AlgoL: Algorithmics of L-functions (2007-2011)

Institut de Mathématiques de Bordeaux Institut Camille Jordan (Lyon 1)

A project supported by the

Summary

We study the so-called L-functions in number theory, from an algorithmic and experimental point of view. L-functions encode delicate arithmetic information, and crucial arithmetic conjectures revolve around them: Riemann Hypotheses, Birch and Swinnerton-Dyer conjecture, Stark conjectures, Bloch-Kato conjectures, etc.

Most of current number theory conjectures originate from (usually mechanized) computations, and have been thoroughly checked numerically. L-functions and their special values are no exception, but available tools and actual computations become increasingly scarce as one goes further away from Dirichlet L-functions. We propose to develop theoretical algorithms and practical tools to study and experiment with (suitable classes of) complex or p-adic L-functions, their coefficients, special or general values, and zeroes. For instance, it is not known whether K-theoretic invariants conjecturally attached to special values are computable in any reasonable complexity model. On the other hand, special values are often readily computable and sometimes provide, albeit conjecturally, the only concrete handle on the said invariants

Our ambition is not only to obtain new theoretical results, but also to translate these to new, or more efficient, functions in the PARI/GP system.