Cyclotomic Hecke L-values of a totally real field

Abstract

It is known that any Fourier coefficient of a newform of weight 2 can be expressed as a polynomial with rational coefficients, of a single algebraic critical value of the corresponding L-function twisted by a Dirichlet character of $p$-power conductor for a rational prime $p$. In the talk, I will discuss a version of this result in terms of Hecke L-function over a totally real field, twisted by Hecke characters of $p$-power conductors. The discussion involves new technical challenges that arise from the presence of the unit group, which are (1) counting lattice points in a cone that $p$-adically close to units and (2) estimating an exponential sum over the unit group. This is joint work with Byungheup Jun and Jungyun Lee.