Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians

TL;DR

This paper investigates the arithmetic invariants of Jacobian varieties of certain hyperelliptic curves, revealing complex high-dimensional phenomena and providing explicit computations for their endomorphism fields and monodromy fields.

Contribution

It computes the decomposition, endomorphism field, and monodromy field of Jacobians of Fermat hyperelliptic curves, offering new examples of intricate algebraic and arithmetic structures.

Findings

Decomposition of Jacobians into simple abelian varieties

Explicit determination of endomorphism fields

Identification of non-trivial monodromy extensions

Abstract

Denote by $J_m$ the Jacobian variety of the hyperelliptic curve defined by the affine equation $y^2=x^m+1$ over $\mathbb{Q}$, where $m \geq 3$ is a fixed positive integer. We compute several interesting arithmetic invariants of $J_m$: its decomposition up to isogeny into simple abelian varieties, the minimal field $\mathbb{Q}(\operatorname{End}(J_m))$ over which its endomorphisms are defined, and its connected monodromy field $\mathbb{Q}(\varepsilon_{J_m})$. Currently, there is no general algorithm that computes the last invariant. For large enough values of $m$, the abelian varieties $J_m$ provide non-trivial examples of high-dimensional phenomena, such as degeneracy and the non-triviality of the extension $\mathbb{Q}(\varepsilon_{J_m})/\mathbb{Q}(\operatorname{End}(J_m))$.