Families of split Jacobians with isogenous components
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 49-77.

We exhibit sufficient and necessary conditions under which, over a field of characteristic zero, an N-isogeny between two elliptic curves without complex multiplication induces curves C of genus two whose Jacobian is (n,n)-isogenous to the product of the two elliptic curves. For n3 and N25, we also present a partial list of one-dimensional families of curves C whose Jacobian is (n,n)-split and has N-isogenous components.

Nous présentons des conditions nécessaires et suffisantes sous lesquelles, sur un corps de caractéristique zéro, une N-isogénie entre deux courbes elliptiques sans multiplication complexe induit des courbes C de genre deux dont la jacobienne est (n,n)-isogène au produit des deux courbes elliptiques. Pour n3 et N25, nous présentons aussi une liste partielle de familles unidimensionnelles des courbes C dont la jacobienne est (n,n)-décomposée et a des composantes qui sont N-isogènes.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1312
Classification : 14H10, 14H40, 14H52, 14K02
Keywords: split Jacobian, elliptic curve, isogeny, binary quadratic form

Martin Djukanović 1

1 Groningen, The Netherlands
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{JTNB_2025__37_1_49_0,
     author = {Martin Djukanovi\'c},
     title = {Families of split {Jacobians} with isogenous components},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {49--77},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {37},
     number = {1},
     year = {2025},
     doi = {10.5802/jtnb.1312},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1312/}
}
TY  - JOUR
AU  - Martin Djukanović
TI  - Families of split Jacobians with isogenous components
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2025
SP  - 49
EP  - 77
VL  - 37
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1312/
DO  - 10.5802/jtnb.1312
LA  - en
ID  - JTNB_2025__37_1_49_0
ER  - 
%0 Journal Article
%A Martin Djukanović
%T Families of split Jacobians with isogenous components
%J Journal de théorie des nombres de Bordeaux
%D 2025
%P 49-77
%V 37
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1312/
%R 10.5802/jtnb.1312
%G en
%F JTNB_2025__37_1_49_0
Martin Djukanović. Families of split Jacobians with isogenous components. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 49-77. doi : 10.5802/jtnb.1312. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1312/

[1] Lubjana Beshaj; Artur Elezi; Tony Shaska Isogenous components of Jacobian surfaces, Eur. J. Math., Volume 6 (2020) no. 4, pp. 1276-1302 | DOI | MR | Zbl

[2] Alex J. Best; Jonathan Bober; Andrew R. Booker; Edgar Costa; John E. Cremona; Maarten Derickx; Min Lee; David Lowry-Duda; David Roe; Andrew V. Sutherland; John Voight Computing classical modular forms, Arithmetic geometry, number theory, and computation (Simons Symposia), Springer, 2021, pp. 131-213 | DOI | Zbl

[3] Modular functions of one variable. IV. Proceedings of the International Summer School on Modular Functions of One Variable and Arithmetical Applications, RUCA, University of Antwerp, Antwerp, July 17–August 3, 1972 (Bryan J. Birch; Willem Kuyk, eds.), Lecture Notes in Mathematics, 476, Springer, 1975

[4] Oskar Bolza On Binary Sextics with Linear Transformations into Themselves, Am. J. Math., Volume 10 (1887) no. 1, pp. 47-70 | DOI | MR | Zbl

[5] Wieb Bosma; John J. Cannon; Catherine Playoust The Magma Algebra System I: The Use Language, J. Symb. Comput., Volume 24 (1997), pp. 235-265 (Magma’s homepage is at http://magma.maths.usyd.edu.au/magma) | DOI | MR | Zbl

[6] 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

[7] John W. S. Cassels; Eugene V. Flynn Prolegomena to a middlebrow arithmetic of curves of genus 2, London Mathematical Society Lecture Note Series, 230, Cambridge University Press, 1996 | DOI | Zbl

[8] David A. Cox Primes of the form x2+ny2: Fermat, class field theory, and complex multiplication, AMS Chelsea Publishing, 387, American Mathematical Society, 2022 | DOI | Zbl

[9] Leonard E. Dickson Introduction to the Theory of Numbers, University of Chicago Press, 1929 | Zbl

[10] Martin Djukanović split-jacobians (GitHub repository, https://github.com/martin-djukanovic/split-jacobians)

[11] Martin Djukanović Families of (3,3)-split Jacobians, Rocky Mt. J. Math., Volume 54 (2024) no. 6, pp. 1621-1654 | DOI | MR | Zbl

[12] Gerhard Frey; Ernst Kani Curves of genus 2 covering elliptic curves and an arithmetical application, Arithmetic algebraic geometry (Texel, 1989) (Progress in Mathematics), Volume 89, Birkhäuser, 1991, pp. 153-176 | DOI | MR | Zbl

[13] Everett W. Howe; Franck Leprévost; Bjorn Poonen Large torsion subgroups of split Jacobians of curves of genus two or three, Forum Math., Volume 12 (2000) no. 3, pp. 315-364 | DOI | MR | Zbl

[14] C. G. J. Jacobi Review of Legendre’s ‘Traité des fonctions elliptiques, troisième supplément’, J. Reine Angew. Math., Volume 8 (1832), pp. 413-417

[15] Ernst Kani The number of curves of genus two with elliptic differentials, J. Reine Angew. Math., Volume 485 (1997), pp. 93-121 | DOI | MR | Zbl

[16] Adolf Krazer Lehrbuch der Thetafunktionen, Chelsea Publishing Company, 1970 | Zbl

[17] Daniel S. Kubert Universal bounds on the torsion of elliptic curves, Compos. Math., Volume 38 (1979) no. 1, pp. 121-128 | DOI | Numdam | MR | Zbl

[18] Robert M. Kuhn Curves of genus 2 with split Jacobian, Trans. Am. Math. Soc., Volume 307 (1988) no. 1, pp. 41-49 | DOI | MR | Zbl

[19] Abhinav Kumar Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields, Res. Math. Sci., Volume 2 (2015), 24, 46 pages | DOI | MR | Zbl

[20] Serge Lang Elliptic functions, Graduate Texts in Mathematics, 112, Springer, 1987 | DOI | MR | Zbl

[21] Barry Mazur; Dorian Goldfeld Rational isogenies of prime degree, Invent. Math., Volume 44 (1978), pp. 129-162 | DOI | Zbl

[22] Bjorn Poonen Rational points on varieties, Graduate Studies in Mathematics, 186, American Mathematical Society, 2017 | DOI | MR | Zbl

Cité par Sources :