arXiv subjects

Andrea Gallese

Publications and source records attributed to Andrea Gallese.

5 recordsLinked to original sources

arXiv2026

Finding the complement of an elliptic curve inside a Jacobian

This note gives a simple algorithm for the following effectivity problem: given a genus $2$ curve $X$ together with a nonconstant map $\pi:X\to E$ to an elliptic curve, determine an elliptic curve $E'$ and a map $\pi':X\to E'$ independent of $\pi$. Equivalently, we compute the complementary elliptic factor in the decomposition of $\operatorname{Jac}(X)$ up to isogeny. While the problem has been studied extensively, and more general ones have been solved by deep and powerful techniques, we are not aware of a reference for the simple explicit procedure described here.

math.NT↗

arXiv2025

Connected monodromy fields of Jacobians with complex multiplication

We describe an algorithm to compute the minimal field of definition of the Tate classes on powers of a Jacobian $J$ with potential complex multiplication. This field arises as a natural invariant of the Galois representations attached to $J$. We also give closed formulas expressing the periods of anti-holomorphic differential forms on $J$ in terms of the periods of the holomorphic ones.

math.NT↗

arXiv2025

Monodromy groups and exceptional Hodge classes, II: Sato-Tate groups

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. In this paper, we compute the Sato-Tate group of $J_m$. Currently, there is no general algorithm that computes this invariant. We also describe the Sato-Tate group of an abelian variety, generalizing existing results that apply only to non-degenerate varieties, and prove an extension of a well-known formula of Gross-Koblitz that relates values of the classical and $p$-adic gamma functions at rational arguments.

math.NT↗

arXiv2024

How to split two-dimensional Jacobians: a geometric construction

Let $\pi\colon Y \to X$ be a branched cover of algebraic curves. Assume that there exists a curve $W$ such that $\operatorname{Jac} Y \sim \operatorname{Jac} X \times \operatorname{Jac} W$. We conjecture that every such isogeny decomposition is induced by an algebraic correspondence of curves that fits in a Galois diagram, and we prove this conjecture when $g(Y)=2$ and $g(X)=1$. Our proof yields a geometric construction of the complementary curve $W$, an explicit correspondence inducing the isogeny, and a general criterion for deciding when an algebraic correspondence of curves fits in a Galois diagram (admits a push-out).

math.AG↗

arXiv2024

Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians

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 $\mathbb{Q}(\varepsilon_{J_m})/\mathbb{Q}(\operatorname{End}(J_m))$.

math.NT↗