Ceresa cycles of bielliptic Picard curves

2023/12/20 by Laga, Jef, Shnidman, Ari · 2 citations
#14C25 (Primary) 14H40 #14H45 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2312.12965

Abstract

We show that the Ceresa cycle κ(C_t) of the genus 3 curve C_t \colon 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. This shows that there are infinitely many smooth plane quartic curves over ℂ (resp. ℚ) with torsion (resp. infinite order) Ceresa cycle. Over ℚ, we show that the Beilinson--Bloch height of κ(C_t) is proportional to the Neron--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 \mathfrakh¹ of an appropriate elliptic curve.

Cited by

• On Modified Diagonal Cycles and the Beauville Decomposition of the Ceresa Cycle
• The modified diagonal cycles of Hypergeometric curves

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