Research

We determine the squarefree part of the scalar factor $S(n)$ that arises when the quartic invariant of the generic binary form of odd degree $2n+1$ is expressed as the discriminant of the unique quadratic covariant. The main theorem identifies $\mathrm{sqf}(S(n))$ with an OEIS sequence: for every prime $p$ and every $n \geq 1$, $v_p(S(n))$ is odd if and only if $p$ is an odd prime and $n+2$ is a power of $p$. The proof uses Gauss's lemma for multivariate polynomials, Kummer's carry-counting theorem, and a $p$-adic deformation argument. The theorem is accompanied by a complete Lean 4 formalization comprising approximately 15,000 lines across 25 modules, produced via an AI-assisted workflow using Claude Code (Anthropic), Codex (OpenAI), and Aristotle (Harmonic).

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

Let $R$ be a Dedekind domain with field of fractions $K$ and $\operatorname{char}(R)\neq3$. In this paper, we generalize Bhargava's parametrization of $3$-torsion ideal classes by binary cubic forms to work over $R$. Specifically, we construct arithmetic subgroups of $\operatorname{GL}_2(K)$ whose actions on certain lattices of binary cubic forms over $K$ parametrize $3$-torsion ideal classes in class groups of quadratic rings over $R$.

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.

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 conjecture of Poonen and Rains.

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 counting integral points on symmetric varieties lying within fundamental domains for coregular representations. As applications, we study the distribution of the $2$-torsion subgroup of the class group in thin families of cubic number fields, as well as the distribution of the $2$-Selmer groups in thin families of elliptic curves over $\mathbb{Q}$. For example, our results suggest that the existence of a generator of the ring of integers with small norm has an increasing effect on the average size of the $2$-torsion subgroup of the class group, relative to the Cohen--Lenstra predictions.

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.

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 $1 + \frac{1}{4} \times \frac{\zeta(2)}{\zeta(4)}$ (resp., $1 + \frac{1}{2} \times \frac{\zeta(2)}{\zeta(4)}$). 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.

Let $\mathcal{F}_g$ be the family of monic odd-degree hyperelliptic curves of genus $g$ over $\mathbb{Q}$. Poonen and Stoll have shown that for every $g \geq 3$, a positive proportion of curves in $\mathcal{F}_g$ have no rational points except the point at infinity. In this note, we prove the analogue for quadratic points: for each $g\geq 4$, a positive proportion of curves in $\mathcal{F}_g$ have no points defined over quadratic extensions except those that arise by pulling back rational points from $\mathbb{P}^1$.

Let $F \in \mathbb{Z}[x,z]$ be a binary form of degree $2n+1 \geq 5$. A result of Darmon and Granville known as "Faltings plus epsilon" states that when $F$ is separable, the superelliptic equation $y^2 = F(x,z)$ has finitely many primitive integer solutions. In this paper, we prove a strong asymptotic version of "Faltings plus epsilon" which states that in families of superelliptic equations of sufficiently large degree and having a fixed non-square leading coefficient, a positive proportion of members have no primitive integer solutions (put another way, a positive proportion of the corresponding binary forms fail to properly represent a square). Moreover, we show that in these families, a positive proportion of everywhere locally soluble members have a Brauer-Manin obstruction to satisfying the Hasse principle. Our result can be viewed as an analogue for superelliptic equations of Bhargava's result that most even-degree hyperelliptic curves over $\mathbb{Q}$ have no rational points.

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 >= 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.

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 Hermite equivalence class of a polynomial has a very natural interpretation in terms of the invariant ring and invariant ideal associated with the polynomial. We apply this interpretation to shed light on the relationship between Hermite equivalence and more familiar notions of polynomial equivalence, such as GL_2(Z)- and Z-equivalence. Specifically, we prove that GL_2(Z)-equivalent polynomials are Hermite equivalent and, for polynomials of degree 2 or 3, the converse is also true. On the other hand, for every n >= 4, we give infinite collections of examples of polynomials f, g in Z[X] of degree n that are Hermite equivalent but not GL_2(Z)-equivalent.

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 Q whose Jacobians have such maximal adelic Galois representations.

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 >= 3, there are infinitely many abelian varieties over Q with adelic Galois representation having image equal to all of GSp_{2g}(Z-hat).

