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arXiv · 2607.18906

Zero-cycles on surfaces dominated by products of hyperelliptic curves

Jean-Louis Colliot-Thélène·Federico Scavia·Alexei Skorobogatov
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Abstract

Conditionally on the finiteness of the relevant Tate-Shafarevich groups, we prove a local-to-global result for zero-cycles of degree $1$ on the surfaces given by $y^2 = f_1(x_1)f_2(x_2)$, where the polynomials $f_1$ and $f_2$ are algebraically general. The proof combines the fibration method, parity results for $2$-Selmer groups in quadratic twist families, and a variant of a theorem of Morgan on the variation of the Cassels-Tate pairing.

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Jean-Louis Colliot-Thélène, Federico Scavia, Alexei Skorobogatov. 2026-07-21. Zero-cycles on surfaces dominated by products of hyperelliptic curves. https://arxiv.org/abs/2607.18906

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