Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold

DOI10.1016/J.JNT.2011.05.005zbMATH Open1277.11037arXiv1003.1157OpenAlexW1985997381MaRDI QIDQ640867

Author name not available (Why is that?)

Publication date: 21 October 2011

Published in: (Search for Journal in Brave)

Abstract: We present here simple trace formulas for Hecke operators

T_{k} (p)

for all

p > 3

on

S_{k} (G a m m a_{0} (3))

and

S_{k} (G a m m a_{0} (9))

, the spaces of cusp forms of weight

k

and levels 3 and 9. These formulas can be expressed in terms of special values of Gaussian hypergeometric series and lend themselves to simple recursive expressions in terms of traces of Hecke operators on spaces of lower weight. Along the way, we show how to express the traces of Frobenius of a family of elliptic curves with 3-torsion as special values of a Gaussian hypergeometric series over

m a t h b b F_{q}

, when

q e q u i v 1 p m o d 3

. We also use these formulas to provide a simple expression for the Fourier coefficients of

e t a (3 z)^{8}

, the unique normalized cusp form of weight 4 and level 9, and then show that the number of points on a certain threefold is expressible in terms of these coefficients.

Full work available at URL: https://arxiv.org/abs/1003.1157

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