In Bernoulli Numbers, Fermat’s Last Theorem, and the Riemann Zeta Function, we introduced the Kubota-Leopold $p$-adic L-function, which encodes the congruences discovered by Kummer between special values of the Riemann zeta function. In this post, we will connect them to Iwasawa theory and $p$-adic modular forms.

Let us start with a little introduction to Iwasawa theory. Consider the Galois group $\text{Gal}(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})$, where $\mathbb{Q}(\mu_{p^{\infty}})$ is the extension of the rational numbers $\mathbb{Q}$ obtained by adjoining all the $p$-th-power roots of unity to $\mathbb{Q}$. This Galois group is isomorphic to $\mathbb{Z}_{p}^{\times}$, the group of units of the $p$-adic integers $\mathbb{Z}_p$.

The group $\mathbb{Z}_{p}^{\times}$ decomposes into the product of a group isomorphic to $1+p\mathbb{Z}_{p}$ and a group isomorphic to $(p-1)$-th roots of unity. Let $\Gamma$ be the subgroup of this Galois group isomorphic to $1+p\mathbb{Z}_{p}$. The Iwasawa algebra is defined to be the group ring $\mathbb{Z}_{p}[[\Gamma]]$, which also happens to be isomorphic to the power series ring $\mathbb{Z}_{p}[[T]]$.

The interest in the Iwasawa algebra comes from the fact that many important objects of interest in number theory are modules over the Iwasawa algebra, and such modules have a structure that makes them easy to study. For instance, the inverse limit of the p-part of the ideal class groups of cyclotomic fields is such a module. The “main conjecture of Iwasawa theory“, a high-powered version of Kummer’s theorem that relates ideal class groups and Bernoulli numbers, describes this module. Namely, the main conjecture of Iwasawa theory states that as a module over the Iwasawa algebra, the inverse limit of the p-part of the ideal class groups of cyclotomic fields has a characteristic ideal generated by none other than the Kubota-Leopoldt $p$-adic L-function!

Let us describe more the relation between the Iwasawa algebra and the Kubota-Leopoldt zeta function by relating them to measures. Our measure here takes functions on the group $\mathbb{Z}_p^{\times}$ and gives an element of $\mathbb{Z}_{p}$. This should remind us of measures and integrals in real analysis, except instead of our functions being on $\mathbb{R}$, they are on the group $\mathbb{Z}_{p}^{\times}$, and instead of taking values in $\mathbb{R}$, they take values in $\mathbb{Z}_{p}$. This is just an example of a more general kind of measure.

Now these measures are actually in one-to-one correspondence with the elements of the Iwasawa algebra!

The Iwasawa algebra is $\mathbb{Z}_{p}[[\Gamma]]$, and note that $\Gamma$ is a subset of $\mathbb{Z}_{p}^{\times}$. Suppose we have an element of the Iwasawa algebra. We define the corresponding measure by saying what it does to a function $f$ on $\mathbb{Z}_{p}^{\times}$. Note that if we extend this function linearly, we can evaluate it on the element of the Iwasawa algebra and get an element of $\mathbb{Z}_{p}^{\times}$. Thus we define our measure by evaluation. The other direction is a bit more involved, but given the measure, we build an element of the Iwasawa algebra by exploiting the profinite nature of $\mathbb{Z}_{p}^{\times}$, which means the measure was built from functions on the finite pieces of it.

Now we know how the Iwasawa algebra and measures are related, what about the Kubota-Leopoldt zeta function? For those we must now take a detour through $p$-adic modular forms, in particular $p$-adic Eisenstein series.

