On special values of Jacobi-sum Hecke L-functions

TL;DR

This paper investigates special values of Jacobi-sum Hecke L-functions associated with Fermat curves, verifying the Beilinson conjecture numerically and deriving formulas for L-values at zero, drawing analogies with classical period formulas.

Contribution

It provides new numerical evidence for the Beilinson conjecture and explicit formulas for L-values at zero related to Fermat curves and hypergeometric functions.

Findings

Numerical verification of the Beilinson conjecture for specific cases.

Explicit formulas for L-values at zero involving hypergeometric functions.

Analogies with the Chowla-Selberg formula for elliptic curves.

Abstract

For motives associated with Fermat curves, there are elements in motivic cohomology whose regulators are written in terms of special values of generalized hypergeometric functions. Using them, we verify the Beilinson conjecture numerically for some cases and find formulae for the values of L-functions at 0. These appear analogous to the Chowla-Selberg formula for the periods of elliptic curves with complex multiplication, which are related with the L-values at 1 by the Birch and Swinnerton-Dyer conjecture.