Refined effective bounds for Bloch-Kato Selmer groups associated to hyperelliptic curves

TL;DR

This paper introduces refined techniques to effectively bound the dimensions of Bloch-Kato Selmer groups for hyperelliptic curves, extending previous work and providing new criteria for finiteness of certain rational points.

Contribution

It extends explicit 2-descent methods to hyperelliptic curves without rational Weierstrass points and develops sharper bounds assuming good ordinary reduction at 2.

Findings

New bounds for Bloch-Kato Selmer groups of hyperelliptic Jacobians.

Criteria for finiteness of the Chabauty-Kim set $X(Q)_2$.

Demonstrated effectiveness on curves from the LMFDB.

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.