Galois action on special theta values

TL;DR

This paper investigates how Galois groups act on special theta function values linked to primitive Dirichlet characters, providing new proofs, experimental insights, and partial results on their non-vanishing at prime conductors.

Contribution

It applies Shimura's reciprocity law to explicitly compute Galois actions on theta values and derives new results confirming and extending previous experimental findings.

Findings

Proved Galois action formulas for special theta values

Confirmed some experimental results of Cohen and Zagier

Obtained partial non-vanishing results for prime conductors

Abstract

Using Shimura's reciprocity law, we calculate the Galois action on the special values of theta functions associated to primitive Dirichlet characters of odd conductor, normalised by the Dedekind eta function, at the point i. As a consequence, we prove some experimental results of Cohen and Zagier and we deduce a partial result on the non-vanishing of these special theta values with prime conductor.