We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let N >= 2, and consider an isolated complete intersection curve singularity germ f : (C^N, 0) -> (C^(N-1), 0). We introduce a numerical function m -> AD_2^m(f) that arises as an error term when counting m-th-order weight-2 inflection points with ramification sequence (0, ..., 0, 2) in a 1-parameter family of curves acquiring the singularity f = 0, and we compute AD_2^m(f) for various (f, m). Particularly, for a node defined by f : (x,y) -> xy, we prove that AD_2^m(xy) = (m+1 choose 4), and we deduce as a corollary that AD_2^m(f) >= (mult_0 Delta_f) * (m+1 choose 4) for any f, where mult_0 Delta_f is the multiplicity of the discriminant Delta_f at the origin in the deformation space. Furthermore, we show that the function m -> AD_2^m(f) - (mult_0 Delta_f) * (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, ..., 0, 1, 1) and for weight-1 inflection points, and we apply our results to solve various related enumerative problems.

More than four decades ago, Eisenbud, Khimshiashvili, 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: R^n -> R^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 A^1-enumerative geometry.

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 enrichment counts the number of lines meeting four lines defined over k in P^3_k, with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in A^1-homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms. In the appendix, the condition that the four lines each be defined over k is relaxed to the condition that the set of four lines being defined over k.

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.

Given an abelian group $G$, it is natural to ask whether there exists a permutation $\pi$ of $G$ that "destroys" all nontrivial 3-term arithmetic progressions (APs), in the sense that $\pi(b) - \pi(a) \neq \pi(c) - \pi(b)$ for every ordered triple $(a,b,c) \in G^3$ satisfying $b-a = c-b \neq 0$. This question was resolved for infinite groups $G$ by Hegarty, who showed that there exists an AP-destroying permutation of $G$ if and only if $G/\Omega_2(G)$ has the same cardinality as $G$, where $\Omega_2(G)$ denotes the subgroup of all elements in $G$ whose order divides $2$. In the case when $G$ is finite, however, only partial results have been obtained thus far. Hegarty has conjectured that an AP-destroying permutation of $G$ exists if $G = \mathbb{Z}/n\mathbb{Z}$ for all $n \neq 2,3,5,7$, and together with Martinsson, he has proven the conjecture for all $n > 1.4 \times 10^{14}$. In this paper, we show that if $p$ is a prime and $k$ is a positive integer, then there is an AP-destroying permutation of the elementary $p$-group $(\mathbb{Z}/p\mathbb{Z})^k$ if and only if $p$ is odd and $(p,k) \not\in \{(3,1),(5,1), (7,1)\}$.

Let $E_1$ and $E_2$ be $\overline{\mathbb{Q}}$-nonisogenous, semistable elliptic curves over $\mathbb{Q}$, having respective conductors $N_{E_1}$ and $N_{E_2}$ and both without complex multiplication. For each prime $p$, denote by $a_{E_i}(p) := p+1-\#E_i(\mathbb{F}_p)$ the trace of Frobenius. Under the assumption of the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power $L$-functions $L(s, \mathrm{Sym}^i E_1\otimes\mathrm{Sym}^j E_2)$ where $i,j\in\{0,1,2\}$, we prove an explicit result that can be stated succinctly as follows: there exists a prime $p\nmid N_{E_1}N_{E_2}$ such that $a_{E_1}(p)a_{E_2}(p)<0$ and $p < \big( (32+o(1))\cdot \log N_{E_1} N_{E_2}\big)^2$. This improves and makes explicit a result of Bucur and Kedlaya. Now, if $I\subset[-1,1]$ is a subinterval with Sato-Tate measure $\mu$ and if the symmetric power $L$-functions $L(s, \mathrm{Sym}^k E_1)$ are functorial and satisfy GRH for all $k \le 8/\mu$, we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime $p\nmid N_{E_1}$ such that $a_{E_1}(p)/(2\sqrt{p})\in I$ and $p < \left((21+o(1)) \cdot \mu^{-2}\log (N_{E_1}/\mu)\right)^2$.

