Siegel modular forms

December 14, 2021December 14, 2021 / Anton Hilado / 2 Comments

In Modular Forms we introduced modular forms as certain holomorphic functions on the upper half-plane following certain transformation properties with respect to the action of the group $\mathrm{SL}_{2}(\mathbb{Z})$ (or more generally its congruence subgroups). We also saw that they are sections of certain sheaves on the compactified moduli space of elliptic curves, possibly together with extra structure, such as a basis of $N$ -torsion, a point of order $N$ , or a cyclic subgroup of order $N$ (see also The Moduli Space of Elliptic Curves).

In this post we shall introduce a higher-dimensional generalization of this idea. Namely, we shall introduce Siegel modular forms, which are to principally polarized abelian varieties as the usual (also called elliptic) modular forms are to elliptic curves.

Let us follow the same approach that we used to introduce modular forms, as certain functions on the upper half-space with certain transformation properties. Therefore the first thing we will need is a higher-dimensional analogue of the upper half-space.

The Siegel upper half-space of degree $g$ (or genus $g$ ), denoted $\mathcal{H}_{g}$ is the set of all $g\times g$ symmetric matrices whose entries are complex numbers with a positive imaginary part. If $g=1$ , then this is the same as the usual upper half-space.

Now we need the analogue of the transformation properties of an elliptic modular form under the modular group $\mathrm{SL}_{2}(\mathbb{Z})$ . We recall that the action of $\mathrm{SL}_{2}(\mathbb{Z})$ on the upper half-plane was inherited from the action of $\mathrm{SL}_{2}(\mathbb{R})$ via Mobius transformations. If $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is an element of $\gamma=\mathrm{SL}_{2}(\mathbb{R})$ , then it maps a point $\tau$ on the upper half-plane to $\displaystyle \gamma(z)=\frac{a\tau+b}{c\tau+d}$ . Then we define a modular form of weight $k$ to be a holomorphic function $f:\mathcal{H}\to\mathbb{C}$ such that $f(\gamma(\tau))=(c\tau+d)^{k}f(\tau)$ and such that $f$ is holomorphic at infinity (it is bounded as the imaginary part of $\tau$ approaches infinity).

For Siegel modular form, our group will be the Siegel modular group $\mathrm{Sp}_{2g}(\mathbb{Z})$ , which is a subgroup of the symplectic group $\mathrm{Sp}_{2g}(\mathbb{R})$ . The elements of the symplectic group are $2g\times 2g$ real matrices which can be written in the form $\begin{pmatrix}A&B\\C&D\end{pmatrix}$ where $A$ , $B$ , $C$ , and $D$ are $g\times g$ real matrices satisfying $AB^{T}=BA^{T}$ , $CD^{T}=DC^{T}$ , and $AD^{T}-DC^{T}=I_{g}$ , where the superscript ${}^{T}$ means taking the transpose and $I_{g}$ is the $g\times g$ identity matrix. Note that if $g=1$ , then the first two conditions are automatically satisfied while the third condition says that the determinant of the matrix must be $1$ . Therefore $\mathrm{Sp}_{2}(\mathbb{R})=\mathrm{SL}_{2}(\mathbb{R})$ .

Now let $\tau$ be an element of the Siegel upper half-plane $\mathcal{H}_{g}$ . Note that $\tau$ is now a $g\times g$ matrix. An element $\gamma$ of $\mathrm{Sp}_{2g}(\mathbb{R})$ sends $\tau$ to the element

$\gamma(\tau)=(A\tau+B)(C\tau+D)^{-1}$ .

We are almost ready to define Siegel modular forms. Although we may define Siegel modular forms as being complex-valued just like elliptic modular forms, and they are in themselves worthwhile objects of study, it is sometimes more natural to consider Siegel modular forms as being vector-valued. This arises for example when we want to obtain Siegel modular forms as sections of the Hodge bundle, which is the pushforward of the sheaf of relative differentials of the universal principally polarized abelian variety over $\mathbb{C}$ on the moduli space of principally polarized abelian varieties over $\mathbb{C}$ (which is obtained as the quotient of $\mathcal{H}_{2g}$ by $\mathrm{Sp}_{2g}(\mathbb{Z})$ ).

Let $V$ be a finite-dimensional vector space over $\mathbb{C}$ , and let $\rho:\mathrm{GL}_{g}(\mathbb{C})\to \mathrm{GL}(V)$ be a representation of $\mathrm{GL}_{g}(\mathbb{C})$ on $V$ . A Siegel modular form of weight $\rho$ is a holomorphic function $f:\mathcal{H}_{g}\to V$ such that

$\displaystyle f(\gamma(\tau))=\rho(C\tau+D)f(\tau)$

for any $g\in\mathrm{SL}_{2}(\mathbb{Z})$ , and which is holomorphic at infinity if $g=1$ . If $g>1$ , the holomorphicity at infinity is automatically taken care of by what is known as Kocher’s principle.

In the special case that $V=\mathbb{C}$ , and $\rho$ is given by taking powers of the determinant, i.e. our Siegel modular form is a holomorphic function $f:\mathcal{H}\to\mathbb{C}$ such that

$f(\gamma(\tau))=\mathrm{det}(C\tau+D)^{k}f(\tau)$

then we say that our Siegel modular form is a classical Siegel modular form. Note that a classical Siegel modular form of degree $1$ is an elliptic modular form.

We may also consider Siegel modular forms for congruence subgroups $\Gamma(N)$ of $\mathrm{Sp}_{2g}(\mathbb{Z})$ , where $\Gamma(N)$ is the subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z})$ consisting of elements that become the identity matrix after reduction mod $N$ .

The theory of Siegel modular forms is more complicated than the theory of elliptic modular forms, but we may use the latter to guide our study of the former. For instance, we may want to consider the Fourier expansion of Siegel modular forms. We may also want to consider its Hecke algebra (see also Hecke Operators). There are also analogues of important examples of elliptic modular forms, such as the Eisenstein series or the discriminant, for Siegel modular forms. We may also use elliptic modular forms to construct explicit examples of Siegel modular forms (a process known as lifting). All these and more will hopefully be discussed in future posts on this blog.

References:

Siegel modular form on Wikipedia

Siegel upper half-space on Wikipedia

Siegel modular variety on Wikipedia

Symplectic group on Wikipedia

Siegel modular forms by Gerard van der Greer

Galois Representations Coming From Weight 2 Eigenforms

April 21, 2021August 9, 2021 / Anton Hilado / 5 Comments

In Galois Representations we mentioned briefly that Galois representations can be obtained from modular forms. In this post we elaborate more on this construction, in the case that the modular form is a weight $2$ eigenform (a weight $2$ cusp form that is a simultaneous eigenfunction for all Hecke operators not dividing the level $N$ ). This specific case is also known as the Shimura construction, after Goro Shimura.

Let $f$ be a weight $2$ Hecke eigenform, of some level $\Gamma_{0}(N)$ (this also works with other level structures). We want to construct a $p$ -adic Galois representation associated to this Hecke eigenform, such that the two are going to be related in the following manner. For every prime $\ell$ not dividing $N$ and not equal to $p$ , the characteristic polynomial of the image of the Frobenius element associated to $\ell$ under this Galois representation will be of the form

$\displaystyle x^{2}-a_{\ell}x+\ell\chi(\ell)$

where $a_{\ell}$ is the eigenvalue of the Hecke operator $T_{\ell}$ and $\chi$ is a Dirichlet character associated to another kind of Hecke operator called the diamond operator $\langle \ell\rangle$ . This diamond operator acts on the argument of the modular form by an upper triangular element of $\mathrm{SL}_{2}(\mathbb{Z})$ whose bottom right entry is $\ell$ mod $N$ . This action is the same as the action of a Dirichlet character $\chi:\mathbb{Z}/N\mathbb{Z}\to\mathbb{C}^{\times}$ . The above polynomial is also known as the Hecke polynomial. All of this comes from what is known as the Eichler-Shimura relation, which relates the Hecke operators and the Frobenius.

The first thing that we will need is the identification of the weight $2$ cusp forms with the holomorphic differentials on the modular curve (as mentioned in Modular Forms in the case of $\mathbb{SL}_{2}(\mathbb{Z})$ , although this is can be done more generally).

The second thing that we will need is the Jacobian. One can think of the Jacobian as the space given by the equivalence classes of all path integrals on a curve (in general we can do this for any algebraic curve, not just modular curves), where two path integrals are to be considered equivalent if they differ by integration along a loop. Since path integration can be considered as a linear functional from holomorphic differentials to the complex numbers, we consider such path integrals as the dual space to the space of holomorphic differentials. However, the loops we wanted to quotient out by can also be expressed as elements of the homology group of the curve (see also Homology and Cohomology)!

