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Title: Vanishing of quartic and sextic twists of L-functions
Abstract

LetEbe an elliptic curve over$${{\mathbb {Q}}}$$Q. We conjecture asymptotic estimates for the number of vanishings of$$L(E,1,\chi )$$L(E,1,χ)as$$\chi $$χvaries over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis onE. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Compared to earlier work by David, Fearnley and Kisilevsky that formulated analogous conjectures for characters of any odd prime order, in the composite order case, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. To do this, we introduce the notion of totally order$$\ell $$characters to quantify how quickly the quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (resp., sextic) characters is much slower than in the sub-family of totally quartic (resp., sextic) characters. We provide a conceptual explanation for this phenomenon by observing that the full family of order$$\ell $$twisted elliptic curveL-functions, with$$\ell $$even and composite, is a mixed family with both unitary and orthogonal aspects.

 
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NSF-PAR ID:
10489049
Author(s) / Creator(s):
; ;
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Research in Number Theory
Volume:
10
Issue:
1
ISSN:
2522-0160
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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