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math.NT2026

Explicit -torsion in the Tate-Shafarevich groups of genus Jacobians

Sam Frengley

Let be a genus curve whose Jacobian has real multiplication by a quadratic order in which splits. We describe an algorithm which outputs twist…

math.NT2026

Galois groups of low dimensional abelian varieties over finite fields

Santiago Arango-Piñeros, Sam Frengley, Sameera Vemulapalli

We consider three isogeny invariants of abelian varieties over finite fields: the Galois group, Newton polygon, and the angle rank. Motivated by work of Dupuy, Kedlaya, and Zureick…

math.NT2026

Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups

Sam Frengley, Dylan Laird

We exhibit examples of geometrically simple abelian surfaces with conductor bounded by whose Tate--Shafarevich groups contain a subgroup isomorphic to…

math.NT2025

Galois groups of simple abelian varieties over finite fields and exceptional Tate classes

Santiago Arango-Piñeros, Sam Frengley, Sameera Vemulapalli

We prove new cases of the Tate conjecture for abelian varieties over finite fields, extending previous results of Dupuy--Kedlaya--Zureick-Brown, Lenstra--Zarhin, Tankeev, and Zarhi…

math.NT2024

Generic models for genus 2 curves with real multiplication

Alex Cowan, Sam Frengley, Kimball Martin

Explicit models of families of genus 2 curves with multiplication by are known for . We obtain generic models for genus 2 curves over with real mu…