The reason modular forms are brought into this is that the value of the zeta function at $1-k$ shows up in the constant term in the Fourier expansion of the Eisenstein series $G_{k}$:

$\displaystyle G_{k}(\tau):=\frac{\zeta(1-k)}{2}+\sum_{n=1}^{\infty}\left(\sum_{d\vert n}d^{k-1}\right)q^{n}$

where $q=e^{2\pi i \tau}$, as is common convention in the theory (hence the Fourier expansion is also known as the $q$-expansion). This Eisenstein series $G_{k}$ is a modular form of weight $k$. A similar relationship holds between the Kubota-Leopoldt $p$-adic L-function and $p$-adic Eisenstein series, the latter of which is an example of a $p$-adic modular form. We will define this now. Let $f$ be a modular form defined over $\mathbb{Q}$. This means that, when we consider its Fourier expansion

$\displaystyle f(\tau)=\sum_{n=0}^{\infty}a_{n}q^{n}$,

the coefficients $a_{n}$ are rational numbers. We define a $p$-adic valuation on the space of modular form by taking the biggest power of $p$ among the coefficients $a_{n}$, i.e.

$\displaystyle v_{p}(f)=\inf_{n} v_{p}(a_{n})$

We recall that the bigger the power of $p$ dividing a rational number, the smaller its $p$-adic valuation. This lets us consider the limit of a sequence. A $p$-adic modular form is the limit of a sequence of classical modular forms.

The weight of a $p$-adic modular form is the limit of the weights of the classical ones of which it is the limit. Serre showed that for classical modular forms $f$ and $g$, if the $p$-adic valuation

$\displaystyle v(f-g)>=v(f)+m$

for some $m$, then the weights of $f$ and $g$ will be congruent mod $(p-1)p^m$.

This implies that the weight of a $p$-adic modular form takes values in the inverse limit of $\mathbb{Z}/(p-1)p^{m}\mathbb{Z}$, which is isomorphic to the product of $\mathbb{Z}_{p}$ and $(p-1)\mathbb{Z}$. Here is where measures come in – this space of weights can be identified with characters of $\mathbb{Z}_{p}^{\times}$, i.e. a weight $k$ is a function on $\mathbb{Z}_{p}^{\times}$and being such a function, it is an input for a measure!

Now, we will create a measure, with a bit of a twist. Given a weight $k$, we can build a $p$-adic Eisenstein series of weight $k$ (recall that this is a limit of classical Eisenstein series):

$\displaystyle G_{k}^{*}:=\varinjlim_{i}G_{k_{i}}$

We think of this as a “measure” that takes a weight $k$ (again recall that the weight $k$ is a character, i.e. a function on $\mathbb{Z}_{p}$) and gives a weight $k$ Eisenstein series, i.e an “Eisenstein measure“. But the value of the Kubota-Leopoldt zeta function at $1-k$ is the constant in the Fourier expansion! Therefore, if we take the constant term of this p-adic Eisenstein series, we have a good old measure, a recipe for taking a function on $\mathbb{Z}_{p}$ (the weight $k$) and giving us an element of $\mathbb{Z}_{p}$. But by our earlier discussion, this is an element of the Iwasawa algebra!

There are some subtleties I swept under the rug, but to summarize – important objects in number theory are modules over the Iwasawa algebra. $p$-adic L-functions which interpolate L-functions at special values are elements of the Iwasawa algebra.

This is a modern, high-powered version of Kummer’s discovery that relates certain ideal class groups and Bernoulli numbers (which are special values of the Riemann zeta function). The Eisenstein measure, which gives a p-adic modular form when evaluated at a certain weight, leads to the notion of a “Hida family“, a “p-adic family” of p-adic modular forms. But that discussion is for another time!

References:

Iwasawa theory on Wikipedia

Iwasawa algebra on Wikipedia

Main conjecture of Iwasawa theory on Wikipedia

An introduction to Eisenstein measures by E. E. Eischen

Modular curves and cyclotomic fields by Romyar Sharifi

Desde Fermat, Lamé y Kummer hasta Iwasawa: Una introducción a la teoría de Iwasawa (in Spanish) by Álvaro Lozano-Robledo

# Bernoulli Numbers, Fermat’s Last Theorem, and the Riemann Zeta Function

The Bernoulli numbers are the Taylor series coefficients of the function

$\displaystyle \frac{x}{e^{x}-1}$.

