory of both formal schemes and rigid analytic spaces, and also provides a natural home for the perfectoid spaces we hinted at in Perfectoid Fields. The theory of adic spaces will hopefully be discussed in some future post on this blog.


Formal Scheme on Wikipedia

Berkeley Lectures on p-adic Geometry by Peter Scholze and Jared Weinstein

Modular Curves at Infinite Level by Jared Weinstein

Adic Spaces by Jared Weinstein

Algebraic Geometry by Robin Hartshorne

p-adic Hodge Theory: An Overview

In The Theory of Motives we discussed the notion of a Weil cohomology, and mentioned four “classical” examples, the singular (also known as Betti) cohomology, the de Rham cohomology, the \ell-adic cohomology, and the crystalline cohomology.

Cohomology theories may be thought of as a way to study geometric objects using linear algebra, by associating vector spaces (or more generally, modules or abelian groups) to such a geometric object. But the four Weil cohomology theories above actually give more than just a vector space:

  • The singular cohomology has an action of complex conjugation.
  • The de Rham cohomology has a Hodge filtration.
  • The \ell-adic cohomology has an action of the Galois group.
  • The crystalline cohomology has an action of Frobenius (and a Hodge filtration as well).

There are relations between these different cohomologies. For example, for a smooth projective variety X over the complex numbers \mathbb{C}, the singular cohomology of the corresponding complex analytic manifold X(\mathbb{C}), with complex coefficients (this can be obtained from singular cohomology with integral coefficients by tensoring with \mathbb{C}) and the de Rham cohomology are isomorphic:

\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=H_{\mathrm{dR}}^{k}(X)

The roots of this idea go back to de Rham’s work on complex manifolds, where chains in singular homology (which is dual to singular cohomology, see also Homology and Cohomology) can be paired with the differential forms of de Rham cohomology (see also Differential Forms), simply by integrating the differential forms along these chains. By the machinery developed by Alexander Grothendieck, this can be ported over into the world of algebraic geometry.

Again borrowing from the world of complex manifolds, the machinery of Hodge theory gives us the following Hodge decomposition (see also Shimura Varieties):

\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=\bigoplus_{i+j=k} H^{i}(X,\Omega_{X/\mathbb{C}}^{j})

Now again for the case of smooth projective varieties over the complex numbers , \ell-adic cohomology also has such an isomorphism with singular cohomology – but this time if it has \ell-adic coefficients (i.e. in \mathbb{Q}_{\ell}).

\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{Q}_{\ell}\simeq H_{\mathrm{et}}^{k}(X,\mathbb{Q}_{\ell})

Such isomorphisms are also known as comparison isomorphisms (or comparison theorems).

More generally, if we have a field B into which we can embed both \mathbb{Q}_{\ell} and \mathbb{C} (for instance \mathbb{C}), we obtain the following comparison theorem:

\displaystyle  H_{\mathrm{et}}^{k}(X,\mathbb{Q}_{\ell}) \otimes_{\mathbb{Q_{\ell}}} B\simeq  H_{\mathrm{dR}}^{k}(X) \otimes_{\mathbb{C}} B

Here is a very interesting thing that these comparison theorems can give us. Let X be a modular curve. Then the Hodge decomposition for the first cohomology gives us

\displaystyle H_{\mathrm{sing}}^{1}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=H^{1}(X,\Omega_{X/\mathbb{C}}^{0})\oplus H^{0}(X,\Omega_{X/\mathbb{C}}^{1})

But the H^{0}(X,\Omega_{X/\mathbb{C}}^{1}) is the cusp forms of weight 2 as per the discussion in Modular Forms (see also Galois Representations Coming From Weight 2 Eigenforms). By the results of Hodge theory, the other summand H^{1}(X,\Omega_{X/\mathbb{C}}^{0}) is just the complex conjugate of H^{0}(X,\Omega_{X/\mathbb{C}}^{1}). But we now also have a comparison with etale cohomology, which has a Galois representation! For this the modular form must lie in the cohomology with \mathbb{Q} coefficients, which happens if it is a Hecke eigenform whose Hecke eigenvalues are in \mathbb{Q}. So one of the great things that these comparison theorems gives us is this way of relating modular forms and Galois representations.

The comparison isomorphisms above work for smooth projective varieties over the complex numbers, but let us now go to the p-adic world, and let us consider smooth projective varieties over the p-adic numbers.

It was observed by John Tate (and later explored by Gerd Faltings) that the p-adic cohomology (i.e. the etale cohomology of a smooth projective variety over \mathbb{Q}_{p}, or more generally some other p-adic field, with p-adic coefficients, distinguishing it from \ell-adic cohomology where another prime \ell different from p must be brought in) can have a decomposition akin to the Hodge decomposition, after tensoring it with the p-adic complex numbers (this is the completion of the algebraic closure of the p-adic numbers):

\displaystyle H^{k}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\mathbb{C}_{p}=\bigoplus_{i+j=k} H^{i}(X,\Omega_{X/\mathbb{Q}}^{j})\otimes_{\mathbb{Q}}\mathbb{C}_{p}(-j)

The p-adic complex numbers here play the role of the complex numbers in the singular cohomology case above or the \ell-adic numbers in the \ell-adic case.

The ideas conjectured by Tate, and later completed by Faltings, was but the prototype of what is now known as p-adic Hodge theory. In its modern form, p-adic Hodge theory concerns comparison isomorphisms between different Weil cohomology theories on smooth projective varieties over the p-adic numbers. However, the role played by the complex numbers, \ell-adic numbers (for the complex case), and p-adic complex numbers (for the p-adic case) must now be played by much more complicated objects called period rings, which were developed by Jean-Marc Fontaine. We will discuss the construction of the period rings at the end of this post, but first let us see how they work.

Let X be a smooth projective variety over \mathbb{Q}_{p} (or more generally some other p-adic field). Let H_{\mathrm{dR}}^{i}(X) and H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p}) be its de Rham cohomology and the p-adic etale cohomology of its base change to the algebraic closure \overline{\mathbb{Q}}_{p} respectively. The comparison isomorphism at the center of p-adic Hodge theory is the following:

\displaystyle H_{\mathrm{dR}}^{i}(X)\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}}=H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}}

The object denoted B_{\mathrm{dR}} here is the aforementioned period ring. It is equipped with both a Galois action and a filtration akin to the Hodge filtration. More than just that isomorphism above, we also have a way of obtaining the de Rham cohomology if we are given the p-adic etale cohomology, simply by taking the part that is invariant under the Galois action:

\displaystyle  \displaystyle H_{\mathrm{dR}}^{i}(X)=(H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}

To go the other way, i.e. to recover the p-adic etale cohomology from the de Rham cohomology, we will need a different kind of period ring. This period ring is B_{\mathrm{cris}}, which aside from having a Galois action and a filtration also has an action of Frobenius. Aside from providing us the same isomorphism between de Rham and p-adic etale cohomology upon tensoring, it also provides us with a solution to our earlier problem (as long as X has a smooth proper integral model) as follows:

\displaystyle H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})= \mathrm{Fil}^{0}(H_{\mathrm{dR}}^{i}(X)\otimes_{\mathbb{Q}_{p}} B_{\mathrm{cris}})^{\varphi=1}

This idea can be further abstracted – since etale cohomology provides Galois representations, we can just take some p-adic Galois representation instead, without caring whether it comes from etale cohomology or not, and tensor it with a period ring, then take Galois invariants. For instance let V be some p-adic Galois representation. Then we can take the tensor product

V_{\mathrm{dR}}=(V\otimes B_{\mathrm{dR}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}

If the dimension of V_{\mathrm{dR}} is equal to the dimension of V, then we say that the Galois representation V is de Rham. Similarly we can tensor with B_{\mathrm{cris}}:

V_{\mathrm{cris}}=(V\otimes B_{\mathrm{cris}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}

If its V_{\mathrm{cris}} is equal to the dimension of V , we say that V is crystalline.

