L-functions : p-adic, analytic and effective aspects – PadLEfAn

The L-function of a mathematical object - a number field, an algebraic variety, or an automorphic representation - is a bridge between that analytic and arithmetical study of this object.

The central motive organizing this project is a concrete, specific incarnation of this classical theme which has been made possible after recent results obtained by its members: special values of L-functions have two components, a transcendental component (the regulator) and a number-theoretic component; each exhaustive and precise result on the arithmetic of the second component therefore has its counterpart in the first, and vice-versa.

Let us open this circle of ideas by noting that the recent proofs of the main conjecture of Iwasawa theory for modular forms and its potential generalization to other automorphic forms, which are p-adic results, allow for the effective computation of p-adic regulators, p-adic periods and p-adic L-functions of these forms, an effective variant of integral p-adic Hodge theory. Effective computation of the p-adic regulator has a counterpart for motives in positive characteristic, where it is known to be intimately linked to the open problem of the existence of a functional equation for the L-function and to the determination of the special values of said L-function.

Likewise, effective computation of regulators has an analytic counterpart, under the guise of effective Brauer-Siegel theorems for number fields as well as abelian varieties; results that yield majoration and minoration of special values of L-functions.

This circle of idea is completed when it is noted that the asymptotic results given by the Brauer-Siegel themselves have applications in the Iwasawa theory of tower of number fields and algebraic varieties.

That mutual interplay between analytic, effective and p-adic properties of special values of L-functions is what we propose to investigate.