Christian Wuthrich (Nottingham): p-adic L-series and Iwasawa theory


Artin and Tate proved a large part of the conjecture of Birch and Swinnerton-Dyer in the function field case in the 60s. Iwasawa theory for elliptic curves as initiated by Mazur tries to use similar tools to approach the p-adic version of the Birch and Swinnerton-Dyer conjecture.

Let E/\mathbb{Q} be an elliptic curve. The traditional conjecture by Birch and Swinnerton-Dyer states that there is a link between the arithmetic invariants of E, like the Mordell-Weil group E(\mathbb{Q}), and the analytically defined complex L-function. In the p-adic BSD, we work with an analytic function L_p(E,s) taking values in the p-adic numbers. It is built on the values of the complex L-function and can be described explicitly using modular symbols. The p-adic conjecture says again that the order of vanishing of L_p(E,s) at s=1 is equal to the rank of the Mordell-Weil group E(\mathbb{Q}). (Except in one special case, namely when the curve has split multiplicative reduction at p.)

The big advantage of the p-adic setting is that we actually know something about it. The p-adic L-function has a natural link to the arithmetic side via the so called "main conjecture" of Iwasawa theory about which we know quite a lot. Iwasawa theory deals with the question of how the arithmetic objects vary as one climbs up the tower of fields K_{\infty}/\mathbb{Q} obtained by adjoining the p-power roots of unity. Similarily one can ask how does the mysterious Tate-Shafarevich group grow (or shrink). Much like the zeta-function for varieties over finite fields, there is a generating function that incodes this information. The main conjecture states that this generating function is equal to the p-adic L-function.

A big and difficult theorem by Kato shows half of this conjecture and Skinner and Urban claim they have shown the other half of it. As a consequence one gets that the order of vanishing of L_p(E,s) is at most the rank of E(\mathbb{Q}). It even says something about the size of the mysterious Tate-Shafarevich group. It also implies that the group E(K_{\infty}) is finitely generated.


Project 1

Use twists by Dirichlet characters on modular symbols and p-adic L-function.

The p-adic L-function of E can be computed using modular symbols. And sage contains already code to do so. But this code could be improved in several direction. There are several subprojects

Project 2

Compute the modular symbols using complex integration

The original definition of the modular symbols [r]^{+} and [r]^{-} is given as an integral in the upper half plane. Sage currently computes the modular symbols attached to an elliptic curve (natively or in eclib) by finding the correct eigenspace in the space of all modular symbols of level N. For large N this is very time consuming or even impossible. When we wish to compute only a few modular symbols, it could be much faster to compute the values of [r]^{+} by the numerical approximation of the complex integrals.

The project proceeds in several steps

This script computes the \gamma transforming a cusp, if possible, to one where the denominator is a divisor of N: find_gamma.sage

Other projects


Lecture Notes

Modular symbols and $p$-adic L-functions

Background reading

days22/wuthrich (last edited 2010-07-02 04:09:03 by Chan-Ho Kim)