Galois action on special theta values

Abstract

For a primitive Dirichlet character χ of conductor N set θ χ (τ)=∑ n∈ℤ n ϵ χ(n)e πin 2 τ/N (where ϵ=0 for even χ, ϵ=1 for odd χ) the associated theta series. Its value at its point of symmetry under the modular transformation τ↦-1/τ is related by θ χ (i)=W(χ)θ χ ¯ (i) to the root number of the L-series of χ and hence can be used to calculate the latter quickly if it does not vanish. Using Shimura’s reciprocity law, we calculate the Galois action on these special values of theta functions with odd N normalised by the Dedekind eta function. 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 N.