Therefore we now define the Jacobian of a curve $X$ as

$\displaystyle J(\Gamma)=\Omega^{\vee}/H_{1}(X,\mathbb{Z})$

where $\Omega$ denotes the holomorphic differentials on $X$ . The notation $\Omega^{\vee}$ denotes the dual to $\Omega$ , since as we said the path integrals form the dual to the holomorphic differentials. The Jacobian can also described in other ways – for instance it is also the connected component of the Picard group (see also Divisors and the Picard Group), and the connection to the description given here is an important classical theorem called the Abel-Jacobi theorem.

The Jacobian is a higher-dimensional complex torus, and actually more is true – it is also an abelian variety, i.e. a projective variety whose points form a group (and hence a generalization of elliptic curves). Note that every complex torus is an elliptic curve, but this is not true in higher dimensions – only certain special kinds of higher dimensional complex tori (namely those with a polarization) are abelian varieties. In this vein the Jacobian of a curve has yet another description – it is “universal” among abelian varieties in that, if there is a morphism from a curve to any abelian variety, it can be expressed as a morphism from the curve to its Jacobian, followed by a morphism to that other abelian variety.

Now we go back to the case of modular curves. Denoting by $S_{2}(\Gamma_{0}(N))$ the space of cusp forms of weight two for the level structure $\Gamma_{0}(N)$ , which as discussed above is isomorphic to the space of holomorphic differentials on the corresponding modular curve $X(\Gamma_{0}(N))$ , we can now define the Jacobian $J(\Gamma_{0}(N))$ as

$\displaystyle J(\Gamma_{0}(N))=S_{2}(\Gamma_{0}(N))^{\vee}/H_{1}(X,\mathbb{Z})$

The third ingredient that we need is a certain ideal of the Hecke algebra (the ring of endomorphisms of $S_{2}(\Gamma_{0}(N))$ generated by the actions of the Hecke operators and diamond operators) corresponding to the weight $2$ Hecke eigenform $f$ (let us denote this ideal by $\mathbb{I}_{f})$ that we want to obtain our Galois representation from. This ideal $\mathbb{I}_{f})$ is defined to be the one generated by all elements of the Hecke algebra whose eigenvalue when acting on $f$ is zero.

Since the Hecke operators and diamond operators act on the Jacobian (we can see this this way – since the Jacobian is the quotient of the linear functionals on $S_{2}(\Gamma_{0}(N))$ , the action is obtained by first applying the Hecke operator or diamond operator to the weight $2$ eigenform, then applying the linear functional), we can use the ideal $\mathbb{I}_{f}$ to cut down a quotient of the Jacobian which is another abelian variety $A_{f}$ :

$\displaystyle A_{f}=J(\Gamma_{0}(N))/\mathbb{I}_{f}J(\Gamma_{0}(N))$

Finally, we can take the Tate module of $A_{f}$ , and this will give us precisely the Galois representation that we want. The abelian variety $A_{f}$ will have dimension equal to the degree of the number field generated by the eigenvalues of the Hecke operators.

If the eigenvalues are all rational, then $A_{f}$ will actually be an elliptic curve – in other words, given an eigenform of weight $2$ whose Hecke eigenvalues are all rational, we can always use it to construct an elliptic curve! This also gives us a map from the modular curve $X(\Gamma_{0}(N))$ to this elliptic curve, called a modular parametrization. The resulting elliptic curve will have the property that its L-function, built from point counts when it is reduced modulo primes, is the same as the L-function of the modular form which is built from its Fourier coefficients! This is because the Frobenius and the Fourier coefficients (which are also the eigenvalues of the Hecke operators) are related, as discussed above. The question of whether, given an elliptic curve, it comes from a modular form in this way, is another restatement of the question of modularity. The affirmative answer to this question, at least for certain elliptic curves over $\mathbb{Q}$ , led to the proof of Fermat’s Last Theorem.

This theory, which is only very roughly sketched here, is just a very special case – one can also obtain, for instance, Galois representations from modular forms which are not of weight $2$ . We leave this for the future.

References:

Jacobian variety on Wikipedia

Abel-Jacobi map on Wikipedia

Modularity theorem on Wikipedia

Course on Mazur’s Theorem Lecture 10: Jacobians by Andrew Snowden

Course on Mazur’s Theorem Lecture 17: Eichler-Shimura by Andrew Snowden

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

More on Galois Deformation Rings

April 17, 2021January 26, 2022 / Anton Hilado / 1 Comment

In Galois Deformation Rings we introduced the concept of a Galois deformation ring, and how it is used to prove “R=T” theorems. In this post we will look at a very simple example to help make things more concrete. Then we will explore more about the structure of Galois deformation rings, in particular we want to relate the tangent space of such a Galois deformation ring to the Selmer group in Galois cohomology (which also shows up in a lot of contexts all over arithmetic geometry and number theory).

Let $F$ be a finite extension of $\mathbb{Q}$ , and let $k$ be some finite field, with ring of Witt vectors $W(k)$ (for example if $k=\mathbb{F}_{p}$ then $W(k)=\mathbb{Z}_{p}$ ). Let our residual representation $\overline{\rho}:\mathrm{Gal}(\overline{F}/F)\to GL_{1}(k)$ be the trivial representation, i.e. the group acts as the identity. A lift will be a Galois representation $\overline{\rho}:\mathrm{Gal}(\overline{F}/F)\to GL_{1}(A)$ , where $A$ is a complete Noetherian algebra over $W(k)$ . Then our Galois deformation ring is given by the completed group ring

$\displaystyle R _{\overline{\rho}}=W(k)[[\mathrm{Gal}(\overline{F}/F)^{\mathrm{ab,p}}]]$

where $\mathrm{Gal}(\overline{F}/F)^{\mathrm{ab,p}}$ means the pro-p completion of the abelianization of the Galois group $\mathrm{Gal}(\overline{F}/F)$ . Using local class field theory, we can express this even more explicitly as

$\displaystyle R_{\overline{\rho}}=W(k)[\mu_{p^{\infty}}(F)][[X_{1},\ldots,X_{[F:\mathbb{Q}]}]]$

Let us now consider a useful fact about the tangent space (see also Tangent Spaces in Algebraic Geometry) of such a deformation ring. Let us first consider the framed deformation ring $R _{\overline{\rho}}^{\Box}$ . It is local, and has a unique maximal ideal $\mathfrak{m}$ . There is only one tangent space, defined to be the dual of $\mathfrak{m}/\mathfrak{m^{2}}$ , but this can also be expressed as the set of its dual number-valued points, i.e. $\mathrm{Hom}(R_{\overline{\rho}}^{\Box},k[\epsilon])$ , which by the definition of the framed deformation functor, is also $D_{\overline{\rho}}(k[\epsilon])^{\Box}$ . Any such deformation must be of the form

$\displaystyle \rho(\sigma)=(1+\varepsilon c(\sigma))\overline{\rho}(\sigma)$

where $c$ is some $n\times n$ matrix with coefficients in $k$ . If $\sigma$ and $\tau$ are elements of $\mathrm{Gal}(\overline{F}/F)$ , if we substitute the above form of $\rho$ into the equation $\rho(\sigma\tau)=\rho(\sigma)\rho(\tau)$ we have

$\displaystyle (1+\varepsilon c(\sigma\tau))\overline{\rho}(\sigma\tau) = (1+\varepsilon c(\sigma))\overline{\rho}(\sigma) (1+\varepsilon c(\tau))\overline{\rho}(\tau)$

from which we can see that

$\displaystyle c(\sigma\tau))\overline{\rho}(\sigma\tau) = c(\sigma)\overline{\rho}(\sigma)\overline{\rho}(\tau)+\overline{\rho}(\sigma)c(\tau)\overline{\rho}(\tau)$

and, multiplying by $\overline{\rho}(\sigma\tau)^{-1}= \overline{\rho}(\tau)^{-1}\overline{\rho}(\sigma)^{-1}$ on the right,

$\displaystyle c(\sigma\tau))=c(\sigma)(\tau)+c(\tau) \overline{\rho}(\sigma)\overline{\rho}(\sigma)^{-1}$

In the language of Galois cohomology, we say that $c$ is a $1$ -cocycle, if we take the $n\times n$ matrices to be a Galois module coming from the “Lie algebra” of $GL_{n}(k)$ . We call this Galois module $\mathrm{Ad}\overline{\rho}$ .

