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 $n\le 3$ and $N\le 25$, 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 $n\le 3$ et $N\le 25$, 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.
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Keywords: split Jacobian, elliptic curve, isogeny, binary quadratic form
Martin Djukanović 1
CC-BY-ND 4.0
Martin Djukanović. Families of split Jacobians with isogenous components. Journal de théorie des nombres de Bordeaux, Volume 37 (2025) no. 1, pp. 49-77. doi: 10.5802/jtnb.1312
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author = {Martin Djukanovi\'c},
title = {Families of split {Jacobians} with isogenous components},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {49--77},
year = {2025},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {37},
number = {1},
doi = {10.5802/jtnb.1312},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1312/}
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