Hostname: page-component-f554764f5-c4bhq Total loading time: 0 Render date: 2025-04-20T23:59:28.930Z Has data issue: false hasContentIssue false
  • English
  • Français

DECOMPOSITION OF THE JACOBIAN OF SOME TWISTS OF A GENUS $2$ CURVE

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  04 October 2024

KEUNYOUNG JEONG Open the ORCID record for KEUNYOUNG JEONG [Opens in a new window] ,
YEONG-WOOK KWON Open the ORCID record for YEONG-WOOK KWON [Opens in a new window]  and
JUNYEONG PARK Open the ORCID record for JUNYEONG PARK [Opens in a new window]
KEUNYOUNG JEONG
Affiliation:
Department of Mathematics Education, Chonnam National University, 77, Yongbong-ro, Buk-gu, Gwangju 61186, Korea e-mail: [email protected]
YEONG-WOOK KWON
Affiliation:
Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, UNIST-gil 50, Ulsan 44919, Korea e-mail: [email protected]
JUNYEONG PARK*
Affiliation:
Department of Mathematics Education, Chonnam National University, 77, Yongbong-ro, Buk-gu, Gwangju 61186, Korea
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Cardona and Lario [‘Twists of the genus 2 curve $y^2 = x^6+1$’, J. Number Theory 209 (2020), 195–211] gave a complete classification of the twists of the curve $y^2 = x^6+1$. In this paper, we study the twists of the curve whose automorphism group is defined over a biquadratic extension of the rationals. If the twists are of type B or C in the Cardona–Lario classification, we find a pair of elliptic curves whose product is isogenous with the Jacobian of the twist.

Keywords

twistsJacobianL-function

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 15
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

K. Jeong was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2024-00341372). Y.-W. Kwon was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2022R1I1A1A01067581). J. Park was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA2001-02 and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2024-00449679).

References

Cardona, G., ‘ ${G}_k$ -groups and twists of the genus two curve ${y}^2={x}^5-x$ ’, J. Algebra 303(2) (2006), 707–721.10.1016/j.jalgebra.2006.02.005CrossRefGoogle Scholar
Cardona, G. and Lario, J.-C., ‘Twists of the genus 2 curve ${y}^2={x}^6+1$ ’, J. Number Theory 209 (2020), 195–211.10.1016/j.jnt.2019.08.017CrossRefGoogle Scholar
Cardona, G. and Quer, J., ‘Curves of genus 2 with group of automorphisms isomorphic to ${D}_8$ or ${D}_{12}$ ’, Trans. Amer. Math. Soc. 359(6) (2007), 2831–2849.10.1090/S0002-9947-07-04111-6CrossRefGoogle Scholar
Cassels, J. W. S. and Flynn, E. V., Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2, London Mathematical Society Lecture Note Series, 230 (Cambridge University Press, Cambridge, 1996).10.1017/CBO9780511526084CrossRefGoogle Scholar
Fité, F. and Sutherland, A. V., ‘Sato–Tate distribution of twists of ${y}^2={x}^5-x$ and ${y}^2={x}^6+1$ ’, Algebra Number Theory 8(3) (2014), 543–585.10.2140/ant.2014.8.543CrossRefGoogle Scholar
Silverman, J. H., Arithmetic of Elliptic Curves, 2nd edn, Graduate Texts in Mathematics, 106 (Springer, Dordrecht, 2009).10.1007/978-0-387-09494-6CrossRefGoogle Scholar