F. Thorne

activity

20112025

most citedSecondary terms in counting functions for cubic fields

64 citations · 140 across the 23 of their papers we have counts for

collaborators

28 papers

math.NT2025

Asymptotics for $6$ -torsion and $D_{6}$ -extensions

Peter Koymans, Robert J. Lemke Oliver, Efthymios Sofos +1

We prove a composite case of the Cohen--Lenstra--Gerth heuristics. Specifically, we establish an asymptotic for the average $6$ -torsion of the class group of quadratic number field…

math.NT2024

Exponential sums over singular binary quartic forms and applications

Yasuhiro Ishitsuka, Takashi Taniguchi, Frank Thorne +1

We investigate exponential sums over singular binary quartic forms, proving an explicit formula for the finite field Fourier transform of this set. Our formula shares much in commo…

math.NT2022

What is the height of two points in the plane?

Jesse Leo Kass, Frank Thorne

Here we describe the distribution of rational points on the Hilbert scheme of two points in the projective plane. More specifically, we explicitly describe a two-parameter family o…

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

Improved bounds on number fields of small degree

Theresa C. Anderson, Ayla Gafni, Kevin Hughes +5

We study the number of degree $n$ number fields with discriminant bounded by $X$ . In this article, we improve an upper bound due to Schmidt on the number of such fields that was pr…

math.NT2021★ 1 cited

Improved error estimates for the Davenport-Heilbronn theorems

Manjul Bhargava, Takashi Taniguchi, Frank Thorne

We improve the error terms in the Davenport-Heilbronn theorems on counting cubic fields to $O (X^{2/3 + ε})$ . This improves on separate and independent results of the authors and Sh…

math.NT2020

Asymptotic identities for additive convolutions of sums of divisors

Robert J. Lemke Oliver, Sunrose T. Shrestha, Frank Thorne

In a 1916 paper, Ramanujan studied the additive convolution $S_{a, b} (n)$ of sum-of-divisors functions $σ_{a} (n)$ and $σ_{b} (n)$ , and proved an asymptotic formula for it when $a$ and $…

math.NT2020★ 3 cited

Upper bounds on number fields of given degree and bounded discriminant

Robert J. Lemke Oliver, Frank Thorne

Let $N_{n} (X)$ denote the number of degree $n$ number fields with discriminant bounded by $X$ . In this note, we improve the best known upper bounds on $N_{n} (X)$ , finding that $N_n(X)…

math.NT2020★ 2 cited

Counting quintic fields with genus number one

Kevin J. McGown, Frank Thorne, Amanda Tucker

We prove several results concerning genus numbers of quintic fields: we compute the proportion of quintic fields with genus number one; we prove that a positive proportion of quint…

math.NT2020★ 4 cited

Malle's Conjecture for $G \times A$ , with $G = S_{3}, S_{4}, S_{5}$

Riad Masri, Frank Thorne, Wei-Lun Tsai +1

We prove Malle's conjecture for $G \times A$ , with $G = S_{3}, S_{4}, S_{5}$ and $A$ an abelian group. This builds upon work of the fourth author, who proved this result with restrictions…

math.NT2019

Upper bounds on polynomials with small Galois group

Robert J. Lemke Oliver, Frank Thorne

When monic integral polynomials of degree $n \geq 2$ are ordered by the maximum of the absolute value of their coefficients, the Hilbert irreducibility theorem implies that asympto…

math.NT2019★ 6 cited

Improved lower bounds for the number of fields with alternating Galois group

Aaron Landesman, Robert J. Lemke Oliver, Frank Thorne

Let $n \geq 6$ be an integer. We prove that the number of number fields with Galois group $A_{n}$ and absolute discriminant at most $X$ is asymptotically at least $X^{1/8 + O (1/ n)}$ .…

math.NT2018

Rank growth of elliptic curves in nonabelian extensions

Robert J. Lemke Oliver, Frank Thorne

Given an elliptic curve $E / Q$ , it is a conjecture of Goldfeld that asymptotically half of its quadratic twists will have rank zero and half will have rank one. Nevertheles…

math.NT2017★ 5 cited

Uniform bounds for lattice point counting and partial sums of zeta functions

David Lowry-Duda, Takashi Taniguchi, Frank Thorne

We prove uniform versions of two classical results in analytic number theory. The first is an asymptotic for the number of points of a complete lattice $Λ \subseteq R^{d}$ in…

math.NT2017

Levels of distribution for sieve problems in prehomogeneous vector spaces

Takashi Taniguchi, Frank Thorne

In a companion paper, we developed an efficient algebraic method for computing the Fourier transforms of certain functions defined on prehomogeneous vector spaces over finite field…

