Infinitely many hyperelliptic curves of small genus and small fixed rank, and of any genus and rank two

TL;DR

The paper develops a general gluing framework to produce infinitely many non‑isomorphic hyperelliptic curves of fixed genus $g\ge2$ over number fields whose Jacobians have rank in {0,1,2}. By decomposing Jacobians as products of lower‑genus Jacobians and applying quadratic twists, it constructs explicit genus‑2 families with prescribed ranks over arbitrary $K$, and extends the method to higher genus by gluing two hyperelliptic curves to form genus $g$ curves whose Jacobians are isogenous to products of the factors. Over $\mathbb{Q}$, it provides explicit infinite families for genus 2 with ranks up to 11, genus 3–4 with ranks 1–4, and genus 5–6 with ranks 1–3, and it gives higher‑genus examples and general results for any number field. The work also connects to recent breakthroughs (KM25, Smith2, KP25) to realize ranks 0,1,2 in broad settings, and, remarkably, confirms the existence of infinitely many absolutely simple Jacobians with fixed rank (notably rank 1) via KM25. Together, these results contribute to arithmetic statistics of Jacobian ranks in infinite families and provide explicit, computable families across genus ranges.

Abstract

We prove that for any number field $K$ and any fixed genus $g \geq 2$, there are infinitely many non-isomorphic hyperelliptic curves of genus $g$ over $K$ whose Jacobians have rank over $K$ equal to each of 0, 1, or 2. As an example of our method, over $\mathbb{Q}$, we prove that there exist infinitely many non-isomorphic hyperelliptic curves of genus two, whose Jacobians have rank equal to a fixed number between $1$ and $11$, genus three and four curves with rank between $1$ and $4$, and genus five and six with rank between $1$ and $3$.

Theorems & Definitions (35)

• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Remark 2.3
• Remark 2.4
• Theorem 2.5
• proof
• Theorem 2.6
• Theorem 2.7