The idea of these “de Rham” and “crystalline” Galois representations is that if they come from the corresponding cohomologies then they will have these properties. But does the converse hold? If they are “de Rham” and “crystalline” does that mean that they come from the corresponding cohomologies (i.e. they “come from geometry”)? This is roughly the content of the Fontaine-Mazur conjecture.

Now let us say a few things about the construction of these period rings. These constructions make use of the concepts we discussed in Perfectoid Fields. We start with the ring A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}}), which, as we recall from Perfectoid Fields, is the ring of Witt vectors of the tilt of \mathcal{O}_{\mathbb{C}_{p}}. By inverting p and taking the completion with respect to the canonical map \theta:  A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}}) \to\mathcal{O}_{\mathbb{C}_{p}}, we obtain a ring which we suggestively denote by B_{\mathrm{dR}}^{+}.

There is a special element t of B_{\mathrm{dR}}^{+} which we think of as the logarithm of the element (1, \zeta^{1/p},\zeta^{1/p},\ldots). Upon inverting this element t, we obtain the field B_{\mathrm{dR}}.

The field B_{\mathrm{dR}} is equipped with a Galois action, carried over from the fields involved in its construction, and a filtration, given by \mathrm{Fil}^{i}B_{\mathrm{dR}}=t^{i}B_{\mathrm{dR}}.

To construct B_{\mathrm{cris}}, we once again start with A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}}) and invert p. However, to have a Frobenius, instead of completing with respect to the kernel of the map \theta, we take a generator of this kernel (which we shall denote by \omega). Then we denote by B_{\mathrm{cris}}^{+} the ring formed by all the power series of the form \sum_{n=0}^{\infty} a_{n}\omega^{n}/n! where the a_{n}‘s are elements of A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})[1/p] which converge as n\to\infty, under the topology of A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})[1/p] (which is not the p-adic topology!). Once again there will be an element t like before; we invert t to obtain B_{\mathrm{cris}}.

There is yet another period ring called B_{\mathrm{st}}, where the subscript stands for semistable; in addition to a Galois action, filtration, and Frobenius, it has a monodromy operator. Since this is less extensively discussed in introductory literature, we follow this lead and leave this topic, and the many other wonderful topics related to p-adic Hodge theory, to future posts on this blog.


p-adic Hodge theory on Wikipedia

de Rham cohomology on Wikipedia

Hodge theory on Wikipedia

An invitation to p-adic Hodge theory, or: How I learned to stop worrying and love Fontaine by Alex Youcis

Seminar on the Fargues-Fontaine curve (Lecture 1 – Overview) by Jacob Lurie

An introduction to the theory of p-adic representations by Laurent Berger

Reciprocity laws and Galois representations: recent breakthroughs Jared Weinstein

Rigid Analytic Spaces

This blog post is inspired by and follows closely an amazing talk given by Ashwin Iyengar at the “What is a…seminar?” online seminar. I hope I can do the talk, and this wonderful topic, some justice in this blog post.

One of the most fascinating and powerful things about algebraic geometry is how closely tied it is to complex analysis, despite what the word “algebraic” might lead one to think. To state one of the more simple and common examples, we have that smooth projective curves over the complex numbers \mathbb{C} are the same thing as compact Riemann surfaces. In higher dimensions we also have Chow’s theorem, which tells us that an analytic subspace of complex projective space which is topologically closed is an algebraic subvariety.

This is all encapsulated in what is known as “GAGA“, named after the foundational work “Géometrie Algébrique et Géométrie Analytique” by Jean-Pierre Serre. We refer to the references at the end of this post for the more precise statement, but for now let us think of GAGA as giving us a fully faithful functor from proper algebraic varieties over \mathbb{C} to complex analytic spaces, which gives us an equivalence of categories between their coherent sheaves.

One may now ask if we can do something similar with the p-adic numbers \mathbb{Q}_{p} (or more generally an extension K of \mathbb{Q}_{p} that is complete with respect to a valuation that extends the one on \mathbb{Q}_{p}) instead of \mathbb{C}. This leads us to the theory of rigid analytic spaces, which was originally developed by John Tate to study a p-adic version of the idea (also called “uniformization”) that elliptic curves over \mathbb{C} can be described as lattices on \mathbb{C}.

Let us start defining these rigid analytic spaces. If we simply naively try to mimic the definition of complex analytic manifolds by having these rigid analytic spaces be locally isomorphic to \mathbb{Q}_{p}^{m}, with analytic transition maps, we will run into trouble because of the peculiar geometric properties of the p-adic numbers – in particular, as a topological space, the p-adic numbers are totally disconnected!

To fix this, we cannot just use the naive way because the notion of “local” would just be too “small”, in a way. We will take a cue from algebraic geometry so that we can use the notion of a Grothendieck topology to fix what would be issues if we were to just use the topology that comes from the p-adic numbers.

The Tate algebra \mathbb{Q}_{p}\langle T_{1},\ldots,T_{n}\rangle is the algebra formed by power series in n variables that converge on the n-dimensional unit polydisc D^{n}, which is the set of all n-tuples (c_{1},\ldots,c_{n}) of elements of \mathbb{Q}_{p} that have p-adic absolute value less than or equal to 1 for all i from 1 to n.

There is another way to define the Tate algebra, using the property of power series with coefficients in p-adic numbers that it converges on the unit polydisc D^{n} if and only if its coefficients go to zero (this is not true for real numbers!). More precisely if we have a power series

\displaystyle f(T_{1},\ldots,T_{n})=\sum_{a} c_{a}T_{1}^{a_{1}}\ldots T_{n}^{a_{n}}

where c_{a}\in \mathbb{Q}_{p} and a=a_{1}+\ldots+a_{n} runs over all n-tuples of natural numbers, then f converges on the unit polydisc D^{n} if and only if \lim_{a\to 0}c_{a}=0.

The Tate algebra has many important properties, for example it is a Banach space (see also Metric, Norm, and Inner Product) with the norm of an element given by taking the biggest p-adic absolute value among its coefficients. Another property that will be very important in this post is that it satisfies a Nullstellensatz – orbits of the absolute Galois group \mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p}) in D^{n} correspond to maximal ideals of the Tate algebra. Explicitly, this correspondence is given by the evaluation map x\in D^{n}\mapsto \lbrace f\vert f(x)=0\rbrace.

A quotient of the Tate algebra by some ideal is referred to as an affinoid algebra. The maximal ideals of the underlying set of an affinoid algebra A will be denoted \mathrm{Max}(A), and this should be reminiscent of how we obtain the closed points of a scheme.

Once we have the underlying set \mathrm{Max}(A) , we need two other things to be able to define “affinoid” rigid analytic spaces, which we shall later “glue” to form more general rigid analytic spaces – a topology, and a structure sheaf (again this should be reminiscent of how schemes are defined).

Again taking a cue from how schemes are defined, we define “rational domains”, which are analogous to the distinguished Zariski open sets of (the underlying topological space of) a scheme. Given elements f_{1},\ldots,f_{r},g of the affinoid algebra A, the rational domain \displaystyle A\left(\frac{f}{g}\right) is the set of all x\in\mathrm{Max}(A) such that f_{i}(x)\leq g(x) for all 1\leq i\leq r.

The rational domains generate a topology, however this is still just the p-adic topology, and so it still does not solve the problems that we originally ran into when we tried to define rigid analytic spaces by mimicking the definition of a complex analytic manifold. The trick will be to make use of something that is more general than just a topology – a Grothendieck topology (see also More Category Theory: The Grothendieck Topos).

Let us now define the particular Grothendieck topology that we will use. Unlike other examples of a Grothendieck toplogy, the covers will involve only subsets of the space being covered (it is also referred to as a mild Grothendieck topology). Let X=\mathrm{Max}(A). A subset U of X is called an admissible open if it can be covered by rational domains \lbrace U_{i}\rbrace_{i\in I} such that for any map Y\to X where Y=\mathrm{Max}(B) for some affinoid algebra B, the covering of Y given by the inverse images of the U_{i}‘s admit a finite subcover.

If U is an admissible open covered by admissible opens \lbrace U_{i}\rbrace_{i\in I}, then this covering is called admissible if for any map Y\to X whose image is contained in U, the covering of Y given by the inverse images of the U_{i}‘s admit a finite subcover. These admissible coverings satisfy the axioms for a Grothendieck topology, which we denote G_{X}.

If A is an affinoid algebra, and f_{1},\ldots,f_{k},g are functions, we let \displaystyle A\left\langle \frac{f}{g}\right\rangle denote the ring A\langle T_{1},\ldots T_{k}\rangle/(gT_{i}-f_{i}). By associating to a rational domain \displaystyle A\left(\frac{f}{g}\right) this ring \displaystyle A\left\langle\frac{f}{g}\right\rangle, we can define a structure sheaf \cal{O}_{X} on this Grothendieck topology.

The data consisting of the set X=\mathrm{Max}(A), the Grothendieck topology G_{X}, and the structure sheaf \mathcal{O}_{X} is what makes up an affinoid rigid analytic space. Finally, just as with schemes, we define a more general rigid analytic space as the data consisting of some set X, a Grothendieck topology G_{X} and a sheaf \mathcal{O}_{X} such that locally, with respect to the Grothendieck topology G_{X}, it is isomorphic to an affinoid rigid analytic space.

Under this definition, we have in fact a version of GAGA that holds for rigid analytic spaces – a fully faithful functor from proper schemes of finite type over \mathbb{Q}_{p} to rigid analytic spaces over \mathbb{Q}_{p} that gives an equivalence of categories between their coherent sheaves.

Finally let us now look at an example. Consider the affinoid rigid analytic space obtained from the affinoid algebra \mathbb{Q}_{p}\langle T\rangle. By the Nullstellensatz the underlying set is the unit disc D. The “boundary” of this is the rational subdomain (and therefore an admissible open) \displaystyle D\left(\frac{1}{T}\right), and its complement, the “interior” is covered by rational subdomains \displaystyle D\left(\frac{T^{n}}{p}\right). With this covering the interior may also be shown to be an admissible open.

While it appears that, since we found two complementary admissible opens, we can disconnect the unit disc, we cannot actually do this in the Grothendieck topology, because the set consisting of the boundary and the interior is not an admissible open! And so in this way we see that the Grothendieck topology is the difference maker that allows us to overcome the obstacles posed by the peculiar geometry of the p-adic numbers.

Since Tate’s innovation, the idea of a p-adic or nonarchimedean geometry has blossomed with many kinds of “spaces” other than the rigid analytic spaces of Tate, for example adic spaces, a special class of which generalize perfectoid fields (see also Perfectoid Fields) to spaces, or Berkovich spaces, which are honest to goodness topological spaces instead of relying on a Grothendieck topology like rigid analytic spaces do. Such spaces will be discussed on this blog in the future.


Rigid analytic space on Wikipedia

Algebraic geometry and analytic geometry on Wikipedia

Several approaches to non-archimedean geometry by Brian Conrad

Reciprocity laws and Galois representations: recent breakthroughs by Jared Weinstein

p-adic families of modular forms [after Coleman, Hida, and Mazur] by Matthew Emerton

Perfectoid Fields

Consider the field of p-adic numbers \mathbb{Q}_{p}. An element of \mathbb{Q}_{p} may be written in the form

\displaystyle \sum_{n=k}^{\infty}a_{n}p^{n}

with each a_{n} being an element of the finite field \mathbb{F}_{p}. Let us compare this with the field of Laurent series \mathbb{F}_{p}((t)) in one variable t over \mathbb{F}_{p}. An element of \mathbb{F}_{p}((t)) may be written in the form

\displaystyle \sum_{m=l}^{\infty}a_{m}t^{m}

We see that they look very similar, even though \mathbb{Q}_{p} is characteristic 0, and \mathbb{F}_{p}((t)) is characteristic p.

How far can we push this analogy? The fact that one is in characteristic 0, and the other is characteristic p means we cannot ask for an isomorphism of fields. However, the Fontaine-Wintenberger theorem gives us another connection between \mathbb{Q}_{p} and \mathbb{F}_{p}((t)) – if we modify them by adjoining p-power roots of p and t respectively. This theorem states that the fields \cup_{n}\mathbb{Q}_{p}(p^{1/p^{n}}) and \cup_{n}\mathbb{F}_{p}((t^{1/p^{n}})) have the same absolute Galois group! By the fundamental theorem of Galois theory, this means the category formed by their extensions will be equivalent as well.

We now let F denote the completion of \cup_{n}\mathbb{Q}_{p}(p^{1/p^{n}}), and we let F^{\flat} suggestively denote the completion of \cup_{n}\mathbb{F}_{p}((t^{1/p^{n}})). Completing these fields does not change their absolute Galois groups, so the absolute Galois groups of F and F^{\flat} remain isomorphic. We say that the characteristic p field F^{\flat} is the tilt of the characteristic 0 field F, and that F^{\flat} is an untilt of F (note the subtle change in our choice of article – untilts are not unique).

In this post, we will explore these kinds of fields – which are called perfectoid fields – and the process of tilting and untilting that bridges the world of characteristic 0 and characteristic p. After Fontaine and Wintenberger came up with their famous theorem their ideas have since been developed into even more general and even more powerful theories of perfectoid rings and perfectoid spaces – but we will leave these to future posts. For now we concentrate on the case of fields.

First let us look at a much more primitive example of bridging the world of characteristic 0 and characteristic p. Consider \mathbb{Q}_{p} (characteristic 0). It has a ring of integers \mathbb{Z}_{p}, whose residue field is \mathbb{F}_{p} (characteristic p). To got the other way, starting from \mathbb{F}_{p} we can take its ring of Witt vectors, which is \mathbb{Z}_{p}. Then we take its field of fractions which is \mathbb{Q}_{p}.

More generally, there is a correspondence between characteristic 0 discretely valued complete fields whose uniformizer is p and characteristic p fields which are perfect, i.e. for which the Frobenius morphism is bijective, and the way to go from one category to the other is as in the previous paragraph.

This is a template for “bridging the world of characteristic 0 and characteristic p“. However, we may want more, something like the Fontaine-Wintenberger theorem where the characteristic 0 object and the characteristic p object have isomorphic absolute Galois groups. We will be tweaking this basic bridge in order to create something like Fontaine-Winterger theorem, and these tweaks will lead us to the notion of a perfectoid field. However, we already have isolated one property that we want from such a “perfectoid” field:

The first property that we want from a perfectoid field is that it has to be nonarchimedean. This allows us to have a “ring of integers” that serves as an intermediary object between the two worlds, as we have seen above.

