Involutions of curves in abelian surfaces and their Jacobians

TL;DR

This paper develops a systematic construction of curves $C$ in a complex abelian surface $A$ with a primitive $(1,d)$-polarization, obtained as preimages of genus $2$ curves under quotients by subgroups $X\le K(L)$, and analyzes how the induced involutions $[-1]$ and translations $t_x$ shape ramification and automorphisms. By relating fixed-point data of involutions to symmetric line bundles and their theta structures, the authors derive precise Jacobian decompositions via Kani–Rosen, including new isogeny relations for cyclic and $(\mathbb{Z}/p\mathmathbb{Z})^2$-type covers and proofs of complete decomposability in new cases. They classify hyperelliptic curves in the linear systems $|L|$ for $d\le4$, describe when such curves arise as covers of genus $2$ curves, and show how, in many settings, $J_C$ splits into products involving $A$ and Jacobians of quotients, sometimes yielding elliptic factors. Overall, the work extends Ekedahl–Serre’s program to broader families of curves on abelian surfaces, providing explicit decompositions and hyperelliptic-curve counts that illuminate the structure of Jacobians in higher-dimensional settings and offering constructive paths to completely decomposable Jacobians.

Abstract

We examine étale covers of genus two curves that occur in the linear system of a polarizing line bundle of type $(1,d)$ on a complex abelian surface. We give results counting fixed points of involutions on such curves as well as decomposing their Jacobians into isogenous products.

Theorems & Definitions (32)

• Theorem 1: \ref{['cyclic']}
• Theorem 2: \ref{['prime']}
• Theorem 3: \ref{['odd']}
• Theorem 4: Propositions \ref{['cyclic.fix']}, \ref{['twelve']}
• Remark 1.2
• Lemma 1.3
• proof
• Remark 1.4
• Proposition 2.1: BirkenhakeLange
• Lemma 2.2