The $n$-th Bernoulli number $B_{n}$ is zero for odd $n$, except for $n=1$, where it is equal to $-1/2$. For the first few even numbers, we have

$\displaystyle B_0=1,\; B_{2}=\frac{1}{6}, \; B_{4}=-\frac{1}{30}, \; B_6=\frac{1}{42}, \; B_{8}=-\frac{1}{30}, \; B_{10}=\frac{5}{66}$.

Bernoulli numbers have many interesting properties, and many mathematicians have studied them for a very long time. They are named after Jacob Bernoulli, but were also studied by Seki Takakazu in Japan at around the same time (end of 17th/beginning of 18th century). In this post I want to focus more on the work of Ernst Edouard Kummer, more than a century after Bernoulli and Takakazu.

We’re going to come back to Bernoulli numbers later, but for now let’s talk about something completely different – Fermat’s Last Theorem, which Kummer was working on. In the time of Kummer, a proposal to study Fermat’s Last Theorem by factoring both sides of the famous equation into linear terms. Just as $x^2+y^2$ factors into

$\displaystyle x^2+y^2=(x+iy)(x-iy)$,

we would have that $x^{p}+y^{p}$ also factors into

$\displaystyle x^{p}+y^{p}=(x+\zeta_{p}y)(x+\zeta_{p}^{2} y)...(x+\zeta_{p}^{p-1} y)$

where $\zeta_{p}$ is a $p$-th root of unity.

However, there is a problem. In these kinds of numbers where $p$-th roots of unity are adjoined, factorization may not be unique! Hence Kummer developed the theory of “ideals” to study this (see also The Fundamental Theorem of Arithmetic and Unique Factorization).

Unique factorization does not work with the numbers themselves, but it works with ideals (this is true for number fields, since they form what is called a “Dedekind domain”). Hence the original name of ideals was “ideal numbers”. To number fields we associate an “ideal class group“. If this group has only one element, unique factorization holds. If not, then things can get complicated. The ideal class group (together with the Galois group) is probably the most important group in number theory.

Kummer found that if $p$ is a “regular prime“, i.e. if p does not divide the number of elements of the ideal class group (also known as the class number) of the “$p$-th cyclotomic field” (the rational numbers with $p$-th roots of unity adjoined), then Fermat’s Last Theorem is true for $p$.

Let’s go back to Bernoulli numbers now – Kummer also found that a prime $p$ is regular if and only if it does not divide the numerator for the nth Bernoulli number, for all $n$ less than $p-1$. In other words, Kummer proved Fermat’s Last Theorem for prime exponents not dividing the numerators of Bernoulli numbers! Fermat’s Last Theorem has now been proved in all cases, but the work of Kummer remains influential.

So we’ve related Bernoulli numbers to ideal class groups and the very famous Fermat’s Last Theorem. Now let us relate Bernoulli numbers to another very famous thing in math – the Riemann zeta function (see also Zeta Functions and L-Functions).

It is known that the Bernoulli numbers are related to values of the Riemann zeta function at the negative integers (so we need the analytic continuation to do this) by the following equation: $B_n=n \zeta(1-n)$ for $n$ greater than or equal to $1$.

Now, Kummer also discovered that Bernoulli numbers satisfy certain congruences modulo powers of a prime $p$, in particular

$\displaystyle \frac{B_{m}}{m}\equiv \frac{B_{n}}{n} \mod p$

where $m$ and $n$ are positive even integers neither of which are divisible by $(p-1)$, and $m\equiv n \mod (p-1)$. Here congruence for two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ means that $ad$ is congruent to $cd$ mod $p$.

We also have a more general congruence for bigger powers of $p$:

$\displaystyle (1-p^{m-1})\frac{B_{m}}{m}\equiv (1-p^{n-1})\frac{B_{n}}{n} \mod p^{a+1}$

where $m$ and $n$ are positive even integers neither of which are divisible by $(p-1)$, and $m\equiv n \mod \varphi(p^{a}+1)$, $\varphi^{a}+1$ being the number of positive integers less than $p^{a+1}$ that are also mutually prime to it.

By by our earlier discussion, this means the special values of the Riemann zeta function also satisfy congruences modulo powers of $p$.

