Galois action on special theta values

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JournalJournal de Théorie des Nombres de Bordeaux
DateAccepted/In press - 5 Sep 2014
DateE-pub ahead of print - 15 Mar 2016
DatePublished (current) - 1 Jun 2016
Issue number2
Volume28
Number of pages14
Pages (from-to)347-360
Early online date15/03/16
Original languageEnglish

Abstract

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.

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© 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

    Research areas

  • Complex multiplication, L-series, Shimura’s reciprocity law, Theta functions

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