Now consider two different lifts (framed deformations) $\rho_{1}$ and $\rho_{2}$ which give rise to the same deformation of $\overline{\rho}$ . Then there exists some $n\times n$ matrix $X$ such that

$\displaystyle \rho_{1}(\sigma)=(1+\varepsilon X)\rho_{2}(\sigma)(1-\varepsilon X)$

Plugging in $\rho_{1}=(1+\varepsilon c_{1})\overline{\rho}$ and $\rho_{2}=(1+\varepsilon c_{2})\overline{\rho}$ we obtain

$\displaystyle (1+\varepsilon c_{1})\overline{\rho}=(1+\varepsilon X) (1+\varepsilon c_{2})\overline{\rho}(1-\varepsilon X)$

which will imply that

$\displaystyle c_{1}(\sigma)=c_{2}(\sigma)+X-\overline{\rho}(\sigma)X\overline{\rho}(\sigma)^{-1}$

In the language of Galois cohomology (see also Etale Cohomology of Fields and Galois Cohomology) we say that $c_{1}$ and $c_{2}$ differ by a coboundary. This means that the tangent space of the Galois deformation ring is given by the first Galois cohomology with coefficients in $\mathrm{Ad}\overline{\rho}$ :

$\displaystyle D_{\overline{\rho}}(k[\epsilon])\simeq H^{1}(\mathrm{Gal}(\overline{F}/F),\mathrm{Ad}\overline{\rho})$

More generally, when our Galois deformation ring is subject to conditions, it will be given by a subgroup of the first Galois cohomology known as the Selmer group (note that the Selmer group shows up in many places in arithmetic geometry and number theory, for instance, in the proof of the Mordell-Weil theorem where the Galois module used comes from the torsion points of an elliptic curve – in this post we are considering the case where the Galois module is $\mathrm{Ad}\overline{\rho}$ , as stated earlier). The advantage of expressing the tangent space in the language of Galois deformation ring using Galois cohomology is that in Galois cohomology there are certain formulas such as Tate duality and the Euler characteristic formula that we can use to perform computations.

Finally to end this post we remark that under certain conditions (namely that for every open subgroup $H$ of $\mathrm{Gal}(\overline{F}/F)$ the space of continuous homomorphisms from $H$ to $\mathbb{F}_{p}$ has finite dimension) this tangent space is going to be a finite-dimensional vector space over $k$ . Then the Galois deformation ring has the following form

$\displaystyle R_{\overline{\rho}}=W(k)[[x_{1},\ldots,x_{g}]]/(f_{1},\ldots,f_{r})$

i.e. it is a quotient of a $W(k)$ -power series in $g$ variables, where the number $g$ is given by the dimension of $H^{1}(\mathrm{Gal}(\overline{F}/F),\mathrm{Ad}\overline{\rho})$ as a $k$ -vector space, while the number of relations $r$ is given by the dimension of $H^{2}(\mathrm{Gal}(\overline{F}/F),\mathrm{Ad}\overline{\rho})$ as a $k$ -vector space.

Knowing the structure of Galois deformation rings is going to be important in proving R=T theorems, since such proofs often reduce to commutative algebra involving these rings. More details will be discussed in future posts on this blog.

References:

Group cohomology on Wikipedia

Galois cohomology on Wikipedia

Selmer group on Wikipedia

Tate duality on Wikipedia

Modularity Lifting Theorems by Toby Gee

Modularity Lifting (Course Notes) by Patrick Allen

Motives and L-functions by Frank Calegari

Beyond the Taylor-Wiles Method by Jack Thorne

Galois Deformation Rings

April 7, 2021April 21, 2021 / Anton Hilado / 6 Comments

In Galois Representations we talked about obtaining continuous Galois representations for example from the $\ell$ -adic etale cohomology of algebraic varieties, and hinted at being able to obtain such Galois representations from modular forms as well. While we postpone the discussion of how to obtain such a Galois representation to some future blog post (hopefully), we now mention the very important topic of modularity – which investigates, given some Galois representation, whether it comes from a modular form, and furthermore whether it provides some other information about the modular form that it comes from.

The topic of modularity is composed of two parts. The first is residual modularity – where we are given a Galois representation over a finite field (we call such a Galois representation a residual representation, in reference to the finite field being the residue field of some other ring) and figure out whether it comes from a modular form (in which case we also say that it is modular). The second part is modularity lifting, where, given a residual representation we know to be modular, we figure out whether it “lifts” to a Galois representation over $\mathbb{Q}_{\ell}$ .

In this post, we focus only on one small ingredient of the approach to proving modularity lifting. Proofs of modularity lifting rely on “R=T” theorems, where R refers to a Galois deformation ring and T comes from a (localization of) a Hecke algebra (see also Hecke Operators). The small ingredient we will focus on in this post is the R, the Galois deformation ring.

A “deformation” in our context is an equivalence class of “lifts” and before we give the precise definitions we give a little bit of intuition about why we are interested in lifts. Roughly, in our context, a lift of some field $\overline{R}$ is a local ring $R$ such that $\overline{R}$ is the residue field of $R$ , i.e. $\overline{R}=R/\mathfrak{m}$ where $\mathfrak{m}$ is the unique maximal ideal of $R$ (since $R$ is a local ring by definition it has a unique maximal ideal).

So now for the intuition. Consider the real numbers $\mathbb{R}$ . The “dual numbers” are defined to be $\mathbb{R}[x]/(x^{2})$ . Its elements are of the form $a+bx$ where $a$ and $b$ are real numbers. We can consider $x$ here to be an “infinitesimal element”. So we may think of an element of the dual numbers to be a number, given by $a$ , but with a “tangent vector” given by the number $b$ . Another way to think about it is that is at “position $a$ “, but it also has a “velocity $b$ “. It’s like numbers, but with a little “wiggle”. Now that we know about the dual numbers $\mathbb{R}[x]/(x^{2})$ , what about elements of $\mathbb{R}[x]/(x^{3})$ ? We may think of such an element, which is of the form $a+bx+cx^{2}$ , to be a position “ $a$ “, with “velocity $b$ “, and “acceleration $c$ “, a kind of “higher wiggle”.

If we continue including higher and higher derivatives, then we have something whose elements are formal power series $a+bx+cx^2+dx^3+\ldots$ . This is the ring $\mathbb{R}[[x]]$ , which is the inverse limit of the rings $\mathbb{R}/(x^{n})$ . Now the ring $\mathbb{R}[[x]]$ is a local ring with maximal ideal $(x)$ , and modding out by this maximal ideal gives $\mathbb{R}$ . So this power series ring is a lift of $\mathbb{R}$ , kind of numbers with “higher wiggles”. This is what the term “deformation” is supposed to bring to mind.

We now give more precise definitions. Let $F$ be a finite extension of $\mathbb{Q}$ , and let $k$ be a finite field. A Galois representation $\overline{\rho}:\text{Gal}(\overline{F}/F)\to \text{GL}_{2}(k)$ is also called a residual representation. Now let $W(k)$ be the ring of Witt vectors of $k$ ; for example, if $k=\mathbb{F}_{p}$ , then $W(k)=\mathbb{Z}_{p}$ . A lift, or framed deformation of the residual representation $\overline{\rho}$ is a Galois representation $\overline{\rho}:\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_{n}(A)$ where $A$ is a complete Noetherian local $W(k)$ -algebra, such that modding out by the unique maximal ideal of $A$ gives the residual representation $\overline{\rho}$ . A deformation of $\overline{\rho}$ is an equivalence class of lifts of $\overline{\rho}$ , where two lifts are considered equivalent if they are conjugates under the kernel of the modding out map.

Consider the functor $\text{Def}_{\overline{\rho}}^{\Box}$ from the category of complete Noetherian local $W(k)$ -algebras to the category of sets, which assigns to a complete Noetherian local $W(k)$ -algebra $A$ the set of all its lifts. This functor happens to be representable, i.e. there is a Galois representation $\overline{\rho}:\text{Gal}(\overline{F}/F)\to \text{GL}_{n}(R_{\overline{\rho}}^{\Box})$ over some ring $R_{\overline{\rho}}^{\Box}$ called the universal framed deformation ring, such that the lifts of $\overline{\rho}$ are given by maps from the Galois deformations to the universal Galois deformation.

We can also do the same for deformations instead of framed deformations, as long as our residual representation satisfies a condition called “Schur’s condition”.

We can also impose conditions on our deformations – for instance, we may want to consider only lifts with a certain fixed determinant. These conditions are also called deformation problems and they are important because it is conjectured that Galois representations coming from modular forms have certain properties, and we want to match up these Galois representations with modular forms.

Roughly, the way these are matched up goes in the following manner. We have said above that deformations of a certain fixed Galois representation $\overline{\rho}$ to $A$ , possibly with some conditions, correspond to maps $R_{\overline{\rho},\mathrm{conditions}}\to A$ . We state that, given an isomorphism between the complex numbers and the p-adic complex numbers we can always construct a map $R_{\overline{\rho}, \mathrm{conditions} }\to \mathbb{C}$ from the preceding map.

