Weil pairing and Galois descent

There is an interesting way in which Weil pairing on Abelian Varieties is nothing more than Galois descent for a particular Galois extension. I have not seen this connection used before in the literature but I have also not seen a lot of literature…

Let $A/k$ be an abelian variety over a field $k$ of characteristic $p$. Suppose $m$ is an integer coprime to $p$ and let $[m]: A \to A$ denote the multiplication by $m$ map with kernel $A[m]$.

Recall that there is a dual abelian variety $A^\vee$ representing the Picard functor for $A$. In particular, $A^\vee[m]$ is the group of line bundles $\mathcal L$ on $A$ such that $[m]^*\mathcal L$ is trivial. $A$ and $A^\vee$ are finite abelian groups and the Weil pairing is a perfect pairing of the form:

$\langle-,-\rangle: A[m] \times A^\vee[m] \to \mu_m.$

It’s definition goes as follows: Let $D$ be a divisor corresponding to $\mathcal L$ and let $g(x)$ be a rational function on $A$ such that $\mathrm{div}\ g(x) = [m]^*D$. Then, for $a \in A[m]$, we define $\langle a,\mathcal L\rangle = g(x+a)/g(x)$.

If $t_a$ denotes translation by $a$, then $g(x+a)$ is the pull back of $g(x)$ along $t_a$. Since $[m]\circ t_a = [m]$, $g(x+a)$ and $g(x)$ have the same divisor and so $g(x+a)/g(x)$ is regular everywhere on $A$ and hence constant by properness of $A$. A little more work shows that this rational functions is in $\mu_m$. We will see below that this is an immediate conclusion of our alternate viewpoint.

This is the standard picture. However, there is also this alternate way of looking at things:

Consider the etale Galois extension $[m]: A \to A$. It’s galois group is canonically identified with $A[m]$. Moreover, $A^\vee[m]$ is precisely the set of line bundles on $A$ trivialized by this etale cover. In other words, it is the set of Galois twists of $\mathcal O_A$ (or even for any line bundle). Since descent is effective for this extension, this will immediately imply that:

$A^\vee[m] \cong H^1(\mathrm{Gal}([m]), \mathcal O_A^\times) = H^1(A([m]), k^\times) = \mathrm{Hom}\ (A[m],k^\times).$

The final equality is because $A[m]$ acts trivially on $\mathcal O_A^\times = k^\times$. We also see immediately that $\mathrm{Hom}\ (A[m],k^\times) = \mathrm{Hom}\ (A[m],\mu_m)$ since $A[m]$ is m-torison as an Abelian group. Therefore, this gives us a pairing:

$A[m]\times A^\vee[m] \to \mu_m.$

Explicitly, the map goes as follows: Given a line bundle $\mathcal L$, we pick an isomorphism $g: O_A \to [m]^*\mathcal L$. Then, the corresponding 1-cocycle for $a \in A[m]$ is defined by $a \to (t_a^*g)^{-1}g \in \mathrm{Aut}(\mathcal O_A)$. It is easily seen that this is the same explicit construction as in the standard viewpoint.

Conclusion:

I find the Galois descent viewpoint conceptually satisfying. The standard treatments of the Weil pairing can seem arbitrary and it not clear why such a pairing should exist or be useful. On the other hand, $[m]:A \to A$ is a perfectly natural Galois extension connected to $A$ and $\mathcal O_A$ torsors are clearly an interesting thing to consider.

Some notes of mine

I wrote these notes a while back for my own reference. I don’t know how useful they will be to others but they are my attempt to understand the subjects they talk about. They have no references to where I got the material since they are only intended for private use (despite my publishing them here…).

1. Modular forms over finite characteristic:

These notes contain material from Serre’s article on modular forms mod-p and at the very end, a little bit about modular forms over the p-adics as in Serre’s paper culminating in a definition of the Kubota-Leopoldt p-adic zeta function using this theory.

After a brief summary of classical modular forms, the results should be more or less self contained. The most interesting section (in my opinion…) is Section 4 where I briefly discuss Katz’ perspective on modular forms and use it to show that the Hasse invariant is a modular form and compute it’s q-expansion using the Tate curve. This section is also scarce on details and I understood this stuff by reading Prof. Emerton’s wonderful expository article here.

