A. Shankar

activity

20102026

most citedThe average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1

45 citations · 246 across the 34 of their papers we have counts for

collaborators

37 papers

math.CO2026★ 1 cited

Remarks on the disproof of the unit distance conjecture

Noga Alon, Thomas F. Bloom, W. T. Gowers +6

We present a short, digested, human-verified version of the recent OpenAI-generated counterexample to the Erdős unit distance conjecture, and a sequence of reflections on it. The a…

math.NT2026

Geometry-of-numbers methods over global fields II: Coregular representations

Manjul Bhargava, Arul Shankar, Xiaoheng Wang

We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ra…

math.NT2026

Counting number fields of fixed degree by their smallest defining polynomial

Santiago Arango-Piñeros, Fabian Gundlach, Robert J. Lemke Oliver +5

When do two irreducible polynomials with integer coefficients define the same number field? One can define an action of $GL_{2} \times GL_{1}$ on the space of polynom…

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

Secondary terms in the counting functions of quartic fields II

Arul Shankar, Jacob Tsimerman

We determine the smoothed counts of $S_{4}$ -quartic fields with bounded discriminant, satisfying any finite specified set of local conditions, as the sum of two main terms with a pow…

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.NT2025

Malle's Conjecture for Galois octic fields over $Q$

Arul Shankar, Ila Varma

We compute the asymptotic number of octic number fields whose Galois groups over $Q$ are isomorphic to $D_{4}$ , the symmetries of a square, when ordering such fields by their…

math.NT2025

Finiteness theorems for some representations of $GL_{3}$

Fatemehzahra Janbazi, Arul Shankar

Let $n \geq 2$ be an integer and let $K$ be a number field with ring of integers $O_{K}$ . We prove that the set of ternary $n$ -ic forms with coefficients in $O_{K}$ …

math.NT2025

Secondary terms in the counting functions of quartic fields

Arul Shankar, Jacob Tsimerman

We prove that the smoothed counting function of the set of quartic fields, satisfying any finite set of local conditions, can be written as a linear combination of $X,X^{5/6}\log X…

math.NT2024

Secondary terms in the first moment of $∣ Sel_{2} (E) ∣$

Arul Shankar, Takashi Taniguchi

We prove the existence of secondary terms of order $X^{3/4}$ , with power saving error terms, in the counting functions of $∣ Sel_{2} (E) ∣$ , the 2-Selmer group of E, for elliptic…

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★ 4 cited

Squarefree values of polynomial discriminants II

Manjul Bhargava, Arul Shankar, Xiaoheng Wang

We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we…

math.NT2022

On the asymptotics of cubic fields ordered by general invariants

Arul Shankar, Frank Thorne

In this article, we introduce a class of invariants of cubic fields termed generalized discriminants. We then obtain asymptotics for the families of cubic fields ordered by these i…

math.NT2022★ 1 cited

An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree

Manjul Bhargava, Arul Shankar, Xiaoheng Wang

We prove an improvement on Schmidt's upper bound on the number of number fields of degree $n$ and absolute discriminant less than X for $6 \leq n \leq 94$ . We carry this out by imp…

math.NT2021

Non-trivial bounds on 2, 3, 4, and 5-torsion in class groups of number fields, conditional on standard $L$ -function conjectures

Arul Shankar, Jacob Tsimerman

We prove new conditional bounds on the the $m$ -torsion of class groups of number fields of any fixed degree, for $m = 2$ , $3$ , $4$ , and $5$ . Our methods first recast the problem in t…

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

Central values of zeta functions of non-Galois cubic fields

Arul Shankar, Anders Södergren, Nicolas Templier

The Dedekind zeta functions of infinitely many non-Galois cubic fields have negative central values.

math.NT2020★ 4 cited

The mean number of 2-torsion elements in the class groups of $n$ -monogenized cubic fields

Manjul Bhargava, Jonathan Hanke, Arul Shankar

We prove that, on average, the monogenicity or $n$ -monogenicity of a cubic field has an altering effect on the behavior of the 2-torsion in its class group.

math.NT2020

Heuristics for the asymptotics of the number of $S_{n}$ -number fields

Arul Shankar, Jacob Tsimerman

We give a heuristic argument supporting conjectures of Bhargava on the asymptotics of the number of $S_{n}$ -number fields having bounded discriminant. We then make our arguments rigo…

math.NT2019★ 13 cited

Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

Ananth N. Shankar, Arul Shankar, Yunqing Tang +1

Given a K3 surface $X$ over a number field $K$ with potentially good reduction everywhere, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite.…

math.NT2019★ 1 cited

Families of elliptic curves ordered by conductor

Ananth N. Shankar, Arul Shankar, Xiaoheng Wang

In this article, we study the family of elliptic curves $E / Q$ , having good reduction at $2$ and $3$ , and whose $j$ -invariants are small. Within this set of elliptic curves…

math.NT2017★ 5 cited

The number of quartic $D_{4}$ -fields ordered by conductor

Salim Ali Altug, Arul Shankar, Ila Varma +1

We consider families of number fields of degree 4 whose normal closures over $Q$ have Galois group isomorphic to $D_{4}$ , the symmetries of a square. To any such field $L$ ,…

math.NT2017★ 13 cited

Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves

Manjul Bhargava, Arul Shankar, Takashi Taniguchi +3

We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields $K$ (the trivial bound being $O_ε(|{…

Remarks on the disproof of the unit distance conjecture

Geometry-of-numbers methods over global fields II: Coregular representations

Counting number fields of fixed degree by their smallest defining polynomial

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

Secondary terms in the counting functions of quartic fields II

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

Malle's Conjecture for Galois octic fields over Q

Finiteness theorems for some representations of GL3​

Secondary terms in the counting functions of quartic fields

Secondary terms in the first moment of ∣Sel2​(E)∣

Counting integral points on symmetric varieties with applications to arithmetic statistics

Squarefree values of polynomial discriminants II

On the asymptotics of cubic fields ordered by general invariants

An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree

Non-trivial bounds on 2, 3, 4, and 5-torsion in class groups of number fields, conditional on standard L-function conjectures

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

Geometry-of-numbers methods in the cusp

Central values of zeta functions of non-Galois cubic fields

The mean number of 2-torsion elements in the class groups of n-monogenized cubic fields

Heuristics for the asymptotics of the number of Sn​-number fields

Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

Families of elliptic curves ordered by conductor

The number of quartic D4​-fields ordered by conductor

Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves

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

Malle's Conjecture for Galois octic fields over $Q$

Finiteness theorems for some representations of $GL_{3}$

Secondary terms in the first moment of $∣ Sel_{2} (E) ∣$

Non-trivial bounds on 2, 3, 4, and 5-torsion in class groups of number fields, conditional on standard $L$ -function conjectures

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

The mean number of 2-torsion elements in the class groups of $n$ -monogenized cubic fields

Heuristics for the asymptotics of the number of $S_{n}$ -number fields

The number of quartic $D_{4}$ -fields ordered by conductor