Now let us concentrate on the Fontaine-Wintenberger theorem. To understand this phenomenon better, we need to make use of a version of the fundamental theorem of Galois theory, which allows us to think in terms of extensions of fields instead of their Galois groups. More properly, we want an equivalence of categories between the “Galois categories” of certain extensions of these “base” fields and this will be the property of these base fields being perfectoid. Now the problem is that the extensions that we are considering may not fit into the primitive correspondence we stated above – for example the corresponding characteristic p object may not be perfect, i.e. the Frobenius morphism may not be surjective.

The fix to this is a kind of “perfection”, which is the tilting functor we mentioned earlier. Let R be a ring. The tilt of R, denoted R^{\flat} is defined to be the inverse limit

\displaystyle R^{\flat}=\varprojlim_{x\mapsto x^{p}}R/pR

In other words, an element x of R^{\flat} is an infinite sequence of elements (x_{0},x_{1},x_{2},\ldots) of the quotient R/pR such that x_{1}\cong x_{0}^{p}\mod p, x_{2}\cong x_{1}^{p}\mod p, and so on. We want R^{\flat} to be a ring, so we define it to have componentwise multiplication, i.e.

\displaystyle (xy)_{i}=x_{i}y_{i}

However the addition is going to be more complicated. We define it, for each component, as follows:

\displaystyle (x+y)_{i}=\lim_{n\to\infty}(x_{i+n}+y_{i+n})^{p^n}

At this point we take the opportunity to define another important concept in the theory of perfectoid fields (and rings). Let W be the Witt vector functor (see also The Field with One Element). Then we give the Witt vectors of the tilt of R, W(R^{\flat}), a special name. We will refer to this ring as A_{\mathrm{inf}}(R). It will make an appearance again later. For now we note that there is going to be a canonical map \theta: A_{\mathrm{inf}}(R)\to R.

As we can see, we have defined the tilt of an arbitrary ring. This is not exclusive to the ones which are “perfectoid” whatever the definition of “perfectoid” may be (we will come to this later of course). Again what makes perfectoid fields (such as our earlier examples) special though, is that if F is a perfectoid field of characteristic 0, then F and its tilt F^{\flat} will have isomorphic absolute Galois groups. This will actually follow from the following statement (together with some technicalities involving fiber functors and so on):

There is an equivalence of categories between the category of finite etale algebras over a perfectoid field F and the category of finite etale algebras over its tilt F^{\flat}.

This in turn will follow from the following two statements:

  1. Finite extensions of perfectoid fields are perfectoid.
  2. There is an equivalence of categories between the category of perfectoid extensions of a perfectoid field F and the category of perfectoid extensions over its tilt F^{\flat}.

This equivalence of categories is given by tilting a perfectoid extension over F. This will actually give us a perfectoid extension over F^{\flat}. However, we need a functor that goes in the other direction, a “quasi-inverse” that when composed with tilting gives us back our original perfectoid extension over F (or at least something isomorphic to it, this is what the “quasi-” part means). However, we also said in an earlier paragraph that the “untilt” of a characteristic p field may not be unique (two different untilts may also not be isomorphic). How do we approach this problem?

We recall again the ring A_{\mathrm{inf}}(R) defined earlier as the ring of Witt vectors of the tilt of R, and we recall that it has a canonical map \theta:A_{\mathrm{inf}}(R)\to R. If we know this map, and if we know that it is surjective, then we can recover R simply by quotienting out by the kernel of the map \theta!

The problem is that (aside from not knowing whether it is in fact surjective or not) is that we only know this map if we know that R^{\flat} was obtained as the tilt of R. If we were simply handed some characteristic p field for instance we would not be able to know this map.

However, note that we are interested in an equivalence of categories between the category of perfectoid extensions over the field F and the corresponding category over its tilt F^{\flat}. By specifying these “bases” F and F^{\flat}, it is in fact enough to specify unique untilts! In other words, if we have say just some perfectoid field A, we cannot determine a unique untilt for it, but if we say in addition that it is a perfectoid extension over F, and we are looking for the unique untilt of it over F^{\flat}, we can in fact find it, as long as the map \theta is surjective.

So now how do we guarantee that \theta is surjective? This brings us to our second property, which is that the Frobenius morphism from \overline{R} to itself must be surjective. This is actually the origin of the word “perfectoid”; since as above a field for which the Frobenius morphism is bijective is called perfect; hence, requiring it to be surjective is a relaxation of this condition. This condition guarantees that the map \phi:A_{\mathrm{inf}}(R)\to R is going to be surjective.

The final property that we want from a perfectoid field is that its valuation must be non-discretely valued. The reason for this is that we want to consider infinitely ramified extensions of \mathbb{Q}_{p}. The two previous conditions that we want can only be found in unramified (discretely valued) or infinitely ramified (non-discretely valued) of \mathbb{Q}_{p}. We have already seen above that if we only look at the ones which are unramified then our corresponding characteristic p objects will be limited to perfect \mathbb{F}_{p}-algebras, and this is not enough to give us the Fontaine-Wintenberger theorem. Therefore we will want infinitely ramified extension of \mathbb{Q}_{p}, and these are non-discretely valued.

These three properties are enough to give us the Fontaine-Wintenberger theorem. To summarize – a perfectoid field is a complete, nonarchimedean field F such that the Frobenius morphism from \mathcal{O}_{F}/\mathfrak{p} to itself is surjective and such that its valuation is non-discretely valued.

We have only attempted to motivate the definition of a perfectoid field in this post, and barely gone into any sort of detail. For that one can only recommend the excellent post by Alex Youcis on his blog The Fontaine-Wintenberger Theorem: Going Full Tilt, which inspired this post, but barely does it any justice.

Aside from the Fontaine-Wintenberger theorem, the concepts we have described here – the idea behind “perfectoid”, the equivalence of categories of perfectoid extensions that gives rise to the Fontaine-Wintenberger theorem, the idea of tilting and untilting which bridges the worlds of characteristic 0 and characteristic p, the ring A_{\mathrm{inf}}(R), and so on, have found much application in many areas of math, from the aforementioned perfectoid rings and perfectoid spaces, to p-adic Hodge theory, and to many others.


Perfectoid Space on Wikipedia

What is…a Perfectoid Space? by Bhargav Bhatt

Berkeley Lectures on p-adic Geometry by Peter Scholze and Jared Weinstein

Reciprocity Laws and Galois Representations: Recent Breakthroughs by Jared Weinstein

The Fontaine-Wintenberger Theorem: Going Full Tilt by Alex Youcis

Galois Representations

The absolute Galois group \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) is one of the most important objects of study in mathematics. However the direct study of this group is very difficult; for instance it is an infinite group, and we know very little about it. To make it easier for us, we will often instead study representations of this group – i.e. group homomorphisms to the group \text{GL}(V) of linear transformations of some vector space V over some field F. When V has finite dimension n, \text{GL}(V) is just \text{GL}_{n}(F), the group of n\times n matrices with entries in F and nonzero determinant. Often we will also want the field F to carry a topology – this will also endow \text{GL}_{n}(F) with a topology. For instance, if F is the p-adic numbers \mathbb{Q}_{p} it has a p-adic topology (see also Valuations and Completions). Since \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) has its own topology, we can then talk about representations which are continuous. In this post we shall consider three examples of these continuous Galois representations.

Our first example of a Galois representation is known as the p-adic cyclotomic character. This is a one-dimensional representation over the p-adic numbers \mathbb{Q}_{p}, i.e. a group homomorphism from \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q} to \text{GL}_{1}(\mathbb{Q}_{p}), which also happens to just be the multiplicative group \mathbb{Q}_{p}^{\times}. Let us explain how to obtain this Galois representation.

