TY - JOUR
T1 - Ceresa cycles of bielliptic Picard curves
AU - Laga, Jef
AU - Shnidman, Ari
N1 - Publisher Copyright:
© 2025 the author(s), published by De Gruyter.
PY - 2025
Y1 - 2025
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.
UR - http://www.scopus.com/inward/record.url?scp=85216363265&partnerID=8YFLogxK
U2 - 10.1515/crelle-2024-0102
DO - 10.1515/crelle-2024-0102
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AN - SCOPUS:85216363265
SN - 0075-4102
JO - Journal fur die Reine und Angewandte Mathematik
JF - Journal fur die Reine und Angewandte Mathematik
ER -