Arithmetic rank bounds for abelian varieties over function fields

TL;DR

The paper develops arithmetic rank bounds for abelian varieties over function fields of curves, extending the GO-S geometric bound to an ℓ-descent framework and yielding bounds $rk_\mathbb{Z} A(k(B)) \le \dim_{\mathbb{F}_\ell} H^1(B,\mathcal{A}[\ell]) + \dim_{\mathbb{F}_\ell} H^0(B,\Phi/\ell\Phi)$ that can be refined in an arithmetic sense. It proves Galois-equivariance of the bound under suitable no-rational-torsion hypotheses and shows that, when $k=\bar{k}$, the arithmetic bound recovers the geometric bound; a key component is the cohomology of $\mathcal{A}[\ell]$ and the Picard groups of curves. The authors develop a higher-dimensional ℓ-descent machinery, including a big monodromy framework, and apply it to Jacobians of hyperelliptic curves via a 2-descent map $\phi_2$, giving explicit connections to theta characteristics and root lattices like $E_8$ and $E_6^*$. They further study explicit genus $2$ families $C: y^2=g(t)+x^d$ with $d=5,6$, obtaining sharp rank bounds, detailed descent correspondences, and illustrating distinct behaviors in the odd vs even degree cases, thereby linking arithmetic geometry with lattice-theoretic Mordell-Weil structures.

Abstract

It follows from the Grothendieck-Ogg-Shafarevich formula that the rank of an abelian variety (with trivial trace) defined over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on bad reduction data. Using a function field version of classical $\ell$-descent techniques, we derive an arithmetic refinement of this bound, extending previous work of the second and third authors from elliptic curves to abelian varieties, and improving on their result in the case of elliptic curves. When the abelian variety is the Jacobian of a hyperelliptic curve, we produce a more explicit $2$-descent map. Then we apply this machinery to studying points on the Jacobians of certain genus $2$ curves over $k(t)$, where $k$ is some perfect base field of characteristic not $2$.

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.2
• Proposition 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof