Algorithmique des fonctions L – ALGOL

This project centers around the so-called L-functions in number theory, from an algorithmic and experimental point of view. L-functions encode delicate arithmetic information, and many crucial conjectures in number theory revolve around them: Riemann Hypotheses, the Birch and Swinnerton Dyer conjecture, Stark conjectures, Bloch-Kato conjecture, etc. Most of current number theory conjectures originate from computations, and have been thoroughly checked numerically. For instance, the conjecture of Birch and Swinnerton-Dyer was formulated after intense numerical experiments; many of the algebraic relations among polylogarithms were discovered by computers. 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) L-functions, their coefficients, special or general values, and zeroes. For instance, it is not known whether K-theoretic invariants conjecturally attached with 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 achieve significant theoretical results, but also to translate these to new, or more efficient, functions in the PARI/GP system, which will be a valuable asset for the whole mathematical community. A subsequent aim is to establish algorithmic number theory as an important theme in several mathematical French departments.