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

Comeau-Lapointe, Antoine

doi:10.1016/j.jnt.2022.03.005

Cited by 5 publications

(3 citation statements)

References 24 publications

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“…This would allow for the more general setting of q being fixed with some other parameter, such as the genus, tending to infinity. Partial results towards questions such as these can be found in [3,8,28]. An important subset of continuous class functions consists of the mixed power trace functions.…”

Section: Patrick Meisner (Montréal)mentioning

confidence: 99%

Moments of traces of Frobenius of higher order Dirichlet $L$-functions over $\mathbb F_q[T]$

Meisner

2023

Acta Arith.

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Section: Patrick Meisner (Montréal)mentioning

confidence: 99%

Moments of traces of Frobenius of higher order Dirichlet $L$-functions over $\mathbb F_q[T]$

Meisner

2023

Acta Arith.

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“…This was studied over number fields and functions fields, for quadratic and higher order twists. For quadratic twists, it is possible to prove results on the one-level density strong enough to deduce that a positive proportion of twists with even (respectively odd) analytic rank do no vanish (respectively vanish of order 1) at the central critical point [HB04,CL22]. The one-level density, or average rank, of higher order twists for elliptic curves L-functions was studied by [Cho] over number fields and [MS,CL22] over function fields.…”

Section: Introductionmentioning

confidence: 99%

“…For quadratic twists, it is possible to prove results on the one-level density strong enough to deduce that a positive proportion of twists with even (respectively odd) analytic rank do no vanish (respectively vanish of order 1) at the central critical point [HB04,CL22]. The one-level density, or average rank, of higher order twists for elliptic curves L-functions was studied by [Cho] over number fields and [MS,CL22] over function fields. Quadratic twists of elliptic curve over functions fields were also studied by [BFKRG20] who obtained results on the correlation of the analytic ranks of two twisted elliptic curves.…”

Section: Introductionmentioning

confidence: 99%

On the vanishing of twisted $L$-functions of elliptic curves over rational function fields

Comeau-Lapointe¹,

David²,

Laĺın³

et al. 2022

Preprint

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We investigate in this paper the vanishing at s = 1 of the twisted L-functions of elliptic curves E defined over the rational function field F q (t) (where F q is a finite field of q elements and characteristic ≥ 5) for twists by Dirichlet characters of prime order ℓ ≥ 3, from both a theoretical and numerical point of view. In the case of number fields, it is predicted that such vanishing is a very rare event, and our numerical data seems to indicate that this is also the case over function fields for non-constant curves. For constant curves, we adapt the techniques of [Li18, DL21] who proved vanishing at s = 1/2 for infinitely many Dirichlet L-functions over F q (t) based on the existence of one, and we can prove that if there is one χ 0 such that L(E, χ 0 , 1) = 0, then there are infinitely many. Finally, we provide some examples which show that twisted L-functions of constant elliptic curves over F q (t) behave differently than the general ones.

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On the vanishing of twisted L-functions of elliptic curves over rational function fields

et al. 2022

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

Cited by 5 publications

References 24 publications

Moments of traces of Frobenius of higher order Dirichlet $L$-functions over $\mathbb F_q[T]$

Moments of traces of Frobenius of higher order Dirichlet $L$-functions over $\mathbb F_q[T]$

On the vanishing of twisted $L$-functions of elliptic curves over rational function fields

On the vanishing of twisted L-functions of elliptic curves over rational function fields

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