The article is here: Modular_Forms_mod_P.

2. Complex Multiplication

This is my attempt to streamline and summarize the main results of complex multiplication as I see them. This is short (about 5-6 pages) and the final section is my answer to the question here.

The article is here: Complex_Multiplication_notes.

3. Growth of Class number in $Z_p$ extensions

A summary of the relevant chapter in Washington’s books. Nothing new here.

The article is: Growth_of_class_groups_in_Z_p_extensions.

Formal Summation and Dirichlet L-functions

Recall the classical Riemann zeta function:

$\zeta(s) = \sum_{n\geq 1}\frac{1}{n^s}$

and the Dirichlet L-functions for a character $\chi: \mathbb Z \to \mathbb C$:

$S(s,\chi) = \sum_{n\geq 1}\frac{\chi(n)}{n^s}.$

defined for $\Re(s) > 1$.  These functions can be analytically continued to the entire complex plane (except a pole at $s=1$ in the case of $\zeta(s)$). In particular, the values at non positive integers carry great arithmetic significance and enjoy many properties.

For instance, $L(s,\chi)$ is an algebraic integer and in fact equal to $B_{k+1,\chi}/(k+1)$ where $B_{k,\chi}$ are the generalized Bernoulli numbers. Moreover, these values satisfy p-adic congruences and integrality properties such as the Kummer congruence (and generalizations to the L-functions).

The standard proof of these proceeds by showing that $L(s,\chi)$ satisfies a functional equation that relates $L(1-s,\chi)$ to $L(s,\overline{\chi})$ and then computing the values $L(n,\chi)$ for $n \geq 1$ integral using analytic techniques (such as Fourier analysis).

Proving the p-adic properties is then by working directly with the definition of the generalized Bernoulli numbers instead of the L-functions. However, the L-functions are clearly the fundamental object here and it would be nice to have a way to directly work with the values at negative integers (without using the functional equation).

One might be tempted to extend the series definition to the negative integers and say:

$\zeta(-k) "=" 1^k + 2^k + 3^k + \dots$

For instance, consider the following (bogus) computation:

$\zeta(0)\frac{t^0}{0!} = 1^0\frac{t^0}{0!} + 2^0\frac{t^0}{0!} + \dots$

$\zeta(-1)\frac{t^1}{1!} = 1^1\frac{t^1}{1!} + 2^0\frac{t^1}{1!} + \dots$

$\zeta(-2)\frac{t^2}{2!} = 1^2\frac{t^2}{2!} + 2^2\frac{t^2}{2!} + \dots$

. . .

and let us “sum” the columns first:

$\sum_{k\geq 0}\zeta(-k)\frac{t^{k}}{k!} = e^{t} + e^{2t} + \dots = \frac{e^t}{1 + e^t}$

which, remarkably enough, is the right generating function for $\zeta(-k)$! We have exchanged the summation over two divergent summations and ended up with the right answer.

In fact, it is possible to rigorously justify this procedure of divergent summation and moreover, one can use it to prove a lot of arithmetic properties of these values rather easily (like the Kummer congruence). I learnt the basic method from some lecture notes of Prof. Akshay Venkatesh here: Section 3, Analytic Class Number formula and L-functions.

I then discovered that one could use these techniques to compute the explicit values (along the outline above) and prove some more stuff. I wrote this up in an article (that also explains the basic technique and should be (almost) self contained) here: Divergent_series_summation.

The Groupoid Cardinality of Finite Semi-Simple Algebras

A groupoid is category where all the morphisms are isomorphisms and groupoid cardinality is a way to assign a notion of size to groupoids. Roughly, the idea is that one should weigh an object inversely by the number of automorphisms it has (and we only count each isomorphic object as one object).

It is important to count only one object from each isomorphism class since we want the notion of groupoid cardinality to be invariant under equivalences of groupoids (in the sense of category theory) and every category is equivalent to it’s skeleton. For further motivation for the idea of a groupoid cardinality, see Qiaochu Yuan’s post on them.

This seems like quite a strange thing to do but it turns out to be quite a useful notion. One of my favorite facts about Elliptic curves is that the groupoid cardinality of the supersingular elliptic curves in characteristic p is $p-1/24$! See the Eichler-Deuring mass formula.

Another interesting computation along these lines is that the number of finite sets is $e$. One can ask this question of various groupoids and the answer is often interesting. I will ask it today of semi-simple finite algebras of order $n$. By an algebra, I will always implicitly mean commutative in this post.

A nice proof that the irreducible characters form a basis for class functions

Warning: This is a very shoddily organized post and can be vastly improved. The method of proof is still nice however.

Let $G$ be a finite group of size $g$ and $k$ an algebraically closed field such that $g \neq 0$ in it. Let $V_1,\dots, V_n$ be the irreducible representations of $G$ over $k$ and $\chi_1,\dots,\chi_n$ the corresponding characters.

A class function $f: G \to k$ is a set function such that $f$ is constant on conjugacy classes. That is, for all $g,h \in G, f(h^{-1}gh) = f(g)$. It is easy to see that any character is a class function by cyclicity of the trace function.

Let us denote by $H$ the vector space of class functions and $W$ the vector space of functions spanned by the $\chi_i$ within $H$. It is natural to wonder about the size of $W$ relative to $H$. In fact, the two vector spaces are equal!

To prove this, let us consider the algebra $k[G]$. By our hypothesis and the usual argument of Maschke’s theorem, this is a semisimple ring. That is, $k[G]$ is semisimple as a $G$ representation. Let $r_G$ be the character attached to this so called regular representation. A simple computation shows us that $r_1 = g$ and $r_G = 0$ otherwise.

Moreover, for any irreducible representation $V_k$, we see from last time that:

$Hom_G(V_k,k[G]) = \frac{1}{g}\sum_g r_G(g^{-1})\chi_k(g) = \chi_k(1) = \dim V_k.$

Therefore, by semisimplicity

$k[G] \cong \oplus_k V_k^{\dim V_k}$

as representations. Moreover, the orthogonality relations from last time also show that the $\chi_k$ are linearly independent. That is, $\dim W = \#$ irreducible characters.

Taking the endomorphism ring on both sides (as modules over $k[G]$), we obtain:

$k[G] \cong M_{\dim V_k}(k)$

as algebras. Considering dimensions, we have incidentally shown that $g = \sum_k \dim V_k^2$. Moreover, let us consider the central elements in $k[G]$. They are easily seen to exactly be the elements of the form $\sum_g f(g)g$ for $f:G \to k$ a class function.

On the other hand, the central elements in $M_n(k)$ are always just $k$. Therefore, the dimension of the center of $k[G]$ (= vector space of class functions) is equal to the number of irreducible representations. Therefore, $\dim W = \dim H$ and we are done.

The above map can be thought of as the representation of a class function in the basis defined by the functions $c_k\overline{\chi_k}$ for some appropriate constants $c_k$. This is because the element:

$\lambda_i = \sum_{g}\overline{\chi_i}g \in k[G]$

maps to a a diagonal element in $M_{\dim V_j}(k)$ that can be calculated by taking the trace. That is to say, the image of $\lambda_i$ in \$latex $M_{\dim V_j}(k)$ is given by:

$\frac{1}{\dim V_k}\sum_{g}\overline{\chi_i(g)}\chi_j(g) = \frac{g}{\dim V_k}\delta_{ij}$

by the orthogonality relations. We see that $c_k = \frac{\dim V_k}{g}$.

Schur’s Lemma and the Schur Orthogonality Relations.

Both Schur’s Lemma and the Schur Orthogonality relations are part of the basic foundation of representation theory. However, the connection between them is not always emphasized and the Orthogonality relations are proven more computationally.

The standard proofs of the relations never made sense to me, however there is very direct way to derive them from Schur’s Lemma (which makes perfect sense to me!) and simple facts about projections on vector spaces. More importantly, it gives a categorical interpretation of the inner product. I think this approach should be emphasized way more than it currently is and I hope this post will go a tiny way towards fixing that.