Consider a primitive p^{n}-th root of unity \zeta_{p^{n}}. Any element \sigma of \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) acts on \zeta_{p^{n}} and sends it to some p^{n}-th root of unity, which amounts to raising it to some integer power between 1 and p^{n}-1, i.e. an element of (\mathbb{Z}/p^{n}\mathbb{Z})^{\times}. We now define the p-adic cyclotomic character \chi to be the map from \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) to \mathbb{Z}_{p}^{\times} which sends the element \sigma to the element of \mathbb{Z}_{p}^{\times} which after modding out by p^{n} is precisely the integer power to which we raised \zeta_{p^{n}}.

Our second example of a Galois representation is known as the Tate module of an elliptic curve. We recall that we also discussed an example of a Galois representation coming from the p-torsion points of an elliptic curve in Elliptic Curves. The Tate module is a way to package the action of the Galois group not only the p-torsion points but also the p^{n}-torsion for any n, by taking an inverse limit over n. Now the p^{n}-torsion points are isomorphic to (\mathbb{Z}/p^{n}\mathbb{Z})^{2}, so the inverse limit is going to be isomorphic to \mathbb{Z}_{p}^{2}. This is not a vector space, since \mathbb{Z}_{p} is not a field, so we take the tensor product with \mathbb{Q}_{p} to get \mathbb{Q}_{p}^{2}, which is a vector space. Therefore we get a Galois representation, i.e. a homomorphism from \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) to \text{GL}_{2}(\mathbb{Q}_{p}). This construction also works for abelian varieties – higher dimensional analogues of elliptic curves – except that the Tate module is now 2g-dimensional, where g is the dimension of the abelian variety.

Our last example of a Galois representation is given by the \ell-adic cohomology (explanation of this terminology to come later) of a smooth proper algebraic variety X over \mathbb{Q}. This is the inverse limit over n of the etale cohomology (see also Cohomology in Algebraic Geometry) of X with coefficients in the constant sheaf \mathbb{Z}/p^{n}\mathbb{Z}. These etale cohomology groups are somewhat confusingly denoted H^{i}(X,\mathbb{Z}_{p}) – note that they are not the etale cohomology of X with \mathbb{Z}_{p} coefficients! Just as in the case of the Tate module, we take the tensor product with \mathbb{Q}_{p} to produce our Galois representation.

These Galois representations coming from the \ell-adic cohomology somewhat subsume the Tate modules discussed earlier – that is because, if X is an elliptic curve or more generally an abelian variety, we have that the \mathbb{Q}_{p}-linear maps from the Tate module (tensored with \mathbb{Q}_{p}) is isomorphic to the first \ell-adic cohomology H_{1}(X,\mathbb{Z}_{p})\otimes\mathbb{Q}_{p}. We say that the first \ell-adic cohomology is the dual of the Tate module.

Although we discussed representations over \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) in this post, it is also often useful to make our study “local” and focus on a single prime \ell, and study \text{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell}) instead. In this case we might as well just have replaced \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) with \text{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell}) in the above discussion, and nothing really changes, as long as the primes \ell and p are different primes. In the case that they are the same prime, things become much more complicated (and the theory is far richer)!

Note: Usually, when discussing “local” Galois representations, the notation for the primes p and \ell are switched! In other words, our local Galois representations are group homomorphisms from \text{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p}) to \text{GL}_{n}(\mathbb{Q}_{\ell}). This is the reason for the terminology “\ell-adic cohomology”. Since we started out just discussing “global” Galois representations, I switched the notation to use p instead for the only instances were we needed a prime. Hopefully this is not overly confusing. We can also study Galois representations more generally for number fields (“global”) and finite extensions of \mathbb{Q}_{p} (“local”).

Finally, although we stated above that we will only discuss three examples here, let us mention a fourth example: Galois representations can also come from modular forms (see also Modular Forms). To discuss these Galois representations would require us to develop some more machinery first, so we leave this to future posts for now.


Cyclotomic character on Wikipedia

Tate module on Wikipedia

Etale cohomology on Wikipedia

Reciprocity laws and Galois representations: recent breakthroughs by Jared Weinstein

Modular Forms

We have previously mentioned modular forms in The Moduli Space of Elliptic Curves and discussed them very briefly in the context of modular curves in Shimura Varieties. In this post, we will discuss this very important and central concept in modern number theory in more detail.

First we recall some facts about the group \text{SL}_{2}(\mathbb{Z}), which is so important that it is given the special name of the modular group. It is defined as the group of 2\times 2 matrices with integer coefficients and determinant equal to 1, and it acts on the upper half-plane (the set of complex numbers with positive imaginary part) in the following manner. Suppose an element \gamma of \text{SL}_{2}(\mathbb{Z}) is written in the form \left(\begin{array}{cc}a&b\\ c&d\end{array}\right). Then for \tau an element of the upper half-plane we write

\displaystyle \gamma(\tau)=\frac{a\tau+b}{c\tau+d}

A modular form (with respect to \text{SL}_{2}(\mathbb{Z})) is a holomorphic function on the upper half-plane such that

\displaystyle f(\gamma(\tau))=(c\tau+d)^{k}f(\tau)

for some k and such that f(\tau) is bounded as the imaginary part of \tau goes to infinity. The number k is called the weight of the modular form. If the function is not required to be bounded as the imaginary part of \tau goes to infinity it is a weakly modular form, and if furthermore it is merely required to be meromorphic, , it is a meromorphic modular form. A meromorphic modular form of weight 0 is just a meromorphic function on the upper half-plane which is invariant under the action of \text{SL}_{2}(\mathbb{Z}) (and bounded as the imaginary part of its argument goes to infinity) – we also call it a modular function.

We denote the set of modular forms of weight k with respect to \text{SL}_{2}(\mathbb{Z}) by \mathcal{M}_{k}(\text{SL}_{2}(\mathbb{Z})). Adding together two modular forms of the same weight gives another modular form of the same weight, and modular forms can be scaled by a complex number, so \mathcal{M}_{k}(\text{SL}_{2}(\mathbb{Z})) actually forms a vector space. We can also multiple a modular form of weight k with a modular form of weight l to get a modular form of weight k+l, so modular forms of a certain weight form a graded piece of a graded ring \mathcal{M}(\text{SL}_{2}(\mathbb{Z}):

\displaystyle \mathcal{M}(\text{SL}_{2}(\mathbb{Z}))=\bigoplus_{k}\mathcal{M}_{k}(\text{SL}_{2}(\mathbb{Z}))

Modular functions are actually functions on the moduli space of elliptic curves – but what about modular forms of higher weight? It turns out that he modular forms of weight 2 correspond to coefficients of differential forms on this space. To see this, consider d\tau and how the group \text{SL}(\mathbb{Z}) acts on it:

\displaystyle d\gamma(\tau)=\gamma'(\tau)d\tau=(c\tau+d)^{-2}d\tau

where \gamma'(\tau) is just the usual derivative of he action of \gamma as describe earlier. For a general differential form given by f(\tau)d\tau to be invariant under the action of \text{SL}(\mathbb{Z}) we must therefore have

\displaystyle f(\gamma(\tau))=(c\tau+d)^{2}f(\tau).

The modular forms of weight greater than 2 arise when we consider products of these differential forms. More technically, modular forms are sections of line bundles on modular curves, which come about when we compactify moduli spaces of elliptic curves (possibly with extra structure).

