CIRM - Videos & books Library - Vanishing of twisted L-functions of elliptic curves over function fields

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Auteurs : Lalin, Matilde (Auteur de la Conférence)
CIRM (Editeur )

00:00

Résumé : Let

E

be an elliptic curve over the rationals, and let

χ

be a Dirichlet character of order

ℓ

for some odd prime

ℓ

. Heuristics based on the distribution of modular symbols and random matrix theory have led to conjectures predicting that the vanishing of the twisted

L

-functions

L (E, χ, s)

s = 1

is a very rare event (David-Fearnley-Kisilevsky and Mazur-Rubin). In particular, it is conjectured that there are only finitely many characters of order

ℓ > 5

such that

L (E, χ, 1) = 0

for a fixed curve

E

.
We investigate the case of elliptic curves over function fields. For Dirichlet

L

-functions over function fields, Li and Donepudi-Li have shown how to use the geometry to produce infinitely many characters of order

l \geq 2

such that the Dirichlet

L

-function

L (χ, s)

vanishes at

s = 1 / 2

, contradicting (the function field analogue of) Chowla's conjecture. We show that their work can be generalized to constant curves

E / F_{q} (t)

, and we show that if there is one Dirichlet character

χ

of order

ℓ

such that

L (E, χ, 1) = 0

, then there are infinitely many, leading to some specific examples contradicting (the function field analogue of) the number field conjectures on the vanishing of twisted

L

-functions. Such a dichotomy does not seem to exist for general curves over

F_{q} (t)

, and we produce empirical evidence which suggests that the conjectures over number fields also hold over function fields for non-constant

E / F_{q} (t)

.

Keywords : non-vanishing of L-functions; twisted L-functions of elliptic curves; function fields; elliptic curve ranks in extensions

Codes MSC :
11G05 - Elliptic curves over global fields
11G40 -

L

-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14H25 - Arithmetic ground fields, See also {11Dxx,11G05,14Gxx}

Informations sur la Vidéo

Réalisateur :

Petit, Jean

Langue :

Date de publication :

Date de captation :

Sous collection :

arXiv category :

Domaine :

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Durée :

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Download :

https://videos.cirm-math.fr/2023-05-15_Lalin.mp4

Informations sur la Rencontre

Nom de la rencontre : Jean-Morlet Chair - Conference - Arithmetic Statistics / Chaire Jean-Morlet - Conférence - Statistiques arithmétiques
Organisateurs de la rencontre : Anni, Samuele ; Lorenzo Garcia, Elisa ; Stevenhagen, Peter ; Vonk, Jan
Dates : 15/05/2023 - 19/05/2023
Année de la rencontre : 2023
URL Congrès : https://conferences.cirm-math.fr/2675.html

Données de citation

DOI : 10.24350/CIRM.V.20045803
Citer cette vidéo: Lalin, Matilde (2023). Vanishing of twisted L-functions of elliptic curves over function fields. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20045803
URI : http://dx.doi.org/10.24350/CIRM.V.20045803

Voir aussi

[Multi angle] Integers expressible as the sum of two rational cubes / Auteur de la Conférence Bhargava, Manjul.
[Multi angle] The average size of 3-torsion in class groups of 2-extensions / Auteur de la Conférence Matchett Wood, Melanie.
[Multi angle] On arithmetic statistics / Auteur de la Conférence Stevenhagen, Peter.
[Multi angle] Quadratic twist families of elliptic curves with unusual $2^{\infty}$ -Selmer groups / Auteur de la Conférence Smith, Alexander.
[Multi angle] Subrings of number fields / Auteur de la Conférence Lenstra, Hendrik.

Bibliographie

COMEAU-LAPOINTE, Antoine, DAVID, Chantal, LALIN, Matilde, et al. On the vanishing of twisted L-functions of elliptic curves over rational function fields. Research in Number Theory, 2022, vol. 8, no 4, p. 76. - https://doi.org/10.48550/arXiv.2207.00197

DAVID, Chantal, FEARNLEY, Jack, et KISILEVSKY, Hershy. On the vanishing of twisted L-functions of elliptic curves. Experimental Mathematics, 2004, vol. 13, no 2, p. 185-198. - https://doi.org/10.1080/10586458.2004.10504532

DAVID, Chantal, FEARNLEY, Jack, et KISILEVSKY, Hershy. Vanishing of L-functions of elliptic curves over number ﬁelds. Ranks of elliptic curves and random matrix theory, 2007, no 341, p. 247. -

DONEPUDI, Ravi et LI, Wanlin. Vanishing of Dirichlet L-functions at the central point over function fields. Rocky Mountain Journal of Mathematics, 2021, vol. 51, no 5, p. 1615-1628. - http://dx.doi.org/10.1216/rmj.2021.51.1615

FEARNLEY, Jack, KISILEVSKY, Hershy, et KUWATA, Masato. Vanishing and non‐vanishing Dirichlet twists of L‐functions of elliptic curves. Journal of the London Mathematical Society, 2012, vol. 86, no 2, p. 539-557. - https://doi.org/10.1112/jlms/jds018

GOUVÊA, Fernando et MAZUR, Barry. The square-free sieve and the rank of elliptic curves. Journal of the American Mathematical Society, 1991, vol. 4, no 1, p. 1-23. - http://www.jstor.org/stable/2939253

LI, Wanlin. Vanishing of hyperelliptic L-functions at the central point. Journal of Number Theory, 2018, vol. 191, p. 85-103. - https://doi.org/10.1016/j.jnt.2018.03.018

MAZUR, Barry, RUBIN, Karl, et LARSEN, Michael. Diophantine stability. American Journal of Mathematics, 2018, vol. 140, no 3, p. 571-616. - https://doi.org/10.1353/ajm.2018.0014

MAZUR, Barry et RUBIN, Karl. Arithmetic conjectures suggested by the statistical behavior of modular symbols. Experimental Mathematics, 2021, p. 1-16. - https://doi.org/10.1080/10586458.2021.1982424

POONEN Bjorn. Squarefree values of multivariable polynomials.Duke Math. J., 2023, 118 (2) p.353 - 373 - https://doi.org/10.1215/S0012-7094-03-11826-8

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