A. Shnidman

activity

20242026

collaborators

7 papers

math.NT2026

Rational points on modular curves: parameterization and geometric explanations

Maarten Derickx, Sachi Hashimoto, Filip Najman +1

We show that, conditional on Zywina's effective version of the Serre uniformity conjecture, there is a natural way to parameterize non-CM $Q$ -rational points on all modula…

math.NT2025

Hecke reciprocity and class groups

Ari Shnidman, Artane Siad

We compute the average size of $Cl_{F} [2]$ in the family of cubic fields $F = Q (3 n)$ . Specifically, as $F$ varies over the subfamily of wildly (resp. tamel…

math.NT2025

Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field

Levent AlpÃ¶ge, Manjul Bhargava, Wei Ho +1

We show that for any quadratic extension of number fields $K / F$ , there exists an abelian variety $A / F$ of positive rank whose rank does not grow upon base change to $K$ . This resul…

math.AG2024

Ceresa cycles of bielliptic Picard curves

Jef Laga, Ari Shnidman

We show that the Ceresa cycle $I^º (C_{t})$ of the genus $3$ curve $C_{t} : y^{3} = x^{4} + 2 t x^{2} + 1$ is torsion if and only if $Q_{t} = (3 t^{2} - 1, t)$ is a torsion point on the el…

math.AG2024

The geometry and arithmetic of bielliptic Picard curves

Jef Laga, Ari Shnidman

We study the geometry and arithmetic of the curves $C : y^{3} = x^{4} + a x^{2} + b$ and their associated Prym abelian surfaces $P$ . We prove a Torelli theorem in this context and gi…

math.NT2024

Arbitrarily large $p$ -torsion in Tate-Shafarevich groups

E. Victor Flynn, Ari Shnidman

We show that, for any prime $p$ , there exist absolutely simple abelian varieties over $Q$ with arbitrarily large $p$ -torsion in their Tate-Shafarevich group. To prove this…