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Vanishing of twisted L-functions of elliptic curves over function fields
Multi angle
Authors : Lalin, Matilde (Author of the conference)
CIRM (Publisher )
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}
CIRM (Publisher )
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Abstract :
Let $E$ be an elliptic curve over the rationals, and let $\chi$ be a Dirichlet character of order $\ell$ for some odd prime $\ell$. 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 $L(E, \chi, s)$ at $s = 1$ 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 $\ell > 5$ such that $L(E, \chi, 1) = 0$ 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 $L(\chi, s)$ 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 $E/\mathbb{F}_q(t)$, and we show that if there is one Dirichlet character $\chi$ of order $\ell$ such that $L(E, \chi, 1) = 0$, 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 $\mathbb{F}_q(t)$, and we produce empirical evidence which suggests that the conjectures over number fields also hold over function fields for non-constant $E/\mathbb{F}_q(t)$.
Keywords : non-vanishing of L-functions; twisted L-functions of elliptic curves; function fields; elliptic curve ranks in extensions
MSC Codes : 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 $L(\chi, s)$ 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 $E/\mathbb{F}_q(t)$, and we show that if there is one Dirichlet character $\chi$ of order $\ell$ such that $L(E, \chi, 1) = 0$, 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 $\mathbb{F}_q(t)$, and we produce empirical evidence which suggests that the conjectures over number fields also hold over function fields for non-constant $E/\mathbb{F}_q(t)$.
Keywords : non-vanishing of L-functions; twisted L-functions of elliptic curves; function fields; elliptic curve ranks in extensions
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}
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Information on the Video
Film maker : Petit, JeanLanguage : English Available date : 30/05/2023 Conference Date : 15/05/2023 Subseries : Research talks arXiv category : Number Theory Mathematical Area(s) : Number Theory Format : MP4 (.mp4) - HD Video Time : 00:45:40 Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students Download : https://videos.cirm-math.fr/2023-05-15_Lalin.mp4 
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Information on the Event
Event Title : Jean-Morlet Chair - Conference - Arithmetic Statistics / Chaire Jean-Morlet - Conférence - Statistiques arithmétiquesEvent Organizers : Anni, Samuele ; Lorenzo Garcia, Elisa ; Stevenhagen, Peter ; Vonk, Jan Dates : 15/05/2023 - 19/05/2023 Event Year : 2023 Event URL : https://conferences.cirm-math.fr/2675.html
Citation Data
DOI : 10.24350/CIRM.V.20045803Cite this video as: 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 |
See Also
- [Multi angle] Integers expressible as the sum of two rational cubes / Author of the conference Bhargava, Manjul.
- [Multi angle] The average size of 3-torsion in class groups of 2-extensions / Author of the conference Matchett Wood, Melanie.
- [Multi angle] On arithmetic statistics / Author of the conference Stevenhagen, Peter.
- [Multi angle] Quadratic twist families of elliptic curves with unusual $2^{\infty }$-Selmer groups / Author of the conference Smith, Alexander.
- [Multi angle] Subrings of number fields / Author of the conference Lenstra, Hendrik.
Bibliography
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- 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
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- 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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