Section: Partnerships and Cooperations

National Initiatives

ANR AlgoL : Algorithmics of $L$-functions

Participants : Bill Allombert, Karim Belabas, Henri Cohen, Jean-Marc Couveignes, Andreas Enge, Pascal Molin.

http://www.math.u-bordeaux1.fr/~belabas/algol/index.html

The AlgoL project comprises research teams in Bordeaux, Montpellier, Lyon, Toulouse and Besançon.

It studies 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 mechanised) 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 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 computed and sometimes provide, albeit conjecturally, the only concrete handle on said invariants.

New theoretical results are translated into new or more efficient functions in the Pari/Gp system.