Now a Hecke algebra $\mathbb{T}$ acts on Hecke eigenforms (which say we want to match up with the Galois representations, to show that these Galois representations come from them) and therefore have associated systems of eigenvalues. It is known that any such system of eigenvalues comes from some Hecke eigenform.

We choose only a localization of the Hecke algebra, which we call $\mathbb{T}_{\mathfrak{m}}$ , corresponding to only the modular forms that are expected to give rise to the Galois representations we are considering (the Eichler-Shimura theorem gives relations between the Fourier coefficients of the Hecke eigenform and the form of the characteristic polynomial of the Frobenius under the Galois representation, restricting it). On the other hand, these systems of eigenvalues corresponds to maps $\mathbb{T}_{\mathfrak{m}}\to \mathbb{C}$ .

So if we can show that $R_{\overline{\rho}, \mathrm{conditions} }=\mathbb{T}_{\mathfrak{m}}$ , then these two sets of maps to $\mathbb{C}$ match up, then we can show that these Galois representations come from modular forms. Showing that $R_{\overline{\rho}, \mathrm{conditions} }=\mathbb{T}_{\mathfrak{m}}$ is itself an elaborate process that involves a fascinating strategy pioneered by Richard Taylor and Andrew Wiles known as patching. We will hopefully discuss R=T theorems, and the method of patching, on this blog in more detail in the future.

References:

Deformation on Wikipedia

Modularity Lifting Theorems by Toby Gee

Modularity Lifting (Course Notes) by Patrick Allen

Motives and L-functions by Frank Calegari

Beyond the Taylor-Wiles Method by Jack Thorne

Perturbations, Deformations, and Variations (and “Near-Misses”) in Geometry, Physics, and Number Theory by Barry Mazur

Modular Forms

May 7, 2020December 13, 2021 / Anton Hilado / 9 Comments

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:

$\Delta(\tau)=(g_{2}(\tau))^{3}-27(g_{3}(\tau))^{2}$ .

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.

References:

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

SEAMS School Manila 2017: Topics on Elliptic Curves

July 30, 2017July 30, 2017 / Anton Hilado / Leave a comment

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

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

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

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

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

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

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

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

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

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

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

Algebraic Spaces and Stacks

June 12, 2017June 12, 2022 / Anton Hilado / 7 Comments

We introduced the concept of a moduli space in The Moduli Space of Elliptic Curves, and constructed explicitly the moduli space of elliptic curves, using the methods of complex analysis. In this post, we introduce the concepts of algebraic spaces and stacks, far-reaching generalizations of the concepts of varieties and schemes (see Varieties and Schemes Revisited), that are very useful, among other things, for constructing “moduli stacks“, which are an improvement over the naive notion of moduli space, namely in that one can obtain from it all “families of objects” by pulling back a “universal object”.

We need first the concept of a fibered category (also spelled fibred category). Given a category $\mathcal{C}$ , we say that some other category $\mathcal{S}$ is a category over $\mathcal{C}$ if there is a functor $p$ from $\mathcal{S}$ to $\mathcal{C}$ (this should be reminiscent of our discussion in Grothendieck’s Relative Point of View).

If $\mathcal{S}$ is a category over some other category $\mathcal{C}$ , we say that it is a fibered category (over $\mathcal{C}$ ) if for every object $U=p(x)$ and morphism $f: V\rightarrow U$ in $\mathcal{C}$ , there is a strongly cartesian morphism $\phi: f^{*}x\rightarrow x$ in $\mathcal{S}$ with $f=p(\phi)$ .

This means that any other morphism $\psi: z\rightarrow x$ whose image $p(\psi)$ under the functor $p$ factors as $p(\psi)=p(\phi)\circ h$ must also factor as $\psi=\phi\circ \theta$ under some unique morphism $\theta: z\rightarrow f^{*}x$ whose image under the functor $p$ is $h$ . We refer to $f^{*}x$ as the pullback of $x$ along $f$ .

Under the functor $p$ , the objects of $\mathcal{S}$ which get sent to $U$ in $\mathcal{C}$ and the morphisms of $\mathcal{S}$ which get sent to the identity morphism $i_{U}$ in $\mathcal{C}$ form a subcategory of $\mathcal{S}$ called the fiber over $U$ . We will also write it as $\mathcal{S}_{U}$ .

An important example of a fibered category is given by an ordinary presheaf on a category $\mathcal{C}$ , i.e. a functor $F:\mathcal{C}^{\text{op}}\rightarrow (\text{Set})$ ; we can consider it as a category fibered in sets $\mathcal{S}_{F}\rightarrow\mathcal{C}$ .

A special kind of fibered category that we will need later on is a category fibered in groupoids. A groupoid is simply a category where all morphisms have inverses, and a category fibered in groupoids is a fibered category where all the fibers are groupoids. A set is a special kind of groupoid, since it may be thought of as a category whose only morphisms are the identity morphisms (which are trivially their own inverses). Hence, the example given in the previous paragraph, that of a presheaf, is also an example of a category fibered in groupoids, since it is fibered in sets.

Now that we have the concept of fibered categories, we next want to define prestacks and stacks. Central to the definition of prestacks and stacks is the concept known as descent, so we have to discuss it first. The theory of descent can be thought of as a formalization of the idea of “gluing”.

Let $\mathcal{U}=\{f_{i}:U_{i}\rightarrow U\}$ be a covering (see Sheaves and More Category Theory: The Grothendieck Topos) of the object $U$ of $\mathcal{C}$ . An object with descent data is a collection of objects $X_{i}$ in $\mathcal{S}_{U}$ together with transition isomorphisms $\varphi_{ij}:\text{pr}_{0}^{*}X_{i}\simeq\text{pr}_{1}^{*}X_{j}$ in $\mathcal{S}_{U_{i}\times_{U}U_{j}}$ , satisfying the cocycle condition

$\displaystyle \text{pr}_{02}^{*}\varphi_{ik}=\text{pr}_{01}^{*}\varphi_{ij}\circ \text{pr}_{12}^{*}\varphi_{jk}:\text{pr}_{0}^{*}X_{i}\rightarrow \text{pr}_{2}^{*}X_{k}$

The morphisms $\text{pr}_{0}:U_{i}\times_{U}U_{j}\rightarrow U_{i}$ and the $\text{pr}_{1}:U_{i}\times_{U}U_{j}\rightarrow U_{j}$ are the projection morphisms. The notations $\text{pr}_{0}^{*}X_{i}$ and $\text{pr}_{1}^{*}X_{j}$ means that we are “pulling back” $X_{i}$ and $X_{j}$ from $\mathcal{S}_{U_{i}}$ and $\mathcal{S}_{U_{j}}$ , respectively, to $\mathcal{S}_{U_{i}\times_{U}U_{j}}$ .

A morphism between two objects with descent data is a a collection of morphisms $\psi_{i}:X_{i}\rightarrow X'_{i}$ in $\mathcal{S}_{U_{i}}$ such that $\varphi'_{ij}\circ\text{pr}_{0}^{*}\psi_{i}=\text{pr}_{1}^{*}\psi_{j}\circ\varphi_{ij}$ . Therefore we obtain a category, the category of objects with descent data, denoted $\mathcal{DD}(\mathcal{U})$ .

We can define a functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ by assigning to each object $X$ of $\mathcal{S}_{U}$ the object with descent data given by the pullback $f_{i}^{*}X$ and the canonical isomorphism $\text{pr}_{0}^{*}f_{i}^{*}X\rightarrow\text{pr}_{1}^{*}f_{j}^{*}X$ . An object with descent data that is in the essential image of this functor is called effective.

Before we give the definitions of prestacks and stacks, we recall some definitions from category theory:

A functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is faithful if the induced map $\text{Hom}_{\mathcal{A}}(x,y)\rightarrow \text{Hom}_{\mathcal{B}}(F(x),F(y))$ is injective for any two objects $x$ and $y$ of $\mathcal{A}$ .

A functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is full if the induced map $\text{Hom}_{\mathcal{A}}(x,y)\rightarrow \text{Hom}_{\mathcal{B}}(F(x),F(y))$ is surjective for any two objects $x$ and $y$ of $\mathcal{A}$ .

A functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is essentially surjective if any object $y$ of $\mathcal{B}$ is isomorphic to the image $F(x)$ of some object $x$ in $\mathcal{A}$ under $F$ .

A functor which is both faithful and full is called fully faithful. If, in addition, it is also essentially surjective, then it is called an equivalence of categories.

Now we give the definitions of prestacks and stacks using the functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ we have defined earlier.

