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Lea Beneish

Publications and source records attributed to Lea Beneish.

16 recordsLinked to original sources
arXiv2026

Effective Bertini theorems and zeros of $p$-adic forms of degrees 7 and 11

We establish an effective Bertini-type theorem for hypersurfaces $X_f \colon f = 0$ defined over a finite field $k$ for which $f$ has no linear factors over the algebraic closure $\overline{k}$. Given a line $L$ defined over $k$ and a nonreduced $\overline{k}$-point $x$ on $X_f \cap L$, we give an upper bound on the number of planes $P$ containing $L$ for which $X_f \cap P$ contains a line through $x$. Underlying this result is a factorization algorithm for bivariate polynomials originally due to Kaltofen, which we present with slightly relaxed hypotheses. Our primary application is to Artin's conjecture on $p$-adic forms of prime degree $d$: if $K/\mathbb{Q}_p$ is a finite extension with residue field isomorphic to $\mathbb{F}_q$ and $F \in K[x_0, \ldots, x_{d^2}]$ is homogeneous of degree $d$, the conjecture states $F$ has a nontrivial zero in $K$. We show this conjecture holds whenever $q > 679$ for $d=7$ and $q > 7393$ for $d=11$, improving upon a result of Wooley.

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arXiv2025

Degrees of points on irreducible hypersurfaces

We study the set of $D$ such that a given irreducible hypersurface $C$ of degree $d$ has infinitely many points of degree $D$ over $\mathbb{Q}$. We give a new explicit proof that this set contains all (positive) multiples of the index of $C$ with finitely many exceptions. When $D$ is sufficiently large and divisible by the index of $C$, we show there are $\gg x^{1/2d^2-ε}$ distinct fields with degree $D$ and discriminant $\leq x$ containing new non-singular points on $C$. Our proof relies on (what we define to be) the index of the Newton polytope $H(C)$ for $C$ which we use as combinatorial proxy for the index of $C$. We conjecture that for almost all $C$ with a given Newton polytope $H$, the index of $H$ equals the index of $C$ and we prove this conjecture for a positive proportion of curves with $H(C)=H$. As an application of our techniques, we prove half of Bhargava's conjecture on the least odd degree of points on a typical hyperelliptic and we recover Springer's theorem and a related statement for rational points on cubic hypersurfaces.

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arXiv2025

Fields generated by points on superelliptic curves

We give an asymptotic lower bound on the number of field extensions generated by algebraic points on superelliptic curves over $\mathbb{Q}$ with fixed degree $n$ and discriminant bounded by $X$. For $C$ a fixed such curve given by an affine equation $y^m = f(x)$ where $m \geq 2$ and $d= \mathrm{deg}\ f (x) \geq m$, we find that for all degrees $n$ divisible by $\gcd(m, d)$ and sufficiently large, the number of such fields is asymptotically bounded below by $X^{δ_n}$, where $δ_n \to 1/m^2$ as $n \to \infty$. We then give geometric heuristics suggesting that for n not divisible by $\gcd(m, d)$, degree $n$ points may be less abundant than those for which $n$ is divisible by $\gcd(m,d)$ and provide an example of conditions under which a curve is known to have finitely many points of certain degrees.

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arXiv2024

How often does a cubic hypersurface have a rational point?

A cubic hypersurface in $\mathbb{P}^n$ defined over $\mathbb{Q}$ is given by the vanishing locus of a cubic form $f$ in $n+1$ variables. It is conjectured that when $n \geq 4$, such cubic hypersurfaces satisfy the Hasse principle. This is now known to hold on average due to recent work of Browning, Le Boudec, and Sawin. Using this result, we determine the proportion of cubic hypersurfaces in $\mathbb{P}^n$, ordered by the height of $f$, with a rational point for $n \geq 4$ explicitly as a product over primes $p$ of rational functions in $p$. In particular, this proportion is equal to 1 for cubic hypersurfaces in $\mathbb{P}^n$ for $n \geq 9$; for $100\%$ of cubic hypersurfaces, this recovers a celebrated result of Heath-Brown that non-singular cubic forms in at least 10 variables have rational zeros. In the $n=3$ case, we give a precise conjecture for the proportion of cubic surfaces in $\mathbb{P}^3$ with a rational point.

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arXiv2024

The Gross--Kohnen--Zagier theorem via $p$-adic uniformization

This article gives a new proof of the Gross--Kohnen--Zagier theorem for Shimura curves which exploits the $p$-adic uniformization of Cerednik--Drinfeld. The explicit description of CM points via this uniformization leads to an expression relating the Gross--Kohnen--Zagier generating series to the ordinary projection of the first derivative, with respect to a weight variable, of a $p$-adic family of positive definite ternary theta series.

