Computing the Cassels-Tate Pairing on the Selmer group of a Richelot Isogeny
Jiali Yan1
1 95, Finborough Road SW10 9DU, London, UK
Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 659-674.

Dans cet article, nous étudions l’accouplement de Cassels-Tate sur les jacobiennes des courbes de genre 2 possédant une isogénie dite de Richelot. Soit ϕ:J→J ^ une isogénie de Richelot entre les jacobiennes de deux courbes de genre 2. Nous donnons une formule explicite et un algorithme pratique pour calculer l’accouplement de Cassels-Tate sur Sel ϕ ^ (J ^)×Sel ϕ ^ (J ^) où ϕ ^ est l’isogénie duale de ϕ. Ces résultats sont obtenus sous l’hypothèse simplificatrice que tous les points de 2-torsion sur J sont définis sur K. Nous donnons un exemple explicite qui montre que nous pouvons transformer la descente par l’isogénie de Richelot en 2-descente en calculant l’accouplement de Cassels-Tate.

In this paper, we study the Cassels-Tate pairing on Jacobians of genus two curves admitting a special type of isogenies called Richelot isogenies. Let ϕ:J→J ^ be a Richelot isogeny between two Jacobians of genus two curves. We give an explicit formula as well as a practical algorithm to compute the Cassels-Tate pairing on Sel ϕ ^ (J ^)×Sel ϕ ^ (J ^) where ϕ ^ is the dual isogeny of ϕ. The formula and algorithm are under the simplifying assumption that all two-torsion points on J are defined over K. We also include a worked example demonstrating we can turn the descent by Richelot isogeny into a 2-descent via computing the Cassels-Tate pairing.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1259
Classification : 11N56, 14G42
Mots clés : Genus 2 Curve, Cassels-Tate Pairing, Richelot isogeny
Jiali Yan 1

1 95, Finborough Road SW10 9DU, London, UK
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JTNB_2023__35_3_659_0,
     author = {Jiali Yan},
     title = {Computing the {Cassels-Tate} {Pairing} on the {Selmer} group of a {Richelot} {Isogeny}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {659--674},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {35},
     number = {3},
     year = {2023},
     doi = {10.5802/jtnb.1259},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1259/}
}
TY  - JOUR
AU  - Jiali Yan
TI  - Computing the Cassels-Tate Pairing on the Selmer group of a Richelot Isogeny
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2023
SP  - 659
EP  - 674
VL  - 35
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1259/
DO  - 10.5802/jtnb.1259
LA  - en
ID  - JTNB_2023__35_3_659_0
ER  - 
%0 Journal Article
%A Jiali Yan
%T Computing the Cassels-Tate Pairing on the Selmer group of a Richelot Isogeny
%J Journal de théorie des nombres de Bordeaux
%D 2023
%P 659-674
%V 35
%N 3
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1259/
%R 10.5802/jtnb.1259
%G en
%F JTNB_2023__35_3_659_0
Jiali Yan. Computing the Cassels-Tate Pairing on the Selmer group of a Richelot Isogeny. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 659-674. doi : 10.5802/jtnb.1259. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1259/

[1] Amod Agashe; William Stein Visibility of Shafarevich–Tate Groups of Abelian Varieties, J. Number Theory, Volume 97 (2002) no. 1, pp. 171-185 | DOI | MR | Zbl

[2] Monique van Beek Computing the Cassels–Tate Pairing, Ph. D. Thesis, University of Cambridge (2015)

[3] Monique van Beek; Tom A. Fisher Computing the Cassels–Tate pairing on 3-isogeny Selmer groups via cubic norm equations, Acta Arith., Volume 185 (2018) no. 4, pp. 367-396 | DOI | MR | Zbl

[4] 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 | DOI | MR | Zbl

[5] Nils Bruin; Kevin Doerksen The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians, Can. J. Math., Volume 63 (2011) no. 5, pp. 992-1024 | DOI | MR | Zbl

[6] John W. S. Cassels Arithmetic on Curves of Genus 1. I. On a conjecture of Selmer, J. Reine Angew. Math., Volume 202 (1959), pp. 52-99 | DOI | MR | Zbl

[7] John W. S. Cassels J. W. S. Cassels, Arithmetic on curves of genus 1, IV. Proof of the Hauptvermutung., J. Reine Angew. Math., Volume 211 (1962), pp. 95-112 | DOI | Zbl

[8] John W. S. Cassels Second Descents for Elliptic Curves, J. Reine Angew. Math., Volume 494 (1998), pp. 101-127 | DOI | MR | Zbl

[9] John W. S. Cassels; E. V. Flynn Prolegomena to a MiddleBrow Arithmetic of Curves of Genus 2, London Mathematical Society Lecture Note Series, 230, Cambridge University Press, 1996 | DOI

[10] S. Donnelly Algorithms for the Cassels–Tate pairing (2015) (preprint)

[11] Tom A. Fisher The Cassels–Tate pairing and the Platonic solids, J. Number Theory, Volume 98 (2003) no. 1, pp. 105-155 | DOI | MR | Zbl

[12] Tom A. Fisher On binary quartics and the Cassels–Tate pairing (2016) (preprint)

[13] Tom A. Fisher; Rachel Newton Computing the Cassels–Tate pairing on the 3-Selmer group of an elliptic curve, J. Number Theory, Volume 10 (2014) no. 7, p. 1881­1907 | DOI | MR | Zbl

[14] Eugene V. Flynn Arbitrarily Large Tate–Shafarevich Group on Abelian Surfaces, J. Number Theory, Volume 186 (2018), pp. 248-258 | DOI | MR | Zbl

[15] Philippe Gille; Tamás Szamuely Central Simple Algebras and Galois Cohomology, Cambridge Studies in Advanced Mathematics, 101, Cambridge University Press, 2006 | DOI

[16] John S. Milne Abelian Varieties, Arithmetic geometry, Springer, 1986, pp. 103-150 | DOI | Zbl

[17] John S. Milne Arithmetic Duality Theorems, BookSurge, 2006, viii+339 pages

[18] John S. Milne Abelian Varieties, 2008, 166+vi pages

[19] Bjorn Poonen; Michael Stoll The Cassels–Tate pairing on polarized abelian varieties, Ann. Math., Volume 150 (1999) no. 3, pp. 1109-1149 | DOI | MR | Zbl

[20] Edward F. Schaefer 2-Descent on the Jacobians of Hyperelliptic Curves, J. Number Theory, Volume 51 (1995) no. 2, pp. 219-232 | DOI | MR | Zbl

[21] Jean-Pierre Serre Local Fields, Graduate Texts in Mathematics, 67, Springer, 1979 | DOI

[22] Katsuyuki Takashima; Reo Yoshida An algorithm for computing a sequence of Richelot isogenies, Bull. Korean Math. Soc., Volume 46 (2009) no. 4, pp. 789-802 | DOI | MR | Zbl

[23] John Tate Duality theorems in Galois cohomology over number fields, Proc. Int. Congr. Math., Stockholm 1962,, 1963, pp. 288-295 | Zbl

[24] Jiali Yan Computing the Cassels–Tate Pairing for Jacobian Varieties of Genus Two Curves, Ph. D. Thesis, University of Cambridge (2021) | DOI

Cité par Sources :