Congruent Numbers and Elliptic Curves

A congruent number $n$ is a positive integer that is the area of a right triangle with three rational number sides. In equations, we are required to find rational positive numbers $a,b,c$ such that:

$\displaystyle a^2+b^2 = c^2$    and    $\displaystyle n = \frac12 ab.$                       (1)

The story of congruent numbers is a very old one, beginning with Diophantus. The Arabs and Fibonacci knew of the problem in the following form:

Find three rational numbers whose squares form an arithmetic progression with common difference $k$.

This is equivalent to finding integers $X,Y,Z,T$ with $T\neq 0$ such that $Y^2 - X^2 = Z^2 - Y^2 = k$ which reduces to finding  a right triangle with rational sides

$\displaystyle \frac{Z+X}{T}, \frac{Z-X}{T}, \frac{2Y}{T}$

with area $k$. This is the congruent number problem for $k$. The Arabs knew several examples of congruent numbers and Fermat stated that no square is a congruent numbers. Since we can scale triangles to assume that $n$ is square free, this is equivalent to saying that $1$ is not a congruent number.

As with many other problems in number theory, the proof of this statement had to wait four centuries for Fermat. The problem led Fermat to discover his method of infinite descent.

In more recent times, the problem has been fruitfully translated into one about Elliptic Curves. We perform a rational transformation of the defining equations (1) for a congruent number in the following way. Set $x = n(a+c)/b$ and $y = 2n^2(a+c)/b$. A calculation shows that:

$\displaystyle y^2 = x^3 - n^2x.$                                          (2)

and $y \neq 0$. If $y = 0$, then $a=-c$ and $b = 0$ but then $n = \frac12 ab = 0$. Conversely, given $x,y$ satisfying (2), we find $a = (x^2-y^2)/y, b = 2nx/y$ and $c = x^2+y^2/n$ and one can check that these numbers satisfy (1).

The projective closure of (2) defines an elliptic curve that we will call $E_n$. We are interested in finding rational points on it that do not satisfy $y=0$. I will prove that $n$ is a congruent number precisely when $E_n$ has positive rank.

The proof is an interesting use of Dirichlet’s Theorem on Arithmetic Progressions and some neat ideas about Elliptic Curves and their reductions modulo primes. I will essentially assume the material in Silverman’s first book and the aforementioned Dirichlet’s Theorem.

Descent on Vector Spaces and Cohomology

It is quite often of interest to study the properties of some variety ${X/\mathbb Q}$. However, it is generally much easier to study varieties over algebraically closed fields and so we need some way of translating a property of ${X_{\overline{\mathbb Q}}/\overline{\mathbb Q}}$ to ${X/\mathbb Q}$. This idea is known as descent and in this post, I would like to say a little bit about the simplest example of descent – over vector spaces.

Let ${L/K}$ be an extension of fields and ${V}$ a vector space over ${K}$. Consider ${W = V\otimes_K L }$. The Galois Group ${G = \mathop{Gal}(L/K)}$ acts on ${W}$ through the second factor (not by linear actions but by semi-linear ones – see below). One can consider the ${K}$-vector space ${W^G}$. This is the vector space fixed by ${G}$.

Theorem 1 (Descent of Vector Spaces) The natural map ${W^G\otimes_K L \rightarrow W}$ is an isomorphism. In particular, if ${W}$ is finite dimensional, then ${\dim_K W^G = \dim_L W}$.

It is not hard to prove this theorem directly (Theorem 2.14 in these notes of K. Conrad) but I would like to relate it to another theorem. This is also well known and is a generalization of the famous Hilbert’s Theorem 90. Let ${{GL}_n(L)}$ be the group of invertible ${n}$-dimensional matrices over ${L}$ and consider the cohomology ${H^1(G,{GL}_n(L))}$. This is not a group unless ${n=1}$ since ${{GL}_n(L)}$ is non-commutative in general. However, it is a pointed set and we have the following theorem:

Theorem 2 (Hilbert’s 90) $\displaystyle H^1(G,{GL}_n(L)) = \{0\}.$

Hilbert stated the above theorem (in a disguised form) for ${n=1}$ and ${L/K}$ a finite cyclic extension. Noether generalized the theorem to arbitrary extensions. I do not know who is responsible for the generalization to general linear groups but I saw this theorem first in Serre’s “Galois Cohomology”.

In this post, I will show that the above theorems are equivalent in the following sense: