Obstructions to Weil Cohomology coming from supersingular elliptic curves and an extension to ordinary elliptic curves

When Serre and Grothendieck were coming up with an extension of the usual cohomology theory to varieties in algebraic geometry, an important example was given by Serre which showed that you couldn’t have a Weil cohomology theory with coefficients in \mathbb Q_p in characteristic p. I will explain the example and extend it to show something about the ordinary case.

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Constructible sets, openness of flat maps and generic freeness.

I will cover a set of results that are connected and illustrate some interesting techniques about working with finiteness conditions (finitely generated as a ring, module etc) in algebraic geometry over a field.

Initially, I will prove Chevalley’s theorem that the image of a constructible set is constructible. This will involve proving Grothendieck’s generic freeness and Noetherian induction. Next, I will give a couple of applications of Chevalley’s theorem – I will prove that flat maps are open and provide a different proof of the Nullstellensatz.

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Irreducibility of Galois representations attached to Elliptic Curves

Epistemic status: I have not checked this carefully for errors, it is entirely possible there are mistakes in this.

 

We will be mostly be concerned with Elliptic curves over number fields E/K. However, let us start with a general lemma about Elliptic curves with endomorphism ring \mathbb Z. In this post, I collect some properties about representations coming from Elliptic curves.

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Weak Mordell-Weil and various approaches to it

I will show a few different approaches to the Weak Mordell-Weil theorem and how they are all really the same proof. The various proofs use the Hermite-Minkowski theorem, Class Field Theory and the Dirichlet’s Unit theorem/finiteness of Class group.

 

Let E/K be an elliptic curve over a number field. The Mordell-Weil theorem says that the group of rational points E(K) is finitely generated. This is usually proven in two steps:

  1. Weak Mordell-Weil Theorem: We prove for some n that E(K)/nE(K) is finite.
  2. Theory of heights: We define the notion of a height of a point on E (roughly, how many bits of information one would need to store the point). Using this and (1), the completion of the proof is quite formal.

I will focus here on the weak Mordell-Weil theorem and in particular, an approach to it using the Hermite-Minkowski theorem. This approach will apply without very little change to the case of Abelian varieties and the general technique seems to be applicable in great generality.

The idea of the proof is as follows:

  1. Reduce the case of general K to supposing that K contains the n-torsion using the Kummer sequence.
  2. Use the Kummer Pairing to reduce to showing that the inverse image of E(K) under the multiplication by n [n] map generates a finite extension of K.
  3. Show that there is a smooth, proper model of E/K over an open subset of \mathcal O_K and hence the inverse image of [n] generates an extension etale over R, not just K.
  4. Apply Hermite-Minkowski.

 

 

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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.