Let $\mathcal{I} \subset \mathbb{N}$ be an infinite subset, and let $\{a_i\}_{i \in \mathcal{I}}$ be a sequence of nonzero real numbers indexed by $\mathcal{I}$ such that there exist positive constants $m, C_1$ for which $|a_i| \leq C_1 \cdot i^m$ for all $i \in \mathcal{I}$. Furthermore, let $c_i \in [-1,1]$ be defined by $c_i = \frac{a_i}{C_1 \cdot i^m}$ for each $i \in \mathcal{I}$, and suppose the $c_i$'s are equidistributed in $[-1,1]$ with respect to a continuous, symmetric probability measure $\mu$. In this paper, we show that if $\mathcal{I} \subset \mathbb{N}$ is not too sparse, then the sequence $\{a_i\}_{i \in \mathcal{I}}$ fails to obey Benford's Law with respect to arithmetic density in any sufficiently large base, and in fact in any base when $\mu([0,t])$ is a strictly convex function of $t \in (0,1)$. Nonetheless, we also provide conditions on the density of $\mathcal{I} \subset \mathbb{N}$ under which the sequence $\{a_i\}_{i \in \mathcal{I}}$ satisfies Benford's Law with respect to logarithmic density in every base. As an application, we apply our general result to study Benford's Law-type behavior in the leading digits of Frobenius traces of newforms of positive, even weight. Our methods of proof build on the work of Jameson, Thorner, and Ye, who studied the particular case of newforms without complex multiplication.

The action of the absolute Galois group Gal(K^sep/K) of a global field K on a tree T(phi, alpha) of iterated preimages of alpha in P^1(K) under phi in K(x) with deg(phi) >= 2 induces a homomorphism rho: Gal(K^sep/K) -> Aut(T(phi, alpha)), which is called an arboreal Galois representation. In this paper, we address a number of questions posed by Jones and Manes about the size of the group G(phi, alpha) := im rho = lim_n Gal(K(phi^{-n}(alpha))/K). Specifically, we consider two cases for the pair (phi, alpha): (1) phi is such that the sequence {a_n} defined by a_0 = alpha and a_n = phi(a_{n-1}) is periodic, and (2) phi commutes with a nontrivial Mobius transformation that fixes alpha. In the first case, we resolve a question posed by Jones about the size of G(phi, alpha), and taking K = Q, we describe the Galois groups of iterates of polynomials phi in Z[x] that have the form phi(x) = x^2 + kx or phi(x) = x^2 - (k+1)x + k. When K = Q and phi in Z[x], arboreal Galois representations are a useful tool for studying the arithmetic dynamics of phi. In the case of phi(x) = x^2 + kx for k in Z, we employ a result of Jones regarding the size of the group G(psi, 0), where psi(x) = x^2 - kx + k, to obtain a zero-density result for primes dividing terms of the sequence {a_n} defined by a_0 in Z and a_n = phi(a_{n-1}). In the second case, we resolve a conjecture of Jones about the size of a certain subgroup C(phi, alpha) of Aut(T(phi, alpha)) that contains G(phi, alpha), and we present progress toward the proof of a conjecture of Jones and Manes concerning the size of G(phi, alpha) as a subgroup of C(phi, alpha).

Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication (CM), and for each prime $p$ of good reduction, let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius. By the Hasse bound, $a_E(p) = 2\sqrt{p} \cos \theta_p$ for a unique $\theta_p \in [0, \pi]$. In this paper, we prove that the least prime $p$ such that $\theta_p \in [\alpha, \beta] \subset [0, \pi]$ satisfies $p \ll \left(\frac{N_E}{\beta - \alpha}\right)^A$, where $N_E$ is the conductor of $E$ and the implied constant and exponent $A > 2$ are absolute and effectively computable. Our result is an analogue for CM elliptic curves of Linnik's Theorem for arithmetic progressions, which states that the least prime $p \equiv a \pmod q$ for $(a,q)=1$ satisfies $p \ll q^L$ for an absolute constant $L > 0$.

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 $\alpha$ of degree $n$ has exactly $m$ real Galois conjugates with probability $n^{-3/4+o(1)}$, when such $\alpha$ are ordered by the heights of their minimal polynomials.

The class of surreal numbers, discovered by John Conway while studying combinatorial games, possesses rich numerical structure sharing many properties with real numbers. This paper extends prior work developing analysis on surreal numbers by treating functions, limits, derivatives, power series, and integrals. The authors propose surreal definitions of arctangent and logarithm functions using Maclaurin series truncations, present a formula for sequence limits characterizing convergent sequences, and evaluate certain series and infinite Riemann sums via extrapolation. They define a new topology on surreal numbers to obtain the Intermediate Value Theorem despite lack of Cauchy completeness, and prove the Fundamental Theorem of Calculus would hold for surreals if consistent integration definitions exist. Open questions include defining other analytic functions, evaluating power series generally, finding consistent integration definitions, proving Stokes' Theorem for surreal integration, and studying differential equations.