A Linear independence result for $p$-adic $L$-values

Abstract

The aim of this paper is to provide an analogue of the Ball-Rivoal theorem for $p$-adic $L$-values of Dirichlet characters. More precisely, we prove for a Dirichlet character $\chi$ and a number field $K$ the formula ${\dim }_K(K+{\sum }_{i=2}^{s+1}{L}_p(i,\chi {\omega }^{1-i})K)\ge \frac{(1-\epsilon )\log (s)}{2[K:\mathbb{Q}](1+\log 2)}$. As a byproduct, we establish an asymptotic linear independence result for the values of the $p$-adic Hurwitz zeta function.

26 pages, final version; Duke Math. J. (accepted)