L functions in families: analysis, interactions, effectiveness – FLAIR

L-functions are ubiquitous objects in Number Theory and Arithmetic Geometry. They are analytic, algebraic, or combinatorial in nature. Recently attempts have been made to formalize the notion of ``good'' family of L-functions. Such families naturally appear in a broad variety of active research fields e.g. automorphic forms and Artin representations, Drinfeld modules, abelian varieties over global fields, inequities in the distribution of sequences indexed by prime numbers or more generally by places of global fields...
The FLAIR project will systematically study both the theoretical and algorithmic aspects of families of L-functions. The main goal is to bring together experts of the aforementioned branches of number theory and arithmetic geometry as well as experts of the computational side of these fields and to have them interact and explore further the richness of the information encoded by L-functions.

Putting emphasis on the unifying role of L-functions and their families, the project is particularly ambitious. Beyond the work of its members, in each of their own fields of expertise, where L-functions naturally come into play, our proposal aims at synthesizing complementary points of view coming from distant fields: the analytic approach in the classical theory of L-functions, the theory of Artin L-functions in connection with the Langlands program, L-functions coming from arithmetic geometry in the spirit of the Weil conjectures and of the pioneering work of Grothendieck, Deligne, and their collaborators, p-adic L-functions, etc. A systematic development of the emerging notion of a good family of L-functions, will allow us to transpose the analytic averaging techniques that have already led to a spectacular progress in numerous concrete problems of arithmetic, geometric or analytic nature, to other situations, where families of L-functions arise.

One of the original aspects of the project lies in the constant interaction between theoretical considerations and numerical and algorithmic features for all the diverse families of L-functions we plan to study. Computer algebra systems have been extensively developed and improved throughout the recent years. More than ever they provide us with a precious tool to test the validity of hypotheses, to give evidence for relevant heuristics and help the whole community by making available powerful, convenient, and effective means to help mathematicians pursuing their research work. The FLAIR project will strongly rely on the established tradition of excellence and expertise in algorithmic aspects of number theory at the ``Institut de mathématiques de Bordeaux''.

As a consequence of the spectacular recent progress in the study of arithmetic problems on average and of families of L-functions, as well as of the ever growing activity in computational methods in number theory, it is a particularly favourable time to launch the FLAIR project.