On the orthogonal symmetry of $L$-functions of a family of Hecke Grössencharacters

J. B. Conrey ¹   ; N. C. Snaith ²

¹ American Institute of Mathematics 360 Portage Ave. Palo Alto, CA 94306, U.S.A. and School of Mathematics University of Bristol Bristol, BS8 1TW, United Kingdom
² School of Mathematics University of Bristol Bristol, BS8 1TW, United Kingdom

Acta Arithmetica, Tome 157 (2013) no. 4, pp. 323-356

Résumé

The family of symmetric powers of an $L$-function associated with an elliptic curve with complex multiplication has received much attention from algebraic, automorphic and $p$-adic points of view. Here we examine one explicit such family from the perspectives of classical analytic number theory and random matrix theory, especially focusing on evidence for the symmetry type of the family. In particular, we investigate the values at the central point and give evidence that this family can be modeled by ensembles of orthogonal matrices. We prove an asymptotic formula with power savings for the average of these L-values, which reproduces, by a completely different method, an asymptotic formula proven by Greenberg and Villegas–Zagier. We give an upper bound for the second moment which is conjecturally too large by just one logarithm. We also give an explicit conjecture for the second moment of this family, with power savings. Finally, we compute the one-level density for this family with a test function whose Fourier transform has limited support. It is known by the work of Villegas–Zagier that the subset of these $L$-functions from our family which have even functional equations never vanish; we show to what extent this result is reflected by our analytic results.

Détail

DOI : 10.4064/aa157-4-2

Keywords: family symmetric powers l function associated elliptic curve complex multiplication has received much attention algebraic automorphic p adic points view here examine explicit family perspectives classical analytic number theory random matrix theory especially focusing evidence symmetry type family particular investigate values central point evidence family modeled ensembles orthogonal matrices prove asymptotic formula power savings average these l values which reproduces completely different method asymptotic formula proven greenberg villegas zagier upper bound second moment which conjecturally too large just logarithm explicit conjecture second moment family power savings finally compute one level density family test function whose fourier transform has limited support known work villegas zagier subset these l functions family which have even functional equations never vanish what extent result reflected analytic results