If the functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ is fully faithful, then the fibered category $\mathcal{S}\rightarrow\mathcal{C}$ is a prestack.

If the functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ is an equivalence of categories, then the fibered category $\mathcal{S}\rightarrow\mathcal{C}$ is a stack.

Going back to the example of a presheaf as a fibered category, we now look at what it means when it satisfies the conditions for being a prestack, or a stack:

(i) $F$ is a prestack if and only if it is a separated functor,

(ii) $F$ is stack if and only if it is a sheaf.

We now have the abstract idea of a stack in terms of category theory. Next we want to have more specific examples of interest in algebraic geometry, namely, algebraic spaces and algebraic stacks. For this we need first the idea of a representable functor (and the closely related idea of a representable presheaf). The importance of representability is that this will allow us to “transfer” interesting properties of morphisms between schemes such as being surjective, etale, or smooth, to functors between categories or natural transformations between functors. Therefore we will be able to say that a functor or natural transformation is surjective, or etale, or smooth, which is important, because we will define algebraic spaces and stacks as functors and categories, respectively, but we want them to still be closely related, or similar enough, to schemes.

A representable functor is a functor from $\mathcal{C}$ to $\textbf{Sets}$ which is naturally isomorphic to the functor which assigns to any object $X$ the set of morphisms $\text{Hom}(X,U)$ , for some fixed object $U$ of $\mathcal{C}$ .

A representable presheaf is a contravariant functor from $\mathcal{C}$ to $\textbf{Sets}$ which is naturally isomorphic to the functor which assigns to any object $X$ the set of morphisms $\text{Hom}(U,X)$ , for some fixed object $U$ of $\mathcal{C}$ . If $\mathcal{C}$ is the category of schemes, the latter functor is also called the functor of points of the object $U$ .

We take this opportunity to emphasize a very important concept in modern algebraic geometry. The functor of points $h_{U}$ of a scheme $U$ may be identified with $U$ itself. There are many advantages to this point of view (which is also known as functorial algebraic geometry); in particular we will need it later when we give the definition of algebraic spaces and stacks.

We now have the idea of a representable functor. Next we want to have an idea of a representable natural transformation (or representable morphism) of functors. We will need another prerequisite, that of a fiber product of functors.

Let $F,G,H:\mathcal{C}^{\text{op}}\rightarrow \textbf{Sets}$ be functors, and let $a:F\rightarrow G$ and $b:H\rightarrow G$ be natural transformations between these functors. Then the fiber product $F\times_{a,G,b}H$ is a functor from $\mathcal{C}^{\text{op}}$ to $\textbf{Sets}$ , and is given by the formula

$\displaystyle (F\times_{a,G,b}H)(X)=F(X)\times_{a_{X},G(X),b_{X}}H(X)$

for any object $X$ of $\mathcal{C}$ .

Let $F,G:\mathcal{C}^{\text{op}}\rightarrow \textbf{Sets}$ be functors. We say that a natural transformation $a:F\rightarrow G$ is representable, or that $F$ is relatively representable over $G$ if for every $U\in\text{Ob}(\mathcal{C})$ and any $\xi\in G(U)$ the functor $h_{U}\times_{G}F$ is representable.

We now let $(\text{Sch}/S)_{\text{fppf}}$ be the site (a category with a Grothendieck topology – see also More Category Theory: The Grothendieck Topos) whose underlying category is the category of $S$ -schemes, and whose coverings are given by families of flat, locally finitely presented morphisms. Any etale covering or Zariski covering is an example of this “fppf covering” (“fppf” stands for fidelement plate de presentation finie, which is French for faithfully flat and finitely presented).

An algebraic space over a scheme $S$ is a presheaf

$\displaystyle F:((\text{Sch}/S)_{\text{fppf}})^{\text{op}}\rightarrow \textbf{Sets}$

with the following properties

(1) The presheaf $F$ is a sheaf.

(2) The diagonal morphism $F\rightarrow F\times F$ is representable.

(3) There exists a scheme $U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}})$ and a map $h_{U}\rightarrow F$ which is surjective, and etale (This is often written simply as $U\rightarrow F$ ). The scheme $U$ is also called an atlas.

The diagonal morphism being representable implies that the natural transformation $h_{U}\rightarrow F$ is also representable, and this is what allows us to describe it as surjective and etale, as has been explained earlier.

An algebraic space is a generalization of the notion of a scheme. In fact, a scheme is simply the case where, for the third condition, we have $U$ is the disjoint union of affine schemes $U_{i}$ and where the map $h_{U}\rightarrow F$ is an open immersion. We recall that a scheme may be thought of as being made up of affine schemes “glued together”. This “gluing” is obtained using the Zariski topology. The notion of an algebraic space generalizes this to the etale topology.

Next we want to define algebraic stacks. Unlike algebraic spaces, which we defined as presheaves (functors), we will define algebraic stacks as categories, so we need to once again revisit the notion of representability in terms of categories.

Let $\mathcal{C}$ be a category. A category fibered in groupoids $p:\mathcal{S}\rightarrow\mathcal{C}$ is called representable if there exists an object $X$ of $\mathcal{C}$ and an equivalence $j:\mathcal{S}\rightarrow \mathcal{C}/X$ (The notation $\mathcal{C}/X$ signifies a slice category, whose objects are morphisms $f:U\rightarrow X$ in $\mathcal{C}$ , and whose morphisms are morphisms $h:U\rightarrow V$ in $\mathcal{C}$ such that $f=g\circ h$ , where $g:U\rightarrow X$ ).

We give two specific special cases of interest to us (although in this post we will only need the latter):

Let $\mathcal{X}$ be a category fibered in groupoids over $(\text{Sch}/S)_{\text{fppf}}$ . Then $\mathcal{X}$ is representable by a scheme if there exists a scheme $U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}})$ and an equivalence $j:\mathcal{X}\rightarrow (\text{Sch}/U)_{\text{fppf}}$ of categories over $(\text{Sch}/S)_{\text{fppf}}$ .

A category fibered in groupoids $p : \mathcal{X}\rightarrow (\text{Sch}/S)_{\text{fppf}}$ is representable by an algebraic space over $S$ if there exists an algebraic space $F$ over $S$ and an equivalence $j:\mathcal{X}\rightarrow \mathcal{S}_{F}$ of categories over $(\text{Sch}/S)_{\text{fppf}}$ .

Next, following what we did earlier for the case of algebraic spaces, we want to define the notion of representability (by algebraic spaces) for morphisms of categories fibered in groupoids (these are simply functors satisfying some compatibility conditions with the extra structure of the category). We will need, once again, the notion of a fiber product, this time of categories over some other fixed category.

Let $F:\mathcal{X}\rightarrow\mathcal{S}$ and $G:\mathcal{Y}\rightarrow\mathcal{S}$ be morphisms of categories over $\mathcal{C}$ . The fiber product $\mathcal{X}\times_{\mathcal{S}}\mathcal{Y}$ is given by the following description:

(1) an object of $\mathcal{X}\times_{\mathcal{S}}\mathcal{Y}$ is a quadruple $(U,x,y,f)$ , where $U\in\text{Ob}(\mathcal{C})$ , $x\in\text{Ob}(\mathcal{X}_{U})$ , $y\in\text{Ob}(\mathcal{Y}_{U})$ , and $f : F(x)\rightarrow G(y)$ is an isomorphism in $\mathcal{S}_{U}$ ,

(2) a morphism $(U,x,y,f) \rightarrow (U',x',y',f')$ is given by a pair $(a,b)$ , where $a:x\rightarrow x'$ is a morphism in $X$ , and $b:y\rightarrow y'$ is a morphism in $Y$ such that $a$ and $b$ induce the same morphism $U\rightarrow U'$ , and $f'\circ F(a)=G(b)\circ f$ .

Let $S$ be a scheme. A morphism $f:\mathcal{X}\rightarrow \mathcal{Y}$ of categories fibered in groupoids over $(\text{Sch}/S)_{\text{fppf}}$ is called representable by algebraic spaces if for any $U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}})$ and any $y:(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{Y}$ the category fibered in groupoids

$\displaystyle (\text{Sch}/U)_{\text{fppf}}\times_{y,\mathcal{Y}}\mathcal{X}$

over $(\text{Sch}/U)_{\text{fppf}}$ is representable by an algebraic space over $U$ .

An algebraic stack (or Artin stack) over a scheme $S$ is a category

$\displaystyle p:\mathcal{X}\rightarrow (\text{Sch}/S)_{\text{fppf}}$

with the following properties:

(1) The category $\mathcal{X}$ is a stack in groupoids over $(\text{Sch}/S)_{\text{fppf}}$ .

(2) The diagonal $\Delta:\mathcal{X}\rightarrow \mathcal{X}\times\mathcal{X}$ is representable by algebraic spaces.

