Ceresa cycles of bielliptic Picard curves

Jef Laga*, Ari Shnidman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We prove that the Chow class Κ(Ct) of the Ceresa cycle of the genus-three curve Ct: y3 = x4 + 2tx2 + 1 is torsion if and only if Qt = (3√t2 - 1, t) is a torsion point on the elliptic curve y2 = x3 + 1. In particular, there are infinitely many plane quartic curves over ℂ with torsion Ceresa cycle. Over ℚ̅, we show that the Beilinson–Bloch height of Κ(Ct) is proportional to the Néron–Tate height of Qt. Thus the height of Κ(Ct) is nondegenerate and satisfies a Northcott property. To prove all this, we show that the Chow motive that controls Κ(Ct) is isomorphic to h1 of an appropriate elliptic curve.

Original languageEnglish
JournalJournal fur die Reine und Angewandte Mathematik
DOIs
StateAccepted/In press - 2025

Bibliographical note

Publisher Copyright:
© 2025 the author(s), published by De Gruyter.

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