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Victoria Cantoral-Farfán

Publications and source records attributed to Victoria Cantoral-Farfán.

7 recordsLinked to original sources
arXiv2026

A Pila--Wilkie theorem for Hensel minimal curves

Recently, a new axiomatic framework for tameness in henselian valued fields was developed by Cluckers, Halupczok, Rideau-Kikuchi and Vermeulen and termed Hensel minimality. In this article we develop Diophantine applications of Hensel minimality. We prove a Pila--Wilkie type theorem for transcendental curves definable in Hensel minimal structures. In order to do so, we introduce a new notion of point counting in this context related to dimension counting over the residue field. We examine multiple classes of examples, showcasing the need for this new dimension counting and prove that our bounds are optimal.

math.NT↗
arXiv2023

Monodromy groups of Jacobians with definite quaternionic multiplication

Let $A$ be an abelian variety over a number field. The connected monodromy field of $A$ is the minimal field over which the images of all the $\ell$-adic torsion representations have connected Zariski closure. We show that for all even $g \geq 4$, there exist infinitely many geometrically nonisogenous abelian varieties $A$ over $\mathbb{Q}$ of dimension $g$ where the connected monodromy field is strictly larger than the field of definition of the endomorphisms of $A$. Our construction arises from explicit families of hyperelliptic Jacobians with definite quaternionic multiplication.

math.NT↗
arXiv2020

Fields of definition of elliptic fibrations on covers of certain extremal rational elliptic surfaces

We study K3 surfaces over a number field $k$ which are double covers of extremal rational elliptic surfaces. We provide a list of all elliptic fibrations on certain K3 surfaces together with the degree of a field extension over which each genus one fibration is defined and admits a section. We show that the latter depends, in general, on the action of the cover involution on the fibers of the genus 1 fibration.

math.AG↗
arXiv2019

Torsion for abelian varieties of type III

Let $A$ be an abelian variety defined over a number field $K$. The number of torsion points that are rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$ of $L$ over $K$. Under the following three conditions, we compute the optimal exponent for this bound, in terms of the dimension of abelian subvarieties and their endomorphism rings. The three hypothesis are the following: $(1)$ $A$ is geometrically isogenous to a product of simple abelian varieties of type I, II or III, according to the Albert classification; $(2)$ $A$ is of "Lefschetz type", that is, the Mumford-Tate group is the group of symplectic or orthogonal similitudes which commute with the endomorphism ring; $(3)$ $A$ satisfies the Mumford-Tate conjecture. This result is unconditional for a product of simple abelian varieties of type I, II or III with specific relative dimensions. Further, building on work of Serre, Pink, Banaszak, Gajda and Krasoń, we also prove the Mumford-Tate conjecture for a few new cases of abelian varieties of Lefschetz type.

math.NT↗
arXiv2018

Chabauty-Coleman experiments for genus 3 hyperelliptic curves

We describe a computation of rational points on genus 3 hyperelliptic curves $C$ defined over $\mathbb{Q}$ whose Jacobians have Mordell-Weil rank 1. Using the method of Chabauty and Coleman, we present and implement an algorithm in Sage to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in $C(\mathbb{Q})$.

math.NT↗
arXiv2017

The twisting Sato-Tate group of the curve $y^2 = x^{8} - 14x^4 + 1$

We determine the twisting Sato-Tate group of the genus $3$ hyperelliptic curve $y^2 = x^{8} - 14x^4 + 1$ and show that all possible subgroups of the twisting Sato-Tate group arise as the Sato-Tate group of an explicit twist of $y^2 = x^{8} - 14x^4 + 1$. Furthermore, we prove the generalized Sato-Tate conjecture for the Jacobians of all $\mathbb Q$-twists of the curve $y^2 = x^{8} - 14x^4 + 1$.

math.NT↗