(3) There exists a scheme $U\in\text{Ob}((\text{Sch/S})_{\text{fppf}})$ and a morphism $(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X}$ which is surjective and smooth (This is often written simply as $U\rightarrow\mathcal{X}$ ). Again, the scheme $U$ is called an atlas.

If the morphism $(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X}$ is surjective and etale, we have a Deligne-Mumford stack.

Just as an algebraic space is a generalization of the notion of a scheme, an algebraic stack is also a generalization of the notion of an algebraic space (recall that that a presheaf can be thought of as category fibered in sets, which themselves are special cases of groupoids). Therefore, the definition of an algebraic stack closely resembles the definition of an algebraic space given earlier, including the requirement that the diagonal morphism (which in this case is a functor between categories) be representable, so that the functor $(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X}$ is also representable, and we can describe it as being surjective and smooth (or surjective and etale).

As an example of an application of the ideas just discussed, we mention the moduli stack of elliptic curves (which we denote by $\mathcal{M}_{1,1}$ – the reason for this notation will become clear later). A family of elliptic curves over some “base space” $B$ is a fibration $\pi:X\rightarrow B$ with a section $O:B\rightarrow X$ such that the fiber $\pi^{-1}(b)$ over any point $b$ of $B$ is an elliptic curve with origin $O(b)$ .

Ideally what we want is to be able to obtain every family $X\rightarrow B$ by pulling back a “universal object” $E\rightarrow\mathcal{M}_{1,1}$ via the map $B\rightarrow\mathcal{M}_{1,1}$ . This is something that even the notion of moduli space that we discussed in The Moduli Space of Elliptic Curves cannot do (we suggestively denote that moduli space by $M_{1,1}$ ). So we need the concept of stacks to construct this “moduli stack” that has this property. A more thorough discussion would need the notion of quotient stacks and orbifolds, but we only mention that the moduli stack of elliptic curves is in fact a Deligne-Mumford stack.

More generally, we can construct the moduli stack of curves of genus $g$ with $\nu$ marked points, denoted $\mathcal{M}_{g,\nu}$ . The moduli stack of elliptic curves is simply the special case $\mathcal{M}_{1,1}$ . Aside from just curves of course, we can construct moduli stacks for many more mathematical objects, such subschemes of some fixed scheme, or vector bundles, also on some fixed scheme.

The subject of algebraic stacks is a vast one, as may perhaps be inferred from the size of one of the main references for this post, the open-source reference The Stacks Project, which consists of almost 6,000 pages at the time of this writing. All that has been attempted in this post is but an extremely “bare bones” introduction to some of its more basic concepts. Hopefully more on stacks will be featured in future posts on the blog.

References:

Stack on Wikipedia

Algebraic Space on Wikipedia

Fibred Category on Wikipedia

Descent Theory on Wikipedia

Stack on nLab

Grothendieck Fibration on nLab

Algebraic Space on nLab

Algebraic Stack on nLab

Moduli Stack of Elliptic Curves on nLab

Stacks for Everybody by Barbara Fantechi

What is…a Stack? by Dan Edidin

Notes on the Construction of the Moduli Space of Curves by Dan Edidin

Notes on Grothendieck Topologies, Fibered Categories and Descent Theory by Angelo Vistoli

Lectures on Moduli Spaces of Elliptic Curves by Richard Hain

The Stacks Project

Algebraic Spaces and Stacks by Martin Olsson

Fundamental Algebraic Geometry: Grothendieck’s FGA Explained by Barbara Fantechi, Lothar Gottsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli

Reduction of Elliptic Curves Modulo Primes

March 7, 2017June 19, 2022 / Anton Hilado / 2 Comments

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

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

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

with $a$ and $b$ satisfying the condition that

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

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

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

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

whose graph in the $x$ – $y$ plane is given by the following figure (plotted using the WolframAlpha software):

msp8421ceh7049d511806600001ha509ah5dee52ed

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

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

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

msp51201b91cb194c07ih2g0000670ibac2gfha7c42

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$Y^{2}Z=X^{3}+a_{2}X^{2}Z+a_{4}XZ^{2}+a_{6}Z^{3}$

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

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

The quantity

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

is called the discriminant of the curve.

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

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

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

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

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

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

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

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

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

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

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

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

where $p$ runs over all prime numbers, and

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

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

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

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

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

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

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

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

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

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

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

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

References:

Elliptic Curve on Wikipedia

Hasse-Weil Zeta Function on Wikipedia

Birch and Swinnerton-Dyer Conjecture on Wikipedia

Modularity Theorem on Wikipedia

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

The Birch and Swinnerton-Dyer Conjecture by Andrew Wiles

A Marvelous Proof by Fernando Q. Gouvea

A Friendly Introduction to Number Theory by Joseph H. Silverman

The Arithmetic of Elliptic Curves by Joseph H. Silverman

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

Invitation to the Mathematics of Fermat-Wiles by Yves Hellegouarch

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

The Moduli Space of Elliptic Curves

February 16, 2017December 13, 2021 / Anton Hilado / 13 Comments

A moduli space is a kind of “parameter space” that “classifies” mathematical objects. Every point of the moduli space stands for a mathematical object, in such a way that mathematical objects which are more similar to each other are closer and those that are more different from each other are farther apart. We may use the notion of equivalence relations (see Modular Arithmetic and Quotient Sets) to assign several objects which are in some sense “isomorphic” to each other to a single point.

We have discussed on this blog before one example of a moduli space – the projective line (see Projective Geometry). Every point on the projective line corresponds to a geometric object, a line through the origin. Two lines which have almost the same value of the slope will be closer on the projective line compared to two lines which are almost perpendicular.

Another example of a moduli space is that for circles on a plane – such a circle is specified by three real numbers, two coordinates for the center and one positive real number for the radius. Therefore the moduli space for circles on a plane will consist of a “half-volume” of some sort, like 3D space except that one coordinate is restricted to be strictly positive. But if we only care about the circles up to “congruence”, we can ignore the coordinates for the center – or we can also think of it as simply sending circles with the same radius to a single point, even if they are centered at different points. This moduli space is just the positive real line. Every point on this moduli space, which is a positive real number, corresponds to all the circles with radius equal to that positive real number.

We now want to construct the moduli space of elliptic curves. In order to do this we will need to first understand the meaning of the following statement:

Over the complex numbers, an elliptic curve is a torus.

We have already seen in Elliptic Curves what an elliptic curve looks like when graphed in the $x$ – $y$ plane, where $x$ and $y$ are real numbers. This gives us a look at the points of the elliptic curve whose coordinates are real numbers, or to put it in another way, these are the real numbers $x$ and $y$ which satisfy the equation of the elliptic curve.

When we look at the points of the elliptic curve with complex coordinates, or in other words the complex numbers which satisfy the equation of the elliptic curve, the situation is more complicated. First off, what we actually have is not what we usually think of as a curve, but rather a surface, in the same way that the complex numbers do not form a line like the real numbers do, but instead form a plane. However, even though it is not easy to visualize, there is a function called the Weierstrass elliptic function which provides a correspondence between the (complex) points of an elliptic curve and the points in the “fundamental parallelogram” of a lattice in the complex plane. We can think of “gluing” the opposite sides of this fundamental parallelogram to obtain a torus. This is what we mean when we say that an elliptic curve is a torus. This also means that there is a correspondence between elliptic curves and lattices in the complex plane.

We will discuss more about lattices later on in this post, but first, just in case the preceding discussion seems a little contrived, we elaborate a bit on the Weierstrass elliptic function. We must first discuss the concept of a holomorphic function. We have discussed in An Intuitive Introduction to Calculus the concept of the derivative of a function. Now not all functions have derivatives that exist at all points; in the case that the derivative of the function does exist at all points, we refer to the function as a differentiable function.

The concept of a holomorphic function in complex analysis (analysis is the term usually used in modern mathematics to refer to calculus and its related subjects) is akin to the concept of a differentiable function in real analysis. The derivative is defined as the limit of a certain ratio as the numerator and the denominator both approach zero; on the real line, there are limited ways in which these quantities can approach zero, but on the complex plane, they can approach zero from several different directions; for a function to be holomorphic, the expression for its derivative must remain the same regardless of the direction by which we approach zero.

In previous posts on topology on this blog we have been treating two different topological spaces as essentially the same whenever we can find a bijective and continuous function (also known as a homeomorphism) between them; similarly, we have been treating different algebraic structures such as groups, rings, modules, and vector spaces as essentially the same whenever we can find a bijective homomorphism (an isomorphism) between two such structures. Following these ideas and applying them to complex analysis, we may treat two spaces as essentially the same if we can find a bijective holomorphic function between them.

