Local heights on hyperelliptic curves and quadratic Chabauty

TL;DR

The paper advances quadratic Chabauty by providing a practical algorithm to compute Nekovár̆ local $p$-adic heights $h_{Z,\ell}$ at odd primes $\ell\neq p$ for hyperelliptic curves of genus $g>1$, thereby broadening the class of curves amenable to this method. The key innovation is to express $h_{Z,\ell}$ via the Coleman–Iovita isomorphism, relating the action of a trace-zero correspondence on $\mathrm{H}^1_{\mathrm{dR}}(X/K)$ to its action on the reduction-graph homology $\mathrm{H}_1(\Gamma,\mathbb{Z})$, and to verify push-pull compatibilities with correspondences. The authors combine cluster pictures and semistable coverings to construct explicit semistable data for hyperelliptic curves, then compute the required actions and traces to obtain exact local heights, with a Magma implementation illustrating numerous examples including genus-7 modular curves. This yields a road map for computing local heights on non-hyperelliptic curves and enables new quadratic-Chabauty computations, including curves with multiple primes of bad reduction and non-trivial local heights. The work thus enhances the practical reach and reliability of quadratic Chabauty in higher genus and broader reduction types, facilitating explicit rational-point determinations in previously intractable cases.

Abstract

Quadratic Chabauty is a $p$-adic method for determining rational points on curves. Local heights are arithmetic invariants used in the quadratic Chabauty method. We present an algorithm to compute these local heights for hyperelliptic curves at odd primes $\ell \neq p$. This algorithm significantly broadens the applicability of quadratic Chabauty to curves which were previously inaccessible due to the presence of non-trivial local heights. We provide numerous examples, including the first quadratic Chabauty computation for a curve having two primes with non-trivial local heights.

Figures (20)

• Figure 1: The wide open described in \ref{['ex:wide_open_in_projective_line']} for $n = 2$. The open discs $D^o$, $D_1^o$, and $D_1^o$ are represented by the dashed lines; the closed subdiscs $D^c$, $D_1^c$, and $D_2^c$ are represented by the solid lines. The open annuli $A$, $A_1$, and $A_2$ are the lightly shaded regions between these open discs and their closed subdiscs, the wide open $U$ is the whole shaded region, and its underlying affinoid $U_0$ is the heavily shaded region.
• Figure 2: Caricature of a semistable covering and its attached graph (\ref{['def:graph_from_covering']}).
• Figure 3: Subdividing the edge $e_{02}$ in the proof of \ref{['lem:coleman-iovita_indept_of_covering']}.
• Figure 4: A picture of the analytic space $U_\mathfrak{s}$.
• Figure 5: Cluster, Berkovich skeleton, special fibre, and reduction map

Theorems & Definitions (136)

• Theorem 1: cf. § \ref{['sec:QC']}
• Theorem 2: Push-pull compatibility of Coleman--Iovita, cf. \ref{['prop:coleman-iovita_functoriality']}
• Theorem 3: cf. \ref{['thm:semistablecoverX']}
• Theorem 4: Boundedness of norms and traces, cf. \ref{['thm:bounds']}
• Proposition 5: cf. \ref{['ex:trivialHeights']}
• Corollary 6: cf. §\ref{['subsec:shimuracurve']}
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4