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(|{…

math.NT2016★ 7 cited

On $D_{ℓ}$ -extensions of odd prime degree $ℓ$

Henri Cohen, Frank Thorne

Generalizing the work of A. Morra and the authors, we give explicit formulas for the Dirichlet series generating function of $D_{ℓ}$ -extensions of odd prime degree $ℓ$ with g…

math.NT2016★ 2 cited

Orbital exponential sums for prehomogeneous vector spaces

Takashi Taniguchi, Frank Thorne

Let (G, V) be a prehomogeneous vector space, let O be any G(F_q)-invariant subset of V(F_q), and let f be the characteristic function of O. In this paper we develop a method for ex…

math.NT2014

The number of ramified primes in number fields of small degree

Robert J. Lemke Oliver, Frank Thorne

In this paper we investigate the distribution of the number of primes which ramify in number fields of degree d <= 5. In analogy with the classical Erdos-Kac theorem, we prove for…

math.NT2013★ 13 cited

Zeros of L-functions outside the critical strip

Andrew R. Booker, Frank Thorne

For a wide class of Dirichlet series associated to automorphic forms, we show that those without Euler products must have zeros within the region of absolute convergence. For insta…

math.NT2013

Dirichlet series associated to cubic fields with given quadratic resolvent

Henri Cohen, Frank Thorne

Let k be a quadratic field. We give an explicit formula for the Dirichlet series enumerating cubic fields whose quadratic resolvent field is isomorphic to k. Our work is a sequel t…

math.NT2012★ 3 cited

On the existence of large degree Galois representations for fields of small discriminant

Jeremy Rouse, Frank Thorne

Let $L / K$ be a Galois extension of number fields. We prove two lower bounds on the maximum of the degrees of the irreducible complex representations of $Gal (L / K)$ , the sharpe…

math.NT2012

An error estimate for counting $S_{3}$ -sextic number fields

Takashi Taniguchi, Frank Thorne

In this note, we prove a power-saving remainder term for the function counting $S_{3}$ -sextic number fields. We also give a prediction on the second main term. We also present numeri…

math.NT2012★ 1 cited

Four Perspectives on Secondary Terms in the Davenport-Heilbronn Theorems

Frank Thorne

This paper is an expanded version of the author's lecture at the Integers Conference 2011. We discuss the secondary terms in the Davenport-Heilbronn theorems on cubic fields and 3-…

Asymptotics for 6-torsion and D6​-extensions

Exponential sums over singular binary quartic forms and applications

What is the height of two points in the plane?

On the asymptotics of cubic fields ordered by general invariants

Improved bounds on number fields of small degree

Improved error estimates for the Davenport-Heilbronn theorems

Asymptotic identities for additive convolutions of sums of divisors

Upper bounds on number fields of given degree and bounded discriminant

Counting quintic fields with genus number one

Malle's Conjecture for G×A, with G=S3​,S4​,S5​

Upper bounds on polynomials with small Galois group

Improved lower bounds for the number of fields with alternating Galois group

Rank growth of elliptic curves in nonabelian extensions

Uniform bounds for lattice point counting and partial sums of zeta functions

Levels of distribution for sieve problems in prehomogeneous vector spaces

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

On Dℓ​-extensions of odd prime degree ℓ

Orbital exponential sums for prehomogeneous vector spaces

The number of ramified primes in number fields of small degree

Zeros of L-functions outside the critical strip

Dirichlet series associated to cubic fields with given quadratic resolvent

On the existence of large degree Galois representations for fields of small discriminant

An error estimate for counting S3​-sextic number fields

Four Perspectives on Secondary Terms in the Davenport-Heilbronn Theorems

Asymptotics for $6$ -torsion and $D_{6}$ -extensions

Malle's Conjecture for $G \times A$ , with $G = S_{3}, S_{4}, S_{5}$

On $D_{ℓ}$ -extensions of odd prime degree $ℓ$

An error estimate for counting $S_{3}$ -sextic number fields