Congruences modulo powers of $p$ is encoded in modern language by the “$p$-adic numbers” (see also Valuations and Completions) introduced by Kurt Hensel near the end of the 19th century. The congruences between the special values of the Riemann zeta function is now similarly encoded in a $p$-adic analytic function known as the Kubota-Leopoldt $p$-adic L-function.

So again, to summarize the story so far – Bernoulli numbers are related to the ideal class group and also to the special values of the Riemann zeta function, and bridge the two subjects.

If this reminds you of the analytic class number formula, well in fact that is one of the ingredients in the proof of Kummer’s result relating regular primes and the Bernoulli numbers. Moreover, the information that they encode is related to divisibility or congruence modulo primes or their powers. This is where the $p$-adic L-functions come in.

The Bernoulli numbers also appear in the constant term of the Fourier expansion of Eisenstein series. The Eisenstein series is an example of a modular form (see also Modular Forms), which gives us Galois representations. The Galois group, on the other hand is related to the ideal class group by class field theory (see also Some Basics of Class Field Theory). So this is one way to create the bridge between the two concepts. In fact, this was used to prove the Herbrand-Ribet theorem, a stronger version of Kummer’s result.

So we also have modular forms in the picture. In modern research all of these are deeply intertwined – ideal class groups, zeta functions, congruences, and modular forms.

References:

Bernoulli number on Wikipedia

Riemann zeta function on Wikipedia

Kummer’s congruence on Wikipedia

Herbrand-Ribet theorem on Wikipedia

Bernoulli numbers, Hurwitz numbers, p-adic L-functions and
Kummer’s criterion
by Alvaro Lozano-Robledo

An introduction to Eisenstein measures by E. E. Eischen

# Modular Forms

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

# Shimura Varieties

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

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

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

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

$Y(N)=\Gamma(N)\backslash\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$

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

$Y_{0}(N)=\Gamma_{0}(N)\backslash\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$

and

$Y_{1}(N)=\Gamma_{1}(N)\backslash\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

References:

Shimura variety on Wikipedia

Reciprocity Laws and Galois Representations: Recent Breakthroughs by Jared Weinstein

Perfectoid Shimura Varieties by Ana Caraiani

Introduction to Shimura Varieties by J.S. Milne

Lecture Notes for Advanced Number Theory by Jared Weinstein

# The Lubin-Tate Formal Group Law

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

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

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

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

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

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

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

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

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

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

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

References:

Lubin-Tate Formal Group Law on Wikipedia

Formal Group Law on Wikipedia

The Geometry of Lubin-Tate Spaces by Jared Weinstein

A Rough Introduction to Lubin-Tate Spaces by Zhiyu Zhang

Formal Groups and Applications by Michiel Hazewinkel

# Arakelov Geometry

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

We also define

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

where

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

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

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

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

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

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

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

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

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

where

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

and

$[g_{Y}]\wedge\delta_{Z}:=q_{*}[q^{*}g_{Y}]$

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

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

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

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

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

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

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

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

$\widehat{c}_{1}(\lambda(E),h_{Q})=f_{*}(\widehat{\text{ch}}(E,h)\widehat{\text{Td}}(Tf,h_{f})-a(\text{ch}(E)_{\mathbb{C}}\text{Td}(Tf_{\mathbb{C}})R(Tf_{\mathbb{C}})))^{(1)}$

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

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

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

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

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

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

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

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

References:

Arakelov Theory on Wikipedia

Arithmetic Intersection Theory by Henri Gillet and Christophe Soule

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

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

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

Explicit Arakelov Geometry by Robin de Jong

Notes on Arakelov Theory by Alberto Camara

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

Introduction to Arakelov Theory by Serge Lang

# SEAMS School Manila 2017: Topics on Elliptic Curves

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

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

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

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

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

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

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

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

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

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

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

# Reduction of Elliptic Curves Modulo Primes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 used Alvaro Lozano Robledo of Wikipedia:

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

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 User Alvano Lozano Robledo of Wikipedia

Image by User Sam Derbyshire of 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

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 $b$ 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