Let us now look at some examples of modular forms. Since modular forms “live on” moduli spaces of elliptic curves, we will keep in mind elliptic curves as we look at these examples. Our first family of examples are Eisenstein series of weight k, denoted by G_{k}(\tau) which is of the form

\displaystyle G_{k}(\tau)=\sum_{(m,n)\in\mathbb{Z}^{2}\setminus (0,0)}\frac{1}{(m+n\tau)^{k}}

Any modular form can in fact be written in terms of Eisenstein series G_{4}(\tau) and G_{6}(\tau).

Now, let us relate this to elliptic curves. An elliptic curve over the complex numbers may be written as a Weierstrass equation

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

The coefficients on the right-hand side g_{2} and g_{3} are in fact modular forms, of weight 4 and weight 6 respectively, given in terms of the Eisenstein series by g_{2}(\tau)=60G_{4}(\tau) and g_{3}(\tau)=140G_{6}(\tau).

Another example of a modular form is the modular discriminant of an elliptic curve, as a modular form denoted \Delta(\tau). It is a modular form of weight 12, and can be expressed via the elliptic curve coefficients that we defined earlier:


Our final example in this post is not of a modular form, but a meromorphic modular form of weight 0, i.e. a modular function. It is holomorphic on the upper half-plane, but goes to infinity as the imaginary part of \tau goes to infinity. It is the j-invariant associated to an elliptic curve. Once again we may express it in terms of the elliptic curve coefficients g_{2} and g_{3}:

\displaystyle j(\tau)=1728\frac{(g_{2}(\tau))^{3}}{(g_{2}(\tau))^{3}-27(g_{3}(\tau))^{2}}

Note that the denominator is also the modular discriminant.  The points of the moduli space of elliptic curves correspond to isomorphism classes of elliptic curves, and since the j-invariant is an honest-to-goodness holomorphic function on the moduli space of elliptic curves over \mathbb{C}, we can see that isomorphic elliptic curves will have the same j-invariant. This is not the case for the other modular forms we described above, which are not modular functions, i.e. they have nonzero weight! Why is this so? Let us recall that an elliptic curve over \mathbb{C} corresponds to a lattice. Acting on a basis of this lattice by an element of \text{SL}_{2}(\mathbb{Z}) changes the basis, but preserves the lattice. This will be reflected as “admissible changes of coordinates” in the Weierstrass equations, and also changes these modular forms associated to the elliptic curves even though the elliptic curves are still isomorphic. But they change in a predictable way, according to the definition of modular forms.

A modular form f(\tau) is also called a cusp form if the limit of f(\tau) is zero as the imaginary part of \tau approaches infinity. We denote the set of cusp forms of weight k by \mathcal{S}_{k}(\text{SL}_{2}(\mathbb{Z}). They are a vector subspace of \mathcal{M}_{k}(\text{SL}_{2}(\mathbb{Z}) and the graded ring formed by their direct sum for all k, denoted \mathcal{S}_{k}(\text{SL}_{2}(\mathbb{Z}), is an ideal of the graded ring \mathcal{M}(\text{SL}_{2}(\mathbb{Z}). Cusp forms form a very important part of modern research, but we will not discuss them much in this introductory post and leave them for the future.

Let us now discuss congruence subgroups of \text{SL}_{2}(\mathbb{Z}) (we have also discussed this briefly in Shimura Varieties), so that we can define more general modular forms with respect to such a congruence subgroup instead of just \text{SL}_{2}(\mathbb{Z}). Given an integer N, the principal congruence subgroup \Gamma(N) of \text{SL}_{2}(\mathbb{Z}) is the subgroup consisting of the elements which reduce to the identity when we reduce the entries modulo N. A congruence subgroup is any subgroup \Gamma that contains the principal congruence subgroup \Gamma(N). We refer to N as the level of the congruence subgroup.

There are two important kinds of congruence subgroups of \text{SL}_{2}(\mathbb{Z}), denoted by \Gamma_{0}(N) and \Gamma_{1}(N). The subgroup \Gamma_{0}(N) consists of the elements that become upper triangular after reduction modulo N, while the subgroup \Gamma_{1}(N) consists of the elements that become upper triangular with ones on the diagonal after reduction modulo N. As we discussed in Shimura Varieties, these are related to moduli spaces of “elliptic curves with level structure”.

Now we can define the modular forms of weight k with respect to such a congruence subgroup \Gamma. We shall once again require them to be holomorphic functions on the upper half-plane, and we require that for \gamma\in \Gamma written as \left(\begin{array}{cc}a&b\\ c&d\end{array}\right) we must have

\displaystyle f(\gamma(\tau))=(c\tau+d)^{k}f(\tau).

However, the condition that the function be bounded as the imaginary part of \tau goes to infinity must be modified. The reason is that the “point at infinity” is a cusp, a point of the modular curve that does not correspond to an elliptic curve over \mathbb{C} but rather to a “degeneration” of it (this point is therefore not a part of the usual moduli space of elliptic curves –  we can think of it as a “puncture” in this space).

We recall that the construction of the moduli space of elliptic curves over \mathbb{C} starts with the upper half-plane, then we quotient out by the action of \text{SL}_{2}(\mathbb{Z}). The cusps come from taking the union of the rational numbers with the upper half-plane, as well as the point at infinity. When we take the quotient by \text{SL}_{2}(\mathbb{Z}) this all gets sent to the same point, therefore the usual moduli space has only one cusp. But if we take the quotient by a congruence subgroup, we may have several cusps. Therefore, what we really require is for the modular form to be “holomorphic at the cusps“. We can still express this condition in familiar terms by requiring that not f(\tau), but rather (c\tau+d)^{-k}f(\gamma(\tau)) for \gamma\in \text{SL}_{2}(\mathbb{Z}) be bounded as the imaginary part of \tau goes to infinity. We can then define cusp forms with respect to \Gamma by requiring vanishing at the cusps instead. The set of modular forms (resp. cusp forms) of weight k with respect to \Gamma are denoted \mathcal{M}_{k}(\Gamma) (resp. \mathcal{S}_{k}(\Gamma)), and they also have the same structures of being vector spaces and being graded pieces of graded rings as the ones for \text{SL}_{2}(\mathbb{Z}).

Having only discussed the very basics of modular forms we end the post here, with the hope  that in the near future we will be able to discuss things such as Hecke operators, modular curves and their Jacobians, and their associated Galois representations. We redirect the interested reader to the references for now.


Modular Form on Wikipedia

Eisenstein Series in Wikipedia

j-invariant on Wikipedia

Modular Form on Wikipedia

Congruence Subgroups on Wikipedia

Image by User Fropuff of Wikipedia

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

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

Shimura Varieties

In The Moduli Space of Elliptic Curves we discussed how to construct a space whose points correspond to isomorphism classes of elliptic curves over \mathbb{C}. This space is given by the quotient of the upper half-plane by the special linear group \text{SL}_{2}(\mathbb{Z}). Shimura varieties kind of generalize this idea. In some cases their points may correspond to isomorphism classes of abelian varieties over \mathbb{C}, which are higher-dimensional generalizations of elliptic curves in that they are projective varieties whose points form a group, possibly with some additional information.

Using the orbit-stabilizer theorem of group theory, the upper half-plane can also be expressed as the quotient \text{SL}_{2}(\mathbb{R})/\text{SO}(2). Therefore, the moduli space of elliptic curves over \mathbb{C} can be expressed as

\displaystyle \text{SL}_{2}(\mathbb{Z})\backslash\text{SL}_{2}(\mathbb{R})/\text{SO}(2).