The Weierstrass elliptic function is not quite holomorphic, but is meromorphic – this means that it would have been holomorphic everywhere if not for the “lattice points” where there exist “poles”. But it is alright for us, because such a lattice point is to be mapped to the “point at infinity”. All in all, this allows us to think of the complex points of the elliptic curve as being essentially the same as a torus, following the ideas discussed in the preceding paragraph.

Moreover, the torus has a group structure of its own, considered as the direct product group $\text{U}(1)\times\text{U}(1)$ where $\text{U}(1)$ is the group of complex numbers of magnitude equal to $1$ with the law of composition given by the multiplication of complex numbers. When the complex points of the elliptic curve get mapped by the Weierstrass elliptic function to the points of the torus, the group structure provided by the “tangent and chord” or “tangent and secant” construction becomes the group structure of the torus. In other words, the Weierstrass elliptic function provides us with a group isomorphism.

All this discussion means that the study of elliptic curves becomes the study of lattices in the complex plane. Therefore, what we want to construct is the moduli space of lattices in the complex plane, up to a certain equivalence relation – two lattices are to be considered equivalent if one can be obtained by multiplying the other by a complex number (this equivalence relation is called homothety). Going back to elliptic curves, this corresponds to an isomorphism of elliptic curves in the sense of algebraic geometry.

Now given two complex numbers $\omega_{1}$ and $\omega_{2}$ , a lattice $\Lambda$ in the complex plane is given by

$\Lambda=\{m\omega_{1}+n\omega_{2}|m,n\in\mathbb{Z}\}$

For example, setting $\omega_{1}=1$ and $\omega_{2}=i$ , gives a “square” lattice. This lattice is also the set of all Gaussian integers. The fundamental parallelogram is the parallelogram formed by the vertices $0$ , $\omega_{1}$ , $\omega_{2}$ , and $\omega_{1}+\omega_{2}$ . Here is an example of a lattice, courtesy of Alvaro Lozano-Robledo:

The fundamental parallelogram is in blue. Here is another, courtesy of Sam Derbyshire:

Because we only care about lattices up to homothety, we can “rescale” the lattice by multiplying it with a complex number equal to $\frac{1}{\omega_{1}}$ , so that we have a new lattice equivalent under homothety to the old one, given by

$\Lambda=\{m+n\omega|m,n\in\mathbb{Z}\}$

where

$\displaystyle \tau=\frac{\omega_{2}}{\omega_{1}}$ .

We can always interchange $\omega_{1}$ and $\omega_{2}$ , but we will fix our convention so that the complex number $\tau=\frac{\omega_{2}}{\omega_{1}}$ , when written in polar form $\tau=re^{i\theta}$ always has a positive angle $\theta$ between 0 and 180 degrees. If we cannot obtain this using our choice of $\omega_{1}$ and $\omega_{2}$ , then we switch the two.

Now what this means is that a complex number $\omega$ , which we note is a complex number in the upper half plane $\mathbb{H}=\{z\in \mathbb{C}|\text{Im}(z)>0\}$ , because of our convention in choosing $\omega_{1}$ and $\omega_{2}$ , uniquely specifies a homothety class of lattices $\Lambda$ . However, a homothety class of lattices may not always uniquely specify such a complex number $\tau$ . Several such complex numbers may refer to the same homothety class of lattices.

What $\omega_{1}$ and $\omega_{2}$ specify is a choice of basis (see More on Vector Spaces and Modules) for the lattice $\Lambda$ ; we may choose several different bases to refer to the same lattice. Hence, the upper half plane is not yet the moduli space of all lattices in the complex plane (up to homothety); instead it is an example of what is called a Teichmuller space. To obtain the moduli space from the Teichmuller space, we need to figure out when two different bases specify lattices that are homothetic.

We will just write down the answer here; two complex numbers $\tau$ and $\tau'$ refer to homothetic lattices if there exists the following relation between them:

$\displaystyle \tau'=\frac{a\tau+b}{c\tau+d}$

for integers $a$ , $b$ , $c$ , and $d$ satisfying the identity

$\displaystyle ad-bc=1$ .

We can “encode” this information into a $2\times 2$ matrix (see Matrices) which is an element of the group (see Groups) called $\text{SL}(2,\mathbb{Z})$ . It is the group of $2\times 2$ matrices with integer entries and determinant equal to $1$ . Actually, the matrix with entries $a$ , $b$ , $c$ , and $d$ and the matrix with entries $-a$ , $-b$ , $-c$ , and $-d$ specify the same transformation, therefore what we actually want is the group called $\text{PSL}(2,\mathbb{Z})$ , also known as the modular group, and also written $\Gamma(1)$ , obtained from the group $\text{SL}(2,\mathbb{Z})$ by considering two matrices to be equivalent if one is the negative of the other.

We now have the moduli space that we want – we start with the upper half plane $\mathbb{H}$ , and then we identify two points if we can map one point into the other via the action of an element of the modular group, as we have discussed earlier. In technical language, we say that they belong to the same orbit. We can write our moduli space as $\Gamma(1)\backslash\mathbb{H}$ (the notation means that the group $\Gamma(1)$ acts on $\mathbb{H}$ “on the left”).

When dealing with quotient sets, which are sets of equivalence classes, we have seen in Modular Arithmetic and Quotient Sets that we can choose from an equivalence class one element to serve as the “representative” of this equivalence class. For our moduli space $\Gamma(1)\backslash\mathbb{H}$ , we can choose for the representative of an equivalence class a point from the “fundamental domain” for the modular group. Any point on the upper half plane can be obtained by acting on a point from the fundamental domain with an element of the modular group. The following diagram, courtesy of user Fropuff on Wikipedia, shows the fundamental domain in gray:

The other parts of the diagram show where the fundamental domain gets mapped to by certain special elements, in particular the “generators” of the modular group, which are the two elements where $a=0$ , $b=-1$ , $c=1$ , and $d=-1$ , and $a=1$ , $b=1$ , $c=1$ , and $d=0$ . We will not discuss too much of these concepts for now. Instead we will give a preview of some concepts related to this moduli space. Topologically, this moduli space looks like a sphere with a missing point; in order to make the moduli space into a sphere (topologically), we take the union of the upper half plane $\mathbb{H}$ with the projective line (see Projective Geometry) $\mathbb{P}^{1}(\mathbb{Q})$ . This projective line may be thought of as the set of all rational numbers $\mathbb{Q}$ together with a “point at infinity.” The modular group also acts on this projective line, so we can now take the quotient of $\mathbb{H}\cup\mathbb{P}^{1}(\mathbb{Q})$ (denoted $\mathbb{H}^{*}$ by the same equivalence relation as earlier; this new space, topologically equivalent to the sphere, is called the modular curve $X(1)$ .

The functions and “differential forms” on the modular curve $X(1)$ are of special interest. They can be obtained from functions on the upper half plane (with the “point at infinity”) satisfying certain conditions related to the modular group. If they are holomorphic everywhere, including the “point at infinity”, they are called modular forms. Modular forms are an interesting object of study in themselves, and their generalizations, automorphic forms, are a very active part of modern mathematical research.

Moduli Space on Wikipedia

Elliptic Curve on Wikipedia

Weierstrass’s Elliptic Functions on Wikipedia

Fundamental Pair of Periods on Wikipedia

Modular Group on Wikipedia

Fundamental Domain on Wikipedia

Modular Form on Wikipedia

Automorphic Form on Wikipedia

Image by Alvano Lozano Robledo on Wikipedia

Image by Sam Derbyshire on Wikipedia

Image by User Fropuff of Wikipedia

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

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

Elliptic Curves

February 12, 2017June 19, 2022 / Anton Hilado / 6 Comments

An elliptic curve (not to be confused with an ellipse) is a certain kind of polynomial equation which can usually be expressed in the form

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

where $a$ and $b$ are numbers (more precisely, elements of some field) which satisfy the condition that the quantity

$\displaystyle 4a^{3}+27b^{2}$

is not equal to zero. This is not the most general form of an elliptic curve, as it will not hold for coefficients of “finite characteristic” equal to $2$ or $3$ ; however, for our present purposes, this definition will suffice.

Examples of elliptic curves are the following:

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

$\displaystyle y^{2}=x^{3}-x+1$

which, for real $x$ and $y$ may be graphed in the “Cartesian” or “ $x$ – $y$ ” plane as follows (image courtesy of user YassineMrabet of Wikipedia):

This rather simple mathematical object has very interesting properties which make it a central object of study in many areas of modern mathematical research.

In this post we focus mainly on one of these many interesting properties, which is the following:

The points of an elliptic curve form a group.

A group is a set with a law of composition which is associative, and the set contains an “identity element” under this law of composition, and every element of this set has an “inverse” (see Groups). Now this law of composition applies whether the points of the elliptic curve have rational numbers, real numbers, or complex numbers for coordinates, and it is always given by the same formula. It is perhaps most visible if we consider real numbers, since in that case we can plot it on the $x$ – $y$ plane as we have done earlier. The law of composition is also often called the “tangent and chord” or “tangent and secant” construction.

We now expound on this construction. Given two points on the elliptic curve $P$ and $Q$ on the curve, we draw a line passing through both of them. In most cases, this line will pass through another point $R$ on the curve. Then we draw a vertical line that passes through the point $R$ . This vertical line will pass through another point $R'$ on the curve. This gives us the law of composition of the points of the elliptic curve, and we write $P+Q=R'$ . Here is an image courtesy of user SuperManu of Wikipedia:

The usual case that we have described is on the left; the other three images show other different cases where the line drawn does not necessarily go through three points. This happens, for example, when the line is tangent to the curve at some point $Q$ , as in the second picture; in this case, we think of the line as passing through $Q$ twice. Therefore, when we compute $P+Q$ , the third point is $Q$ itself, and it is through $Q$ that we draw our vertical line to locate $Q'$ , which is equal to $P+Q$ .

The second picture also shows another computation, that of $Q+Q$ , or $2Q$ . Again, since this necessitates taking a line that passes through the point $Q$ twice, this means that the line must be tangent to the elliptic curve at $Q$ . The third point that it passes through is the point $P$ , and we draw the vertical line through $P$ to find the point $P'$ , which is equal to $2Q$ .

Now we discuss the case described by the third picture, where the line going through the two points $P$ and $Q$ which we want to “add” is a vertical line. To explain what happens, we need the notion of a “point at infinity” (see Projective Geometry). We write the point at infinity as $0$ , expressing the idea that it is the identity element of our group. We cannot find this point at infinity in the $x$ – $y$ plane, but we can think of it as the third point that the vertical line passes through aside from $P$ and $Q$ . In this case, of course, there is no need to draw another vertical line – we simply write $P+Q=0$ .

Finally we come to the case described by the fourth picture; this is simply a combination of the earlier cases we have described above. The vertical line is tangent to the curve at the point $P$ , so we can think of it as passing through $P$ twice, and the third point is passes through is the point at infinity $0$ , so we can write $2P=0$ .

We will not prove explicitly that the points form a group under this law of composition, i.e. that the conditions for a set to form a group are satisfied by our procedure, but it is an interesting exercise to attempt to do so; readers may try it out for themselves or consult the references provided at the end of the post. It is worth mentioning that our group is also an abelian group, i.e. we have $P+Q=Q+P$ , and hence we have written our law of composition “additively”.

Now, to make the group law apply even when $x$ and $y$ are not real numbers, we need to write this procedure algebraically. This is a very powerful approach, since this allows us to operate with mathematical concepts even when we cannot visualize them.

Let $x_{P}$ and $y_{P}$ be the $x$ and $y$ coordinates of a point $P$ , and let $x_{Q}$ and $y_{Q}$ be the $x$ and $y$ coordinates of another point $Q$ . Let

$\displaystyle m=\frac{y_{Q}-y_{P}}{x_{Q}-x_{P}}$

be the slope of the line that connects the points $P$ and $Q$ . Then the point $P+Q$ has $x$ and $y$ coordinates given by the following formulas:

$\displaystyle x_{P+Q}=m^{2}-x_{P}-x_{Q}$

$\displaystyle y_{P+Q}=-y_{P}-m(x_{P+Q}-x_{P})$

In the case that $Q$ is the same point as $P$ , then we define the slope of the tangent line to the elliptic curve at the point $P$ using the formula

$\displaystyle m=\frac{3x_{P}^{2}+a}{2y_{P}}$

where $a$ is the coefficient of $x$ in the formula, of the elliptic curve, i.e.

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

Then the $x$ and $y$ coordinates of the point $2P$ are given by the same formulas as above, appropriately modified to reflect the fact that now the points $P$ and $Q$ are the same:

$\displaystyle x_{2P}=m^{2}-2x_{P}$

$\displaystyle y_{2P}=-y_{P}-m(x_{2P}-x_{P})$

This covers the first two cases in the image above; for the third case, when $P$ and $Q$ are distinct points and $y_{P}=-y_{Q}$ , we simply set $P+Q=0$ . For the fourth case, when $P$ and $Q$ refer to the same point, and $y_{P}=0$ , we set $2P=0$ . The point at infinity itself can be treated as a mere point and play into our computations, by setting $P+0=P$ , reflecting its role as the identity element of the group.

The group structure on the points of elliptic curves have practical applications in cryptography, which is the study of “encrypting” information so that it cannot be deciphered by parties other than the intended recipients, for example in military applications, or when performing financial transactions over the internet.

On the purely mathematical side, the study of the group structure is currently a very active field of research. An important theorem called the Mordell-Weil theorem states that even though there may be an infinite number of points whose coordinates are given by rational numbers (called rational points), these points may all be obtained by performing the “tangent and chord” or “tangent and secant” construction on a finite number of points. In more technical terms, the group of rational points on an elliptic curve is finitely generated.

There is a theorem concerning finitely generated abelian groups stating that any finitely generated abelian group $G$ is isomorphic to the direct sum of $r$ copies of the integers and a finite abelian group called the torsion subgroup of $G$ . The number $r$ is called the rank of $G$ . The famous Birch and Swinnerton-Dyer conjecture, which currently carries a million dollar prize for its proof (or disproof), concerns the rank of the finitely generated abelian group of rational points on an elliptic curve.

Another thing that we can do with elliptic curves is use them to obtain representations of Galois groups (see Galois Groups). A representation of a group $G$ on a vector space $V$ over a field $K$ is a homomorphism from $G$ to $GL(V)$ , the group of bijective linear transformations of the vector space $V$ to itself. We know of course from Matrices that linear transformations of vector spaces can always be written as matrices (in our case the matrices must have nonzero determinant to ensure that the linear transformations are bijective). Representation theory allows us to study the objects of abstract algebra using the methods of linear algebra.

To any elliptic curve we can associate a certain algebraic number field (see Algebraic Numbers). The elements of these algebraic number fields are “generated” by the algebraic numbers that provide the coordinates of “ $p$ -torsion” points of the elliptic curve, i.e. those points $P$ for which $pP=0$ for some prime number $p$ .

The set of $p$ -torsion points of the elliptic curve is a $2$ -dimensional vector space over the finite field $\mathbb{Z}/p\mathbb{Z}$ (see Modular Arithmetic and Quotient Sets), also written as $\mathbb{F}_{p}$ . Among other things this means that we can choose two $p$ -torsion points $P$ and $Q$ of the elliptic curve such that any other $p$ -torsion point can be written as $aP+bQ$ for integers $a$ and $b$ between $0$ and $p-1$ . When an element of the Galois group of the algebraic number field generated by the coordinates of the $p$ -torsion points of the elliptic curve permutes the elements of the algebraic number field, it also permutes the $p$ -torsion points of the elliptic curve. This permutation can then be represented by a $2\times 2$ matrix with coefficients in $\mathbb{F}_{p}$ .

The connection between Galois groups and elliptic curves is a concept that is central to many developments and open problems in mathematics. It plays a part, for example in the proof of the famous problem called Fermat’s Last Theorem. It is also related to the open problem called the Kronecker Jugendtraum (which is German for Kronecker’s Childhood Dream, and named after the mathematician Leopold Kronecker), also known as Hilbert’s Twelfth Problem, which seeks a procedure for obtaining all field extensions of algebraic number fields whose Galois group is an abelian group. This problem has been solved only in the special case of imaginary quadratic fields, and the solution involves special kinds of “symmetries” of elliptic curves called complex multiplication (not to be confused with the multiplication of complex numbers). David Hilbert, who is one of the most revered mathematicians in history, is said to have referred to the theory of complex multiplication as “…not only the most beautiful part of mathematics but of all science.”

References:

Elliptic Curve on Wikipedia

Mordell-Weil Theorem on Wikipedia

Birch and Swinnerton-Dyer Conjecture on Wikipedia

Wiles’ Proof of Fermat’s Last Theorem on Wikipedia

Hilbert’s Twelfth Problem on Wikipedia

Complex Multiplication on Wikipedia

Image by User YassineMrabet of Wikipedia

Image by User SuperManu of Wikipedia

Fearless Symmetry: Exposing the Hidden Patterns of Numbers by Avner Ash and Robert Gross

Elliptic Tales: Curves, Counting, and Number Theory by Avner Ash and Robert Gross

Rational Points on Elliptic Curves by Joseph H. Silverman