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arXiv2023

Replicable functions arising from code-lattice VOAs fixed by automorphisms

We ascertain properties of the algebraic structures in towers of codes, lattices, and vertex operator algebras (VOAs) by studying the associated subobjects fixed by lifts of code automorphisms. In the case of sublattices fixed by subgroups of code automorphisms, we identify replicable functions that occur as quotients of the associated theta functions by suitable eta products. We show that these lattice theta quotients can produce replicable functions not associated to any individual automorphisms. Moreover, we show that the structure of the fixed subcode can induce certain replicable lattice theta quotients and we provide a general code theoretic characterization of order doubling for lifts of code automorphisms to the lattice-VOA. Finally, we prove results on the decompositions of characters of fixed subVOAs.

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arXiv2022

On the proportion of locally soluble superelliptic curves

We investigate the proportion of superelliptic curves that have a $\mathbb{Q}_p$ point for every place $p$ of $\mathbb{Q}$. We show that this proportion is positive and given by the product of local densities, we provide lower bounds for this proportion in general, and for superelliptic curves of the form $y^3 = f(x,z)$ for an integral binary form $f$ of degree 6, we determine this proportion to be 96.94%. More precisely, we give explicit rational functions in $p$ for the proportion of such curves over $\mathbb{Z}_p$ having a $\mathbb{Q}_p$-point.

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arXiv2022

Generalized Ramanujan-Sato Series Arising from Modular Forms

Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanujan-Sato series for $1/π$. We then use it to construct explicit examples related to non-compact arithmetic triangle groups, as classified by Takeuchi. Some of our examples are new, and some reproduce existing examples.

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arXiv2021

Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves of Prime Level and Genus 4, 5, 6

We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus four, five, and six. We find that the only such curves with exceptional rational points are of levels $137$ and $311$. In particular there are no exceptional rational points on those curves of genus five and six. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.

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arXiv2019

Module constructions for certain subgroups of the largest Mathieu group

For certain subgroups of $M_{24}$, we give vertex operator algebraic module constructions whose associated trace functions are meromorphic Jacobi forms. These meromorphic Jacobi forms are canonically associated to the mock modular forms of Mathieu moonshine. The construction is related to the Conway moonshine module and employs a technique introduced by Anagiannis--Cheng--Harrison. With this construction we are able to give concrete vertex algebraic realizations of certain cuspidal Hecke eigenforms of weight two. In particular, we give explicit realizations of trace functions whose integralities are equivalent to divisibility conditions on the number of $\mathbb{F}_p$ points on the Jacobians of modular curves.

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arXiv2018

Quasimodular moonshine and arithmetic connections

We prove the existence of a module for the largest Mathieu group, whose trace functions are weight two quasimodular forms. Restricting to the subgroup fixing a point, we see that the integrality of these functions is equivalent to certain divisibility conditions on the number of $\mathbb{F}_p$ points on Jacobians of modular curves. Extending such expressions to arbitrary primes, we find trace functions for modules of cyclic groups of prime order with similar connections. Moreover, for cyclic groups, we give an explicit vertex operator algebra construction whose trace functions are given only in terms of weight two Eisenstein series.

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arXiv2018

Moonshine modules and a question of Griess

We consider the situation in which a finite group acts on an infinite-dimensional graded module in such a way that the graded trace functions are weakly holomorphic modular forms. Under a mild hypothesis we completely describe the asymptotic module structure of the homogeneous subspaces. As a consequence we find that moonshine for a group gives rise to partial orderings on its irreducible representations. This serves as a first answer to a question posed by Griess. In particular, we show that our hypothesis holds for umbral moonshine and for automorphism groups of certain vertex operator algebras.

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arXiv2014

Traces of singular values of Hauptmoduln

In an important paper, Zagier proved that certain half-integral weight modular forms are generating functions for traces of polynomials in the $j$-function. It turns out that Zagier's work makes it possible to algorithmically compute Hilbert class polynomials using a canonical family of modular forms of weight $\frac{3}{2}$. We generalize these results and consider Haupmoduln for levels $1, 2, 3, 5, 7,$ and $13$. We show that traces of singular values of polynomials in Haupmoduln are again described by coefficients of half-integral weight modular forms. This realization makes it possible to algorithmically compute class polynomials.

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