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Refined effective bounds for Bloch-Kato Selmer groups associated to hyperelliptic curves (2502.11154v2)

Published 16 Feb 2025 in math.NT

Abstract: We develop refined methods to effectively bound the dimension of Bloch-Kato Selmer groups associated to the higher Chow group $\mathrm{CH}2(J,1)$, where $J$ is the Jacobian of a hyperelliptic curve $X$. This extends the recent work of Dogra on explicit $2$-descent for these Selmer groups to include cases where $X$ does not have a rational Weierstrass point. Additionally, we develop methods for obtaining sharper dimension bounds under the assumption that $X$ has good ordinary reduction at $2$. As a consequence, we establish new criteria for deducing finiteness of the depth $2$ Chabauty-Kim set $X(\mathbb{Q}_2)_2$, and demonstrate the efficacy of these criteria on curves from the LMFDB.

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