A. Swaminathan

activity

20132026

most citedSurjectivity of Galois Representations in Rational Families of Abelian Varieties

16 citations · 48 across the 25 of their papers we have counts for

collaborators

28 papers

math.NT2026

Local Factors in the BSD Conjecture: A Unified Statistical View

David Kurniadi Angdinata, Kenny Lau, Ken Ono +2

For elliptic curves $E / Q$ in short Weierstrass form \[ E=E(a_4,a_6): y^2=x^3+a_4x+a_6, \] we derive a multivariable Euler product generating function which encodes the Tam…

math.NT2026

Beyond Mock Modularity: Elliptic Corrections for Higher Dyson Ranks

Claudia Alfes, Ken Ono, Ashvin Swaminathan

When $m = 1$ , the Dyson rank generating function is a classical bridge between partition theory, Ramanujan's mock theta functions, and the theory of harmonic Maass forms and nonhol…

math.NT2026

On the Quartic Invariant of Odd Degree Binary Forms

Ashvin Swaminathan

We determine the squarefree part of the scalar factor that arises when the quartic invariant of the generic binary form $F$ of odd degree $2 n + 1$ is expressed as the discriminant of…

math.NT2026

The existence of infinitely many cubic fields with class group of exact 2-rank 1

Manjul Bhargava, Arul Shankar, Artane Siad +1

We show that infinitely many cubic fields have class group of 2-rank 1.

math.NT2025

Parity of the partition function in quadratic progressions

Ken Ono, Ashvin Swaminathan

The parity of the partition function $p (n)$ is one of the most stubborn problems in number theory. In 2010, the first author conjectured, for square-free $1 < D \equiv 23 (mod 24)$ , t…

math.NT2025

A parametrization of $3$ -class groups of quadratic rings over Dedekind domains

Eliot Hodges, Ashvin A. Swaminathan

Let $R$ be a Dedekind domain with field of fractions $K$ and $char (R) \neq = 3$ . In this paper, we generalize Bhargava's parametrization of $3$ -torsion ideal classes by…

math.NT2025

The second moment of the size of the $2$ -class group of monogenized cubic fields

Manjul Bhargava, Arul Shankar, Ashvin Swaminathan

We prove that when totally real (resp., complex) monogenized cubic number fields are ordered by height, the second moment of the size of the $2$ -class group is at most $3$ (resp.,…

math.PR2024

Universality theorems for zeros of random real polynomials with fixed coefficients

Matthew C. King, Ashvin Swaminathan

Consider a monic polynomial of degree $n$ whose subleading coefficients are independent, identically distributed, nondegenerate random variables having zero mean, unit variance, an…

math.NT2024

A positive proportion of monic odd-degree hyperelliptic curves of genus $g \geq 4$ have no unexpected quadratic points

Jef Laga, Ashvin A. Swaminathan

Let $F_{g}$ be the family of monic odd-degree hyperelliptic curves of genus $g$ over $Q$ . Poonen and Stoll have shown that for every $g \geq 3$ , a positive proport…

math.NT2023★ 1 cited

Counting integral points on symmetric varieties with applications to arithmetic statistics

Arul Shankar, Artane Siad, Ashvin A. Swaminathan

In this article, we combine Bhargava's geometry-of-numbers methods with the dynamical point-counting methods of Eskin--McMullen and Benoist--Oh to develop a new technique for count…

math.NT2022

The mean number of $2$ -torsion elements in the class groups of cubic orders

Ashvin Swaminathan

We determine the mean number of 2-torsion elements in class groups of cubic orders, when such orders are enumerated by discriminant. Specifically, we prove that when isomorphism cl…

math.NT2021★ 1 cited

The second moment of the size of the $2$ -Selmer group of elliptic curves

Manjul Bhargava, Arul Shankar, Ashvin Swaminathan

In this paper, we prove that when elliptic curves over $Q$ are ordered by height, the second moment of the size of the $2$ -Selmer group is at most $15$ . This confirms a co…

math.NT2021

Geometry-of-numbers methods in the cusp

Arul Shankar, Artane Siad, Ashvin Swaminathan +1

In this article, we develop new methods for counting integral orbits having bounded invariants that lie inside the cusps of fundamental domains for coregular representations. We il…

math.NT2021★ 3 cited

Hermite equivalence of polynomials

Manjul Bhargava, Jan-Hendrik Evertse, Kálmán Győry +2

In this paper, we resurrect a long-forgotten notion of equivalence for univariate polynomials with integral coefficients introduced by Hermite in the 1850s. We show that the Hermit…

math.NT2020

A new parametrization for ideal classes in rings defined by binary forms, and applications

