One-level density of the family of twists of an elliptic curve over function fields

We fix an elliptic curve E/Fq(t) and consider the family {E⊗χD} of E twisted by quadratic Dirichlet characters. The one-level density of their L-functions is shown to follow orthogonal symmetry for test functions with Fourier transform supported inside (−1,1). As an application, we obtain an upper b...

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Bibliographic Details
Published in:Journal of number theory 2022-12, Vol.241, p.165-197
Main Author: Comeau-Lapointe, Antoine
Format: Article
Language:English
Subjects:
Average rank
Dirichlet characters
Elliptic curves
Finite fields
Function fields
L-function
One-level density
Twisted L-functions
Zeta functions
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Summary:We fix an elliptic curve E/Fq(t) and consider the family {E⊗χD} of E twisted by quadratic Dirichlet characters. The one-level density of their L-functions is shown to follow orthogonal symmetry for test functions with Fourier transform supported inside (−1,1). As an application, we obtain an upper bound of 3/2 on the average analytic rank. By splitting the family according to the sign of the functional equation, we obtain that at least 12.5% of the family have rank zero, and at least 37.5% have rank one. The Katz and Sarnak philosophy predicts that those percentages should both be 50% and that the average analytic rank should be 1/2. We finish by computing the one-level density of E twisted by Dirichlet characters of order ℓ≠2 coprime to q. We obtain a restriction of (−1/2,1/2) on the support with a unitary symmetry.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2022.03.005