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

Description

Artin and Tate have shown 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/Q be an elliptic curve. The traditional conjecture by Birch and Swinnerton-Dyer states that there is a link between the arithmetic invariance, like the Mordell-Weil group E(Q), and the analytically defined complex L-function. In the p-adic BSD, we work with an analytic function Lp(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 Lp(E,s) at s=1 is equal to the rank of the Mordell-Weil group E(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 how so the arithmetic objects vary as one climbs up the tower of fields K∞/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 Lp(E,s) is at most the rank of E(Q). It even says something about the size of the mysterious Tate-Shafarevich group. It also implies that the rank of E(K∞) is finitely generated over the field K∞.

Projects

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.

• Allow to twist the function by Dirichlet characters. In particular with the Teichmüllers.

• Implement a function that extracts the λ and μ invariant and which decides it the growth of the Selmer group is due to the growth of the Tat-Shafarevich group or due to the increase of the rank. Statistics on the values of these fundamental Iwasawa theoretic invariants. A question I was often asked by Iwasawa theorists is: Are the μ-invariants over Q(ζp) zero, too.

• Can we compute the modular symbols using complex integration ?
• Look at overconvergent modular symbols
• What happens for primes of additive reduction ?

References

• Mazur, Tate, Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84 (1986), no. 1, 1--48. At mathscinet or gdz.

• Greenberg Ralph, Introduction to Iwasawa Theory for Elliptic Curves, (paper) on his web page full of Iwasawa theory. Also there is the more advanced Iwasawa Theory for Elliptic Curves (paper).

• Stein and Wuthrich, Computations About Tate-Shafarevich Groups Using Iwasawa Theory, preprint.

• Sage Reference Manual on p-adic L-functions of elliptic curves: http://sagemath.org/doc/reference/sage/schemes/elliptic_curves/padic_lseries.html. See also nearby sections.

Backgrond reading

• As for other lectures, Silverman's book "The arithmetic of elliptic curves" contains a good background for elliptic curve, especially chapter III.
  • We will use some further results on elliptic curves over global fields, mainly in chapter VII and X, but by far not everything there is needed.

• A more concise introduction to the subject, including the discussion of how elliptic cures over Q are linked to modular forms is in the first

  • two chapters of Darmon's book "Rational Point on Modular Elliptic Curves".