If we wanted to parametrize “level structures” as well, we could replace \text{SL}_{2}(\mathbb{Z}) with a congruence subgroup \Gamma(N), a subgroup which contains the matrices in \text{SL}_{2}(\mathbb{Z}) which reduce to an identity matrix when we mod out b some natural number N which is greater than 1. Now we obtain a moduli space of elliptic curves over \mathbb{C} together with a basis of their N-torsion:


We could similarly consider the subgroup \Gamma_{0}(N), the subgroup of \text{SL}_{2}(\mathbb{Z}) containing elements that reduce to an upper-triangular matrix mod N, to parametrize elliptic curves over \mathbb{C} together with a cyclic N-subgroup, or \Gamma_{1}(N), the subgroup of \text{SL}_{2}(\mathbb{Z}) which contains elements that reduce to an upper-triangular matrix with 1 on every diagonal entry mod N, to parametrize elliptic curves over \mathbb{C} together with a point of order N. These give us




Let us discuss some important properties of these moduli spaces, which will help us generalize them. The space \text{SL}_{2}(\mathbb{R})/\text{SO}(2), i.e. the upper-half plane, is an example of a Riemannian symmetric space. This means it is a Riemannian manifold whose group of automorphisms act transitively – in layperson’s terms, every point looks like every other point – and every point has an associated involution fixing only that point in its neighborhood.

These moduli spaces almost form smooth projective curves, but they have missing points called “cusps” that do not correspond to an isomorphism class of elliptic curves but rather to a “degeneration” of such. We can fill in these cusps to “compactify” these moduli spaces, and we get modular curves X(N), X_{0}(N), and X_{1}(N). On these modular curves live cusp forms, which are modular forms satisfying certain conditions at the cusps. Traditionally these modular forms are defined as functions on the upper-half plane satisfying certain conditions under the action of \text{SL}_{2}(\mathbb{Z}), but when they are cusp forms we may also think of them as sections of line bundles on these modular curves. In particular the cusp forms of “weight 2” are the differential forms on a modular curve.

These modular curves are equipped with Hecke operators, T_{p} and \langle p\rangle for every p not equal to N. These are operators on modular forms, but may also be thought of in terms of Hecke correspondences. We recall that elliptic curves over \mathbb{C} are lattices in \mathbb{C}. Take such a lattice \Lambda. The p-th Hecke correspondence is a sum over all the index p sublattices of \Lambda. It is a multivalued function from the modular curve to itself, but the better way to think of such a multivalued function is as a correspondence, a curve inside the product of the modular curve with itself.

With these properties as our guide, let us now proceed to generalize these concepts. One generalization is through the concept of an arithmetic manifold. This is a double coset space

\Gamma\backslash G(\mathbb{R})/K

where G is a semisimple algebraic group over \mathbb{Q}, K is a maximal compact subgroup of G(\mathbb{R}), and \Gamma is an arithmetic subgroup, which means that it is intersection with G(\mathbb{Z}) has finite index in both \Gamma and G(\mathbb{Z}). A theorem of Margulis says that, with a handful of exceptions, G(\mathbb{R})/K is a Riemannian symmetric space. Arithmetic manifolds are equipped with Hecke correspondences as well.

Arithmetic manifolds can be difficult to study. However, in certain cases, they form algebraic varieties, in which case we can use the methods of algebraic geometry to study them. For this to happen, the Riemannian symmetric space G(\mathbb{R})/K must have a complex structure compatible with its Riemannian structure, which makes it into a Hermitian symmetric space. The Baily-Borel theorem guarantees that the quotient of a Hermitian symmetric space by an arithmetic subgroup of G(\mathbb{Q}) is an algebraic variety. This is what Shimura varieties accomplish.

To motivate this better, we discuss the idea of Hodge structures. Let V be an n-dimensional real vector space. A (real) Hodge structure on V is a decomposition of its complexification V\otimes\mathbb{C} as follows:

\displaystyle V\otimes\mathbb{C}=\bigoplus_{p,q} V^{p,q}

such that V^{q,p} is the complex conjugate of V^{p,q}. The set of pairs (p,q) for which V^{p,q} is nonzero is called the type of the Hodge structure. Letting V_{n}=\bigoplus_{p+q=n} V^{p,q}, the decomposition V=\bigoplus_{n} V_{n} is called the weight decomposition. An integral Hodge structure is a \mathbb{Z}-module V together with a Hodge structure on V_{\mathbb{R}} such that the weight decomposition is defined over \mathbb{Q}. A rational Hodge structure is defined similarly but with V a finite-dimensional vector space over \mathbb{Q}.

An example of a Hodge structure is given by the singular cohomology of a smooth projective variety over \mathbb{C}:

\displaystyle H^{n}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=\bigoplus_{i+j=n}H^{j}(X,\Omega_{X/\mathbb{C}}^{i})

In particular for an abelian variety A, the integral Hodge structure of type (1,0),(0,1) given by the first singular cohomology H^{1}(A(\mathbb{C}),\mathbb{Z}) gives an integral Hodge structure of type (-1,0),(0,-1) on its dual, the first singular homology H_{1}(A(\mathbb{C}),\mathbb{Z}). Specifying such an integral Hodge structure of type (-1,0),(0,-1) on H_{1}(A(\mathbb{C}),\mathbb{Z}) is also the same as specifying a complex structure on H_{1}(A(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}} \mathbb{R}. In fact, the category of integral Hodge structures of type (-1,0),(0,-1) is equivalent to the category of complex tori.

Let \mathbb{S} be the group \text{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{\text{m}}. It is the Tannakian group for Hodge structures on finite-dimensional real vector spaces, which basically means that the category of Hodge structures on finite-dimensional real vector spaces are equivalent to the category of representations of \mathbb{S} on finite-dimensional real vector spaces. This lets us redefine Hodge structures as a pair (V,h) where V is a finite-dimensional real vector space and h is a map from \mathbb{S} to \text{GL}(V).

We have earlier stated that the category of integral Hodge structures of type (-1,0),(0,-1) is equivalent to the category of complex tori. However, not all complex tori are abelian varieties. To obtain an equivalence between some category of Hodge structures and abelian varieties, we therefore need a notion of polarizable Hodge structures. We let \mathbb{R}(n) denote the Hodge structure on \mathbb{R} of type (-n,-n) and define \mathbb{Q}(n) and \mathbb{Z}(n) analogously. A polarization on a real Hodge structure V of weight n is a morphism \Psi of Hodge structures from V\times V to \mathbb{R}(-n) such that the bilinear form defined by (u,v)\mapsto \Psi(u,h(i)v) is symmetric and positive semidefinite.

A polarizable Hodge structure is a Hodge structure that can be equipped with a polarization, and it turns out that the functor that assigns to an abelian variety A its first singular homology H_{1}(X,\mathbb{Z}) defines an equivalence of categories between the category of abelian varieties over \mathbb{C} and the category of polarizable integral Hodge structures of type (-1,0),(0,-1).

A Shimura datum is a pair (G,X) where G is a connected reductive group over \mathbb{Q}, and X is a G(\mathbb{R}) conjugacy class of homomorphisms from \mathbb{S} to G, satisfying the following conditions:

  • The composition of any h\in X with the adjoint action of G(\mathbb{R}) on its Lie algebra \mathfrak{g} induces a Hodge structure of type (-1,1)(0,0)(1,-1) on \mathfrak{g}.
  • For any h\in X, h(i) is a Cartan involution on G(\mathbb{R})^{\text{ad}}.
  • G^{\text{ad}} has no factor defined over \mathbb{Q} whose real points form a compact group.

Let (G,X) be a Shimura datum. For K a compact open subgroup of G(\mathbb{A}_{f}) where \mathbb{A}_{f} is the finite adeles (the restricted product of completions of \mathbb{Q} over all finite places, see also Adeles and Ideles), the Shimura variety \text{Sh}_{K}(G,X) is the double quotient

\displaystyle G(\mathbb{Q})\backslash (X\times G(\mathbb{A}_{f})/K)

The introduction of adeles serves the purpose of keeping track of the level structures all at once. The space \text{Sh}_{K}(G,X) is a disjoint union of locally symmetric spaces of the form \Gamma\backslash X^{+}, where X^{+} is a connected component of X and \Gamma is an arithmetic subgroup of G(\mathbb{Q})^{+}. By the Baily-Borel theorem, it is an algebraic variety. Taking the inverse limit of over compact open subgroups K gives us the Shimura variety at infinite level \text{Sh}(G,X).

Let us now look at some examples. Let G=\text{GL}_{2}, and let X be the conjugacy class of the map

\displaystyle h:a+bi\to\left(\begin{array}{cc}a&b\\ -b&a\end{array}\right)

There is a G(\mathbb{R})-equivariant bijective map from X to \mathbb{C}\setminus \mathbb{R} that sends h to i. Then the Shimura varieties \text{Sh}_{K}(G,X) are disjoint copies of modular curves and the Shimura variety at infinite level \text{Sh}(G,X) classifies isogeny classes of elliptic curves with full level structure.

Let’s look at another example. Let V be a 2n-dimensional symplectic space over \mathbb{Q} with symplectic form \psi. Let G be the group of symplectic similitudes \text{GSp}_{2n}, i.e. for k a \mathbb{Q}-algebra

\displaystyle G(k)=\lbrace g\in \text{GL}(V\otimes k)\vert \psi(gu,gv)=\nu(g)\psi(u,v)\rbrace

where \nu:G\to k^{\times} is called the similitude character. Let J be a complex structure on V_{\mathbb{R}} compatible with the symplectic form \psi and let X be the conjugacy class of the map h that sends a+bi to the linear transformation v\mapsto av+bJv. Then the conjugacy class X is the set of complex structures polarized by \pm\psi. The Shimura varieties Sh_{K}(G,X) are called Siegel modular varieties and they parametrize isogeny classes of n-dimensional principally polarized abelian varieties with level structure.

There are many other kinds of Shimura varieties, which parametrize abelian varieties with other kinds of extra structure. Just like modular curves, Shimura varieties also have many interesting aspects, from Galois representations (related to their having Hecke correspondences), to certain special points related to the theory of complex multiplication, to special cycles with height pairings generalizing results such as the Gross-Zagier formula in the study of special values of L-functions and their derivatives. There is also an analogous local theory; in this case, ideas from p-adic Hodge theory come into play, where we can further relate the p-adic analogue of Hodge structures and Galois representations. The study of Shimura varieties is a very fascinating aspect of modern arithmetic geometry.


Shimura variety on Wikipedia

Reciprocity Laws and Galois Representations: Recent Breakthroughs by Jared Weinstein

Perfectoid Shimura Varieties by Ana Caraiani

Introduction to Shimura Varieties by J.S. Milne

Lecture Notes for Advanced Number Theory by Jared Weinstein

The Lubin-Tate Formal Group Law

A (one-dimensional, commutative) formal group law f(X,Y) over some ring A is a formal power series in two variables with coefficients in A satisfying the following axioms that among other things makes it behave like an abelian group law:

  • f(X,Y)=X+Y+\text{higher order terms}
  • f(X,Y)=f(Y,X)
  • f(f(X,Y),Z)=f(X,f(Y,Z))

A homomorphism of formal group laws g:f_{1}(X,Y)\to f_{2}(X,Y) is another formal power series in two variable such f_{1}(g(X,Y))=g(f_{2}(X,Y)). An endomorphism of a formal group law is a homomorphism of a formal group law to itself.

As basic examples of formal group laws, we have the additive formal group law \mathbb{G}_{a}(X,Y)=X+Y, and the multiplicative group law \mathbb{G}_{m}(X,Y)=X+Y+XY. In this post we will focus on another formal group law called the Lubin-Tate formal group law.

Let F be a nonarchimedean local field and let \mathcal{O}_{F} be its ring of integers. Let A be an \mathcal{O}_{F}-algebra with i:\mathcal{O}_{F}\to A its structure map. A formal \mathcal{O}_{F}-module law over A over A is a formal group law f(X,Y) such that for every element a of \mathcal{O}_{F} we have an associated endomorphism [a] of f(X,Y), and such that the linear term of this endomorphism as a power series is i(a)X.

Let \pi be a uniformizer (generator of the unique maximal ideal) of \mathcal{O}_{F}. Let q=p^{f} be the cardinality of the residue field of \mathcal{O}_{F}. There is a unique (up to isomorphism) formal \mathcal{O}_{F}-module law over \mathcal{O}_{F} such that as a power series its linear term is \pi X and such that it is congruent to X^{q} mod \pi. It is called the Lubin-Tate formal group law and we denote it by \mathcal{G}(X,Y).

The Lubin-Tate formal group law was originally studied by Jonathan Lubin and John Tate for the purpose of studying local class field theory (see Some Basics of Class Field Theory). The results of local class field theory state that the Galois group of the maximal abelian extension of F is isomorphic to the profinite completion \widehat{F}^{\times}. This profinite completion in turn decomposes into the product \mathcal{O}_{F}^{\times}\times \pi^{\widehat{\mathbb{Z}}}.

The factor isomorphic to \mathcal{O}_{F}^{\times} fixes the maximal unramified extension F^{\text{nr}} of F, the factor isomorphic to \pi^{\widehat{\mathbb{Z}}} fixes an infinite, totally ramified extension F_{\pi} of F, and we have that F=F^{\text{nr}}F_{\pi}. The theory of the Lubin-Tate formal group law was developed to study F_{\pi}, taking inspiration from the case where F=\mathbb{Q}_{p}. In this case \pi=p and the infinite totally ramified extension F_{p} is obtained by adjoining to \mathbb{Q}_{p} all p-th power roots of unity, which is also the p-th power torsion of the multiplicative group \mathbb{G}_{m}. We want to generalize \mathbb{G}_{m}, and this is what the Lubin-Tate formal group law accomplishes.

Let \mathcal{G}[\pi^{n}] be the set of all elements in the maximal ideal of some separable extension \mathcal{O}_{F} such that its image under the endomorphism [\pi^{n}] is zero. This takes the place of the p-th power roots of unity, and adjoining to F all the \mathcal{G}[\pi^{n}] for all n gives us the field F_{\pi}.

Furthermore, Lubin and Tate used the theory they developed to make local class field theory explicit in this case. We define the \pi-adic Tate module T_{\pi}(\mathcal{G}) as the inverse limit of \mathcal{G}[\pi^{n}] over all n. This is a free \mathcal{O}_{F}-module of rank 1 and its automorphisms are in fact isomorphic to \mathcal{O}_{F}^{\times}. Lubin and Tate proved that this is isomorphic to the Galois group of F_{\pi} over F and explicitly described the reciprocity map of local class field theory in this case as the map from F^{\times } to \text{Gal}(F_{\pi}/F) sending \pi to the identity and an element of \mathcal{O}_{F}^{\times} to the image of its inverse under the above isomorphism.

To study nonabelian extensions, one must consider deformations of the Lubin-Tate formal group. This will lead us to the study of the space of these deformations, called the Lubin-Tate space. This is intended to be the subject of a future blog post.


Lubin-Tate Formal Group Law on Wikipedia

Formal Group Law on Wikipedia

The Geometry of Lubin-Tate Spaces by Jared Weinstein

A Rough Introduction to Lubin-Tate Spaces by Zhiyu Zhang

Formal Groups and Applications by Michiel Hazewinkel