Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces

Timo Keller; Michael Stoll

doi:10.5802/crmath.313

Géométrie algébrique

Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces
[Vérification exacte de la conjecture BSD forte pour certaines variétés abéliennes absolument simples]

Timo Keller ¹ ; Michael Stoll ¹

¹ Lehrstuhl Mathematik II (Computeralgebra), Universität Bayreuth, Universitätsstraße 30, 95440 Bayreuth, Germany

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 483-489.

Résumé
Abstract

Soit $X$ un des $28$ quotients d’Atkin–Lehner d’une courbe $X_{0} (N)$ tel que $X$ est de genre $2$ et sa jacobienne $J$ est absolument simple. On démontre que le groupe de Shafarevich–Tate $Ш (J / ℚ)$ est trivial. Ceci vérifie la conjecture BSD forte pour $J$ .

Let $X$ be one of the $28$ Atkin–Lehner quotients of a curve $X_{0} (N)$ such that $X$ has genus $2$ and its Jacobian variety $J$ is absolutely simple. We show that the Shafarevich–Tate group $Ш (J / ℚ)$ is trivial. This verifies the strong BSD conjecture for $J$ .

Reçu le : 2021-06-30
Révisé le : 2021-11-05
Accepté le : 2021-12-09
Publié le : 2022-05-23

DOI : 10.5802/crmath.313

Classification : 11G40, 11-04, 11G10, 11G30, 14G35

Affiliations des auteurs :

Timo Keller ¹ ; Michael Stoll ¹

¹ Lehrstuhl Mathematik II (Computeralgebra), Universität Bayreuth, Universitätsstraße 30, 95440 Bayreuth, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2022__360_G5_483_0,
     author = {Timo Keller and Michael Stoll},
     title = {Exact verification of the strong {BSD} conjecture for some absolutely simple abelian surfaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {483--489},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.313},
     language = {en},
}

TY  - JOUR
AU  - Timo Keller
AU  - Michael Stoll
TI  - Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 483
EP  - 489
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.313
LA  - en
ID  - CRMATH_2022__360_G5_483_0
ER  -

%0 Journal Article
%A Timo Keller
%A Michael Stoll
%T Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces
%J Comptes Rendus. Mathématique
%D 2022
%P 483-489
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.313
%G en
%F CRMATH_2022__360_G5_483_0

Timo Keller; Michael Stoll. Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 483-489. doi : 10.5802/crmath.313. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.313/

Bibliographie
Cité par

[1] Raymond van Bommel Numerical verification of the Birch and Swinnerton–Dyer conjecture for hyperelliptic curves of higher genus over $ℚ$ up to squares, Exp. Math., Volume 31 (2022) no. 1, pp. 138-145 | DOI | MR

[2] Wieb Bosma; John Cannon; Catherine Playoust The Magma algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3–4, pp. 235-265 Computational algebra and number theory (London, 1993) | DOI | MR | Zbl

[3] Christophe Breuil; Brian Conrad; Fred Diamond; Richard Taylor On the modularity of elliptic curves over $Q$ : wild 3-adic exercises, J. Am. Math. Soc., Volume 14 (2001) no. 4, pp. 843-939 | DOI | MR | Zbl

[4] Daniel Bump; Solomon Friedberg; Jeffrey Hoffstein A nonvanishing theorem for derivatives of automorphic $L$ -functions with applications to elliptic curves, Bull. Am. Math. Soc., Volume 21 (1989) no. 1, pp. 89-93 | DOI | MR | Zbl

[5] Francesc Castella; Mirela Çiperiani; Christopher Skinner; Florian Sprung On the Iwasawa main conjectures for modular forms at non-ordinary primes, 2018 (https://arxiv.org/abs/1804.10993)

[6] Alina Carmen Cojocaru On the surjectivity of the Galois representations associated to non-CM elliptic curves, Can. Math. Bull., Volume 48 (2005) no. 1, pp. 16-31 (With an appendix by Ernst Kani) | DOI | MR | Zbl

[7] Brendan Creutz; Robert L. Miller Second isogeny descents and the Birch and Swinnerton–Dyer conjectural formula, J. Algebra, Volume 372 (2012), pp. 673-701 | DOI | MR | Zbl

[8] Luis V. Dieulefait Explicit determination of the images of the Galois representations attached to abelian surfaces with $End (A) = ℤ$ , Exp. Math., Volume 11 (2002) no. 4, pp. 503-512 | DOI | MR | Zbl

[9] E. Victor Flynn; Franck Leprévost; Edward F. Schaefer; William A. Stein; Michael Stoll Empirical evidence for the Birch and Swinnerton–Dyer conjectures for modular Jacobians of genus 2 curves, Math. Comput., Volume 70 (2001) no. 236, pp. 1675-1697 | DOI | MR | Zbl

[10] Grigor Grigorov; Andrei Jorza; Stefan Patrikis; William A. Stein; Corina Tarniţǎ Computational verification of the Birch and Swinnerton–Dyer conjecture for individual elliptic curves, Math. Comput., Volume 78 (2009) no. 268, pp. 2397-2425 | DOI | MR | Zbl

[11] Benedict H. Gross; Don B. Zagier Heegner points and derivatives of $L$ -series, Invent. Math., Volume 84 (1986) no. 2, pp. 225-320 | DOI | MR | Zbl

[12] Yuji Hasegawa Table of quotient curves of modular curves $X_{0} (N)$ with genus $2$ , Proc. Japan Acad., Ser. A, Volume 71 (1995) no. 10, pp. 235-239 | MR | Zbl

[13] Viktor A. Kolyvagin; Dmitri Yu. Logachëv Finiteness of the Shafarevich–Tate group and the group of rational points for some modular abelian varieties, Algebra Anal., Volume 1 (1989) no. 5, pp. 171-196 | MR | Zbl

[14] Barry Mazur Modular curves and the Eisenstein ideal, Publ. Math., Inst. Hautes Étud. Sci. (1977) no. 47, pp. 33-186 (With an appendix by Mazur and M. Rapoport) | DOI | Numdam | MR | Zbl

[15] Robert L. Miller; Michael Stoll Explicit isogeny descent on elliptic curves, Math. Comput., Volume 82 (2013) no. 281, pp. 513-529 | DOI | MR | Zbl

[16] Jan Steffen Müller; Michael Stoll Canonical heights on genus-2 Jacobians, Algebra Number Theory, Volume 10 (2016) no. 10, pp. 2153-2234 | DOI | MR | Zbl

[17] Edward F. Schaefer; Michael Stoll How to do a $p$ -descent on an elliptic curve, Trans. Am. Math. Soc., Volume 356 (2004) no. 3, pp. 1209-1231 | DOI | MR | Zbl

[18] Christopher Skinner Multiplicative reduction and the cyclotomic main conjecture for ${GL}_{2}$ , Pac. J. Math., Volume 283 (2016) no. 1, pp. 171-200 | DOI | MR | Zbl

[19] Christopher Skinner; Eric Urban The Iwasawa main conjectures for ${GL}_{2}$ , Invent. Math., Volume 195 (2014) no. 1, p. 1–277 | DOI | Zbl

[20] Michael Stoll Implementing 2-descent for Jacobians of hyperelliptic curves, Acta Arith., Volume 98 (2001) no. 3, pp. 245-277 | DOI | MR | Zbl

[21] Richard Taylor; Andrew Wiles Ring-theoretic properties of certain Hecke algebras, Ann. Math., Volume 141 (1995) no. 3, pp. 553-572 | DOI | MR | Zbl

[22] The LMFDB collaboration Genus 2 curves over $ℚ$ (from the database “L-functions and Modular Forms Database”, https://www.lmfdb.org/Genus2Curve/Q/)

[23] Jean-Loup Waldspurger Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie, Compos. Math., Volume 54 (1985) no. 2, pp. 173-242 | Numdam | Zbl

[24] Xiang Dong Wang $2$ -dimensional simple factors of $J_{0} (N)$ , Manuscr. Math., Volume 87 (1995) no. 2, p. 179–197 | DOI | MR | Zbl

[25] Andrew Wiles Modular elliptic curves and Fermat’s last theorem, Ann. Math., Volume 141 (1995) no. 3, pp. 443-551 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique