Galois action on special theta values

TY - JOUR

T1 - Galois action on special theta values

AU - Bengoechea, Paloma

N1 - © Société Arithmétique de Bordeaux, 2016. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

PY - 2016/6/1

Y1 - 2016/6/1

N2 - For a primitive Dirichlet character χ of conductor N set θχ(τ) = ∑n ∈ℤ n∈ χ(n) eπin2τ/N (where ∈ = 0 for even χ, ∈ = 1 for odd χ) the associated theta series. Its value at its point of symmetry under the modular transformation τ(image found)−1/τ is related by θχ(i) = W(χ)θ(image found) (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.

AB - For a primitive Dirichlet character χ of conductor N set θχ(τ) = ∑n ∈ℤ n∈ χ(n) eπin2τ/N (where ∈ = 0 for even χ, ∈ = 1 for odd χ) the associated theta series. Its value at its point of symmetry under the modular transformation τ(image found)−1/τ is related by θχ(i) = W(χ)θ(image found) (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.

KW - Complex multiplication

KW - L-series

KW - Shimura’s reciprocity law

KW - Theta functions

UR - http://www.scopus.com/inward/record.url?scp=84973305016&partnerID=8YFLogxK

U2 - 10.5802/jtnb.943

DO - 10.5802/jtnb.943

M3 - Article

AN - SCOPUS:84973305016

SN - 1246-7405

VL - 28

SP - 347

EP - 360

JO - Journal de Théorie des Nombres de Bordeaux

JF - Journal de Théorie des Nombres de Bordeaux

IS - 2

ER -