Ceresa cycles of bielliptic Picard curves

Jef Laga; Ari Shnidman

doi:10.1515/crelle-2024-0102

Ceresa cycles of bielliptic Picard curves

Jef Laga, Ari Shnidman

Einstein Institute of Mathematics

The Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that the Chow class Κ_∞(C_t) of the Ceresa cycle of the genus-three curve C_t: y³ = x⁴ + 2tx² + 1 is torsion if and only if Q_t = (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_∞(C_t) is proportional to the Néron–Tate height of Q_t. Thus the height of Κ_∞(C_t) is nondegenerate and satisfies a Northcott property. To prove all this, we show that the Chow motive that controls Κ_∞(C_t) is isomorphic to h¹ of an appropriate elliptic curve.

Original language	English
Journal	Journal fur die Reine und Angewandte Mathematik
DOIs	https://doi.org/10.1515/crelle-2024-0102
State	Accepted/In press - 2025

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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10.1515/crelle-2024-0102

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title = "Ceresa cycles of bielliptic Picard curves",

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 N{\'e}ron–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.",

author = "Jef Laga and Ari Shnidman",

note = "Publisher Copyright: {\textcopyright} 2025 the author(s), published by De Gruyter.",

year = "2025",

doi = "10.1515/crelle-2024-0102",

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T1 - Ceresa cycles of bielliptic Picard curves

AU - Laga, Jef

AU - Shnidman, Ari

PY - 2025

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N2 - 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.

AB - 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.

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U2 - 10.1515/crelle-2024-0102

DO - 10.1515/crelle-2024-0102

M3 - مقالة

SN - 0075-4102

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

ER -

Ceresa cycles of bielliptic Picard curves

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