Clayton Williams

Affiliation: University of Illinois at Urbana-Champaign

Title Of Talk: The Kohnen-Zagier Formula in Level 1 for Eta Multipliers

URLS:
TBA

Abstract: Kohnen-Zagier formulae give an explicit identity between the coefficients of half-integer weight modular forms at discriminants to central values of twisted L-functions through the Shimura correspondence, which is for modular forms transforming with respect to the theta multiplier. They have a number of arithmetic applications. Recently Ahlgren, Andersen, and Dicks, building on work of Yang, found a Shimura Correspondence for eta multipliers, $\mathscr{S}_t:S_{\lambda+1/2}(\Gamma_0(N),\ u_\eta^r\chi)\to S_{2\lambda}^{\operatorname{new}2,3}(6N,\chi^2)$ for $N$ with $\operatorname{gcd}(N,6)=\operatorname{gcd}(r,2)=1$ and $t>0$, with precise information at the primes 2 and 3. Here $\chi$ is a character modulo $N$. When $N=1$ Yang proved that a linear combination of the $\mathscr{S}_t$ maps is an isomorphism of Hecke modules. In this talk I will present a proof of a new Kohnen-Zagier formula using the new Shimura correspondence for eta multipliers when $N=1$. I will also discuss the main technical theorem in the proof, which requires relating Kloosterman sums to quadratic Weyl sums using the Weil representation for quadratic modules.

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