A. Shnidman

activity

20122026

most citedThe average size of the 3-isogeny Selmer groups of elliptic curves

y^{2} = x^{3} + k

23 citations · 108 across the 30 of their papers we have counts for

collaborators

32 papers

math.NT2026

Exceptional points on Atkin--Lehner quotients

Eran Assaf, Sachi Hashimoto, Ari Shnidman

We study the rational points on the star curve $X_{0}^{*} (N) := X_{0} (N) / W (N)$ , the quotient of the classical modular curve $X_{0} (N)$ by the full group of Atkin--Lehner involutions, for s…

math.NT2026

Rational torsion on simple genus two Jacobians

Jennifer S. Balakrishnan, Filip Najman, Ari Shnidman +1

We exhibit new subgroups of rational torsion points in geometrically simple Jacobians of genus-two curves over $Q$ . The largest group, which has order 96 and invariants [2,…

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★ 1 cited

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

Levent Alpöge, 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.NT2024

Derivatives of Rankin-Selberg $L$ -functions and heights of generalized Heegner cycles

David T. -B. G. Lilienfeldt, Ari Shnidman

Let $f$ be a newform of weight $2 k$ and let $χ$ be an unramified imaginary quadratic Hecke character of infinity type $(2 t, 0)$ , for some integer $0 < t \leq k - 1$ . We show that the…

math.AG2024★ 3 cited

Vanishing criteria for Ceresa cycles

Jef Laga, Ari Shnidman

Let $C$ be a smooth projective curve, and let $J$ be its Jacobian. We prove vanishing criteria for the Ceresa cycle $κ (C) \in CH_{1} (J) \otimes Q$ in the Chow group…

math.AG2023

Ceresa cycles of bielliptic Picard curves

Jef Laga, Ari Shnidman

We show that the Ceresa cycle $κ (C_{t})$ 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 ell…

math.NT2023★ 1 cited

Rational torsion points on abelian surfaces with quaternionic multiplication

Jef Laga, Ciaran Schembri, Ari Shnidman +1

Let $A$ be an abelian surface over $Q$ whose geometric endomorphism ring is a maximal order in a non-split quaternion algebra. Inspired by Mazur's theorem for elliptic cur…

math.AG2023★ 1 cited

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.NT2023

Normal distribution of bad reduction

Robert J. Lemke Oliver, Daniel Loughran, Ari Shnidman

We prove normal distribution laws for primes of bad semistable reduction in families of curves. As a consequence, we deduce that when ordered by height, $100%$ of curves in these…

math.NT2022★ 4 cited

Integers expressible as the sum of two rational cubes

Levent Alpöge, Manjul Bhargava, Ari Shnidman

We prove that a positive proportion of integers are expressible as the sum of two rational cubes, and a positive proportion are not so expressible, thus proving a conjecture of Dav…

math.NT2022

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…

math.NT2021★ 7 cited

Rank growth of elliptic curves over $N$ -th root extensions

Ari Shnidman, Ariel Weiss

Fix an elliptic curve $E$ over a number field $F$ and an integer $n$ which is a power of $3$ . We study the growth of the Mordell--Weil rank of $E$ after base change to the fields $…

math.NT2021★ 7 cited

Experiments with Ceresa classes of cyclic Fermat quotients

David T. -B. G. Lilienfeldt, Ari Shnidman

We give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, the central value of the $L$ -function of…

math.NT2021

Sandpile groups of supersingular isogeny graphs

Nathanaël Munier, Ari Shnidman

Let $p$ and $q$ be distinct primes, and let $X_{p, q}$ be the $(q + 1)$ -regular graph whose nodes are supersingular elliptic curves over $F_{p}$ and whose edges are…

math.NT2021

A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so

Levent Alpöge, Manjul Bhargava, Ari Shnidman

We show that a positive proportion of quartic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof builds on the corresponding result for cub…

math.NT2021★ 3 cited

Ranks of abelian varieties in cyclotomic twist families

Ari Shnidman, Ariel Weiss

Let $A$ be an abelian variety over a number field $F$ , and suppose that $Z [ζ_{n}]$ embeds in $End_{F ˉ} A$ , for some root of unity $ζ_{n}$ of order $n = 3^{m}$ . Assum…

math.NT2021★ 5 cited

Elements of prime order in Tate-Shafarevich groups of abelian varieties over $Q$

Ari Shnidman, Ariel Weiss

For each prime $p$ , we show that there exist geometrically simple abelian varieties $A / Q$ with non-trivial $p$ -torsion in their Tate-Shafarevich groups. Specifically, for a…

math.NT2021★ 3 cited

Genus two curves with full $3$ -level structure and Tate-Shafarevich groups

Nils Bruin, E. Victor Flynn, Ari Shnidman

We give an explicit rational parameterization of the surface $H_{3}$ over $Q$ whose points parameterize genus 2 curves~ $C$ with full $3$ -level structure on…

math.NT2020★ 2 cited

A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so

Levent Alpöge, Manjul Bhargava, Ari Shnidman

We show that a positive proportion of cubic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof involves the comparison of $2$ -descent and $…

math.NT2020

Manin-Drinfeld cycles and derivatives of $L$ -functions

Ari Shnidman

We study algebraic cycles in the moduli space of $PGL_{2}$ -shtukas, arising from the diagonal torus. Our main result shows that their intersection pairing with the Heegner-D…

math.NT2019★ 9 cited

Elements of given order in Tate-Shafarevich groups of abelian varieties in quadratic twist families

Manjul Bhargava, Zev Klagsbrun, Robert J. Lemke Oliver +1

Let $A$ be an abelian variety over a number field $F$ and let $p$ be a prime. Cohen-Lenstra-Delaunay-style heuristics predict that the Tate-Shafarevich group of $A_{s}$ should contai…

math.NT2017

Quadratic twists of abelian varieties with real multiplication

Ari Shnidman

Let $F$ be a totally real number field and $A / F$ a principally polarized abelian variety with real multiplication by the ring of integers $O$ of a totally real field. Ass…

Exceptional points on Atkin--Lehner quotients

Rational torsion on simple genus two Jacobians

Rational points on modular curves: parameterization and geometric explanations

Hecke reciprocity and class groups

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

Derivatives of Rankin-Selberg L-functions and heights of generalized Heegner cycles

Vanishing criteria for Ceresa cycles

Ceresa cycles of bielliptic Picard curves

Rational torsion points on abelian surfaces with quaternionic multiplication

The geometry and arithmetic of bielliptic Picard curves

Normal distribution of bad reduction

Integers expressible as the sum of two rational cubes

Arbitrarily large p-torsion in Tate-Shafarevich groups

Rank growth of elliptic curves over N-th root extensions

Experiments with Ceresa classes of cyclic Fermat quotients

Sandpile groups of supersingular isogeny graphs

A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so

Ranks of abelian varieties in cyclotomic twist families

Elements of prime order in Tate-Shafarevich groups of abelian varieties over Q

Genus two curves with full 3​-level structure and Tate-Shafarevich groups

A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so

Manin-Drinfeld cycles and derivatives of L-functions

Elements of given order in Tate-Shafarevich groups of abelian varieties in quadratic twist families

Quadratic twists of abelian varieties with real multiplication

Derivatives of Rankin-Selberg $L$ -functions and heights of generalized Heegner cycles

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

Rank growth of elliptic curves over $N$ -th root extensions

Elements of prime order in Tate-Shafarevich groups of abelian varieties over $Q$

Genus two curves with full $3$ -level structure and Tate-Shafarevich groups

Manin-Drinfeld cycles and derivatives of $L$ -functions