Ashvin Swaminathan

We give a parametrization of square roots of the ideal class of the inverse different of rings defined by binary forms in terms of the orbits of a coregular representation. This pa…

math.NT2019★ 1 cited

Most Odd-Degree Binary Forms Fail to Primitively Represent a Square

Ashvin Swaminathan

Let $F$ be a separable integral binary form of odd degree $N \geq 5$ . A result of Darmon and Granville known as ``Faltings plus epsilon'' implies that the degree- $N$ \emph{superell…

math.AG2019

On the EKL-Degree of a Weyl Cover

Joseph Knight, Ashvin Swaminathan, Dennis Tseng

More than four decades ago, Eisenbud, Khimšiašvili, and Levine introduced an analogue in the algebro-geometric setting of the notion of local degree from differential topology. The…

math.AG2018★ 7 cited

An arithmetic count of the lines meeting four lines in P^3

Padmavathi Srinivasan, Kirsten Wickelgren

We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field $k$ , this enrichme…

math.NT2017★ 6 cited

Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians

Aaron Landesman, Ashvin Swaminathan, James Tao +1

In this article, we show that in each of four standard families of hyperelliptic curves, there is a density- $1$ subset of members with the property that their Jacobians have adelic…

math.AG2017

Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

Anand Patel, Ashvin Swaminathan

We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let $N \geq 2$ , and consider an…

math.NT2016★ 16 cited

Surjectivity of Galois Representations in Rational Families of Abelian Varieties

Aaron Landesman, Ashvin Swaminathan, James Tao +1

In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with…

math.GR2016★ 2 cited

Lifting Subgroups of Symplectic Groups over $Z / ℓ Z$

Aaron Landesman, Ashvin Swaminathan, James Tao +1

For a positive integer $g$ , let $Sp_{2 g} (R)$ denote the group of $2 g \times 2 g$ symplectic matrices over a ring $R$ . Assume $g \geq 2$ . For a prime number $ℓ$ , we give a…

math.NT2016

Permutations that Destroy Arithmetic Progressions in Elementary $p$ -Groups

Noam D. Elkies, Ashvin Swaminathan

Given an abelian group $G$ , it is natural to ask whether there exists a permutation $π$ of $G$ that "destroys" all nontrivial 3-term arithmetic progressions (APs), in the sense tha…

math.NT2015★ 2 cited

Elliptic Curve Variants of the Least Quadratic Nonresidue Problem and Linnik's Theorem

Evan Chen, Peter S. Park, Ashvin Swaminathan

Let $E_{1}$ and $E_{2}$ be $Q$ -nonisogenous, semistable elliptic curves over $Q$ , having respective conductors $N_{E_{1}}$ and $N_{E_{2}}$ and both without com…

Local Factors in the BSD Conjecture: A Unified Statistical View

Beyond Mock Modularity: Elliptic Corrections for Higher Dyson Ranks

On the Quartic Invariant of Odd Degree Binary Forms

The existence of infinitely many cubic fields with class group of exact 2-rank 1

Parity of the partition function in quadratic progressions

A parametrization of 3-class groups of quadratic rings over Dedekind domains

The second moment of the size of the 2-class group of monogenized cubic fields

Universality theorems for zeros of random real polynomials with fixed coefficients

A positive proportion of monic odd-degree hyperelliptic curves of genus g≥4 have no unexpected quadratic points

Counting integral points on symmetric varieties with applications to arithmetic statistics

The mean number of 2-torsion elements in the class groups of cubic orders

The second moment of the size of the 2-Selmer group of elliptic curves

Geometry-of-numbers methods in the cusp

Hermite equivalence of polynomials

A new parametrization for ideal classes in rings defined by binary forms, and applications

Most Odd-Degree Binary Forms Fail to Primitively Represent a Square

On the EKL-Degree of a Weyl Cover

An arithmetic count of the lines meeting four lines in P^3

Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians

Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

Surjectivity of Galois Representations in Rational Families of Abelian Varieties

Lifting Subgroups of Symplectic Groups over Z/ℓZ

Permutations that Destroy Arithmetic Progressions in Elementary p-Groups

Elliptic Curve Variants of the Least Quadratic Nonresidue Problem and Linnik's Theorem

A parametrization of $3$ -class groups of quadratic rings over Dedekind domains

The second moment of the size of the $2$ -class group of monogenized cubic fields

A positive proportion of monic odd-degree hyperelliptic curves of genus $g \geq 4$ have no unexpected quadratic points

The mean number of $2$ -torsion elements in the class groups of cubic orders

The second moment of the size of the $2$ -Selmer group of elliptic curves

Lifting Subgroups of Symplectic Groups over $Z / ℓ Z$

Permutations that Destroy Arithmetic Progressions in Elementary $p$ -Groups