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Ashvin Swaminathan

Publications and source records attributed to Ashvin Swaminathan.

At least 19 recordsLinked to original sources
arXiv2026

Local Factors in the BSD Conjecture: A Unified Statistical View

For elliptic curves $E/\mathbb{Q}$ 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 Tamagawa product $Tam(E)=\prod_p c_p(E)$. Using this generating function, we compute limiting distributions, exact covariances, and moment and tail bounds for four important statistics on the Tamagawa number; for example we show that more than half of all curves in this height ordering have trivial Tamagawa product, about $42.2\%$ have exactly one prime with nontrivial local Tamagawa number, and only about $6.8\%$ have exactly two such primes. The product is obtained by specializing an Euler product indexed by local reduction data that we derive from Tate's algorithm. The results in this paper were autoformalized in Lean by AxiomProver.

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arXiv2026

On a conjecture of Browning and Sawin on random hypersurfaces with sign coefficients

Browning and Sawin conjectured that random hypersurfaces with sign coefficients are smooth with probability tending to one as the degree grows. We prove this conjecture and obtain a quantitative bound. For each $n\geq1$, a degree $d$ form in $n+1$ variables, with independent uniform coefficients in $\{-1,1\}$, defines a singular complex hypersurface with probability $O_n(d^{-1/2})$. The positive-dimensional singular loci occur with exponentially small probability. For $n\geq3$, the same exponential bound holds for failure of absolute irreducibility. These results have been formalized in Lean by AxiomProver assuming existing literature.

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arXiv2026

Parity of the partition function in quadratic progressions

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 $1<D\equiv 23\pmod{24}$, that the values $p\!\left(\frac{Dm^2+1}{24}\right)$, as $m$ ranges over positive integers with $(m,6)=1$, include infinitely many even and infinitely many odd terms. We prove this conjecture. The key new idea is geometric. Logarithmic derivatives of twisted Borcherds products built from Ramanujan's third-order mock theta functions recast the problem in terms of CM points of discriminant $-D$ on $X_0(6)$. The uniqueness of canonical lifts from characteristic $2$ to characteristic zero shows that the CM points supporting the poles remain distinct after reduction modulo $2$. This fact, combined with an Eisenstein series comparison and a Galois representation argument, rules out constant parity and gives infinitely many values of each parity. The method applies to the coefficients of analogous generalized twisted Borcherds products. These results imply that \[\#\{0\leq n\leq X: p(n)\text{ is odd}\}\gg\sqrt X,\] which is now the best known lower bound for odd values of $p(n)$. The algebraic identities at the heart of this paper, as well as the improved lower bound for odd partition numbers, have been formalized in Lean by AxiomProver.

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arXiv2026

Beyond Mock Modularity: Elliptic Corrections for Higher Dyson Ranks

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 nonholomorphic Jacobi forms. The rank is a statistic on partitions, and the higher Dyson systems, for $m \geq 2$, are a natural multivariable refinement of it, combining $m$ graded rank contributions. Unlike the classical case, these higher systems are not expected to fit the mock-modular framework, which raises the question of what analytic structure governs them. We show that their root-of-unity specializations carry a hidden elliptic structure. A finite $q$-difference recurrence produces an explicit polynomial obstruction to the expected index $m$ elliptic transformation law, and because the obstruction is finite, its partial fractions canonically determine finitely many Appell--Lerch correction terms that remove it. The corrected functions satisfy a twisted index $m$ elliptic law; a natural translation removes the twist, and their holomorphic finite parts admit finite theta decompositions. Thus, the natural analogue of Dyson's mock-modular phenomenon at higher $m$ is not mock modularity but a finite theta decomposition governed by an index $m$ elliptic transformation law. These results grew out of a human--AI collaboration, and the key new formulas were formalized and machine-verified in Lean/Mathlib by AxiomProver.

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arXiv2026

On the Quartic Invariant of Odd Degree Binary Forms

We determine the squarefree part of the scalar factor that arises when the quartic invariant of the generic binary form $F$ of odd degree $2n+1$ is expressed as the discriminant of the unique quadratic covariant $(F,F)_{2n}$. This squarefree part is exactly $p$ when $n+2$ is a power of an odd prime $p$, and $1$ otherwise. Equivalently, for each prime $p$: $v_2(S(n))$ is always even, and for odd $p$, $v_p(S(n))$ is odd if and only if $n+2$ is a power of $p$. This generalizes the classical identity $\operatorname{disc}(H(F))=-3\cdot\operatorname{disc}(F)$ for binary cubics, which dates back to the work of Cayley and Sylvester in the 1850s. The proof, which involves substantial explicit coefficient analysis and $p$-adic deformation arguments, was developed using an AI-assisted research workflow: the author's earlier partial attempts were completed through systematic collaboration with Claude Code (Anthropic) and Codex (OpenAI), and key arithmetic lemmas were formally verified in Lean~4 using Aristotle (Harmonic). We describe this workflow in detail as a case study in AI-assisted mathematical research. We also discuss representation-theoretic, geometric, and arithmetic interpretations of the quadratic covariant.

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arXiv2025

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

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., at most $6$). In the totally real case, we further prove that the second moment of the size of the narrow $2$-class group is at most $9$. This result gives further evidence in support of the general observation, first made in work of Bhargava--Hanke--Shankar and recently formalized into a set of heuristics in work of Siad--Venkatesh, that monogenicity has an altering effect on class group distributions. All of the upper bounds we obtain are tight, conditional on tail estimates.

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arXiv2024

Universality theorems for zeros of random real polynomials with fixed coefficients

Consider a monic polynomial of degree $n$ whose subleading coefficients are independent, identically distributed, nondegenerate random variables having zero mean, unit variance, and finite moments of all orders, and let $m \geq 0$ be a fixed integer. We prove that such a random monic polynomial has exactly $m$ real zeros with probability $n^{-3/4+o(1)}$ as $n\to \infty$ through integers of the same parity as $m$. More generally, we determine conditions under which a similar asymptotic formula describes the corresponding probability for families of random real polynomials with multiple fixed coefficients. Our work extends well-known universality results of Dembo, Poonen, Shao, and Zeitouni, who considered the family of real polynomials with all coefficients random. As a number-theoretic consequence of these results, we deduce that an algebraic integer $α$ of degree $n$ has exactly $m$ real Galois conjugates with probability $n^{-3/4+o(1)}$, when such $α$ are ordered by the heights of their minimal polynomials.

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arXiv2024

Geometry-of-numbers methods in the cusp

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 illustrate these methods for a representation of cardinal interest in number theory, namely that of the split orthogonal group acting on the space of quadratic forms.

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arXiv2024

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

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{superelliptic equation} $y^2 = F(x,z)$ has finitely many primitive integer solutions. In this paper, we consider the family $\mathscr{F}_N(f_0)$ of degree-$N$ superelliptic equations with fixed leading coefficient $f_0 \in \mathbb{Z} \smallsetminus \pm\mathbb{Z}^2$, ordered by height. For every sufficiently large $N$, we prove that among equations in the family $\mathscr{F}_N(f_0)$, more than $74.9\%$ are insoluble, and more than $71.8\%$ are everywhere locally soluble but fail the Hasse principle due to the Brauer--Manin obstruction. We further show that these proportions rise to at least $99.9\%$ and $96.7\%$, respectively, when $f_0$ has sufficiently many prime divisors of odd multiplicity. Our result can be viewed as a strong asymptotic form of ``Faltings plus epsilon'' for superelliptic equations and constitutes an analogue of Bhargava's result that most hyperelliptic curves over $\mathbb{Q}$ have no rational points.

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arXiv2024

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

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 parametrization, which can be construed as a new integral model of a ``higher composition law'' discovered by Bhargava and generalized by Wood, was the missing ingredient needed to solve a range of previously intractable open problems concerning distributions of class groups, Selmer groups, and related objects. For instance, in this paper, we apply the parametrization to bound the average size of the $2$-class group in families of number fields defined by binary $n$-ic forms, where $n \geq 3$ is an arbitrary integer, odd or even; in the paper [41], we applied it to prove that most integral odd-degree binary forms fail to primitively represent a square; and in the paper [11], joint with Bhargava and Shankar, we applied it to bound the second moment of the size of the $2$-Selmer group of elliptic curves.

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arXiv2024

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

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 classes of totally real (resp., complex) cubic orders are enumerated by discriminant, the average $2$-torsion in the class group is $1 + \frac{1}{4} \times \frac{ζ(2)}{ζ(4)}$ (resp., $1 + \frac{1}{2} \times \frac{ζ(2)}{ζ(4)}$). In particular, we find that the average $2$-torsion in the class group increases when one ranges over all orders in cubic fields instead of restricting to the subfamily of rings of integers of cubic fields, where the average $2$-torsion in the class group was first determined in work of Bhargava to be $\frac{5}{4}$ (resp., $\frac{3}{2}$). By work of Bhargava--Varma, proving this result amounts to obtaining an asymptotic count of the number of "reducible" $\operatorname{SL}_3(\mathbb{Z})$-orbits on the space $\mathbb{Z}^2 \otimes_{\mathbb{Z}} \operatorname{Sym}^2 \mathbb{Z}^3$ of $3 \times 3$ symmetric integer matrices having bounded invariants and satisfying local conditions. In this paper, we resolve the generalization of this orbit-counting problem where the dimension $3$ is replaced by any fixed odd integer $N \geq 3$. More precisely, we determine asymptotic formulas for the number of reducible $\operatorname{SL}_N(\mathbb{Z})$-orbits on $\mathbb{Z}^2 \otimes_{\mathbb{Z}} \operatorname{Sym}^2 \mathbb{Z}^N$ satisfying general infinite sets of congruence conditions.

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arXiv2022

Surjectivity of Galois Representations in Rational Families of Abelian Varieties

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 image as large as possible form a density-$1$ subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension $g \geq 3$, there are infinitely many abelian varieties over $\mathbb Q$ with adelic Galois representation having image equal to all of $\operatorname{GSp}_{2g}(\widehat{\mathbb Z})$.

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arXiv2022

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

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 Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations.

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arXiv2020

On the EKL-Degree of a Weyl Cover

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. Their notion of degree, which we call the EKL-degree, can be thought of as a refinement of the usual notion of local degree in algebraic geometry that works over non-algebraically closed base fields, taking values in the Grothendieck-Witt ring. In this note, we compute the EKL-degree at the origin of certain finite covers $f\colon \mathbb{A}^n\to \mathbb{A}^n$ induced by quotients under actions of Weyl groups. We use knowledge of the cohomology ring of partial flag varieties as a key input in our proofs, and our computations give interesting explicit examples in the field of $\mathbb{A}^1$-enumerative geometry.

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arXiv2020

Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

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 isolated complete intersection curve singularity germ $f \colon (\mathbb{C}^N,0) \to (\mathbb{C}^{N-1},0)$. We introduce a numerical function $m \mapsto \operatorname{AD}_{(2)}^m(f)$ that arises as an error term when counting $m^{\mathrm{th}}$-order weight-$2$ inflection points with ramification sequence $(0, \dots, 0, 2)$ in a $1$-parameter family of curves acquiring the singularity $f = 0$, and we compute $\operatorname{AD}_{(2)}^m(f)$ for various $(f,m)$. Particularly, for a node defined by $f \colon (x,y) \mapsto xy$, we prove that $\operatorname{AD}_{(2)}^m(xy) = {{m+1} \choose 4},$ and we deduce as a corollary that $\operatorname{AD}_{(2)}^m(f) \geq (\operatorname{mult}_0 Δ_f) \cdot {{m+1} \choose 4}$ for any $f$, where $\operatorname{mult}_0 Δ_f$ is the multiplicity of the discriminant $Δ_f$ at the origin in the deformation space. Furthermore, we show that the function $m \mapsto \operatorname{AD}_{(2)}^m(f) -(\operatorname{mult}_0 Δ_f) \cdot {{m+1} \choose 4}$ is an analytic invariant measuring how much the singularity "counts as" an inflection point. We obtain similar results for weight-$2$ inflection points with ramification sequence $(0, \dots, 0, 1,1)$ and for weight-$1$ inflection points, and we apply our results to solve various related enumerative problems.

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arXiv2017

Lifting Subgroups of Symplectic Groups over $\mathbb{Z} / \ell \mathbb{Z}$

For a positive integer $g$, let $\mathrm{Sp}_{2g}(R)$ denote the group of $2g \times 2g$ symplectic matrices over a ring $R$. Assume $g \ge 2$. For a prime number $\ell$, we give a self-contained proof that any closed subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z}_\ell)$ which surjects onto $\mathrm{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$ must in fact equal all of $\mathrm{Sp}_{2g}(\mathbb{Z}_\ell)$. The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties.

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