Frank Thorne
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View article: On the asymptotics of cubic fields ordered by general invariants
On the asymptotics of cubic fields ordered by general invariants Open
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 invariants. In addition, we determine which of these famil…
View article: Identitites for Field Extensions Generalizing the Ohno–Nakagawa Relations
Identitites for Field Extensions Generalizing the Ohno–Nakagawa Relations Open
In previous work, Ohno [Ohn97] conjectured, and Nakagawa [Nak98] proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakaga…
View article: Exponential sums over singular binary quartic forms and applications
Exponential sums over singular binary quartic forms and applications Open
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 common with analogous formulas proved previously for other vecto…
View article: What is the height of two points in the plane?
What is the height of two points in the plane? Open
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 of height functions $H_{s, t}$, such that the height functio…
View article: On the asymptotics of cubic fields ordered by general invariants
On the asymptotics of cubic fields ordered by general invariants Open
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 invariants. In addition, we determine which of these familie…
View article: Improved bounds on number fields of small degree
Improved bounds on number fields of small degree Open
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 previously the best known upper bound for $6 \leq n \leq 94$.
View article: Improved error estimates for the Davenport-Heilbronn theorems
Improved error estimates for the Davenport-Heilbronn theorems Open
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 Shankar and Tsimerman. The present paper uses the analytic th…
View article: Improved lower bounds for the number of fields with alternating Galois group
Improved lower bounds for the number of fields with alternating Galois group Open
Let $n \\geq 6$ be an integer. We prove that the number of number fields with\nGalois group $A_n$ and absolute discriminant at most $X$ is asymptotically at\nleast $X^{1/8 + O(1/n)}$. For $n \\geq 8$ this improves upon the previously best\…
View article: UPPER BOUNDS ON POLYNOMIALS WITH SMALL GALOIS GROUP
UPPER BOUNDS ON POLYNOMIALS WITH SMALL GALOIS GROUP Open
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 asymptotically 100% are irreducible and have Galois group isomorph…
View article: Asymptotic identities for additive convolutions of sums of divisors
Asymptotic identities for additive convolutions of sums of divisors Open
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 $b$ are positive odd integers. He also conjectured that his …
View article: Counting quintic fields with genus number one
Counting quintic fields with genus number one Open
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 quintic fields have arbitrarily large genus number; and we compu…
View article: Upper bounds on number fields of given degree and bounded discriminant
Upper bounds on number fields of given degree and bounded discriminant Open
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 $N_n(X) = O(X^{ c (\log n)^2})$ for an explicit constant $c$.
View article: Malle's Conjecture for $G \times A$, with $G = S_3, S_4, S_5$
Malle's Conjecture for $G \times A$, with $G = S_3, S_4, S_5$ Open
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 on the primes dividing $A$.
View article: Rank Growth of Elliptic Curves in Non-Abelian Extensions
Rank Growth of Elliptic Curves in Non-Abelian Extensions Open
Given an elliptic curve $E/\mathbb{Q}$, it is a conjecture of Goldfeld that asymptotically half of its quadratic twists will have rank zero and half will have rank one. Nevertheless, higher rank twists do occur: subject to the parity conje…
View article: The distribution of $G$-Weyl CM fields and the Colmez conjecture
The distribution of $G$-Weyl CM fields and the Colmez conjecture Open
Let $G$ be a transitive subgroup of $S_d$ and $E$ be a CM field of degree $2d$ with a maximal totally real $G$-field. If the Galois group of the Galois closure of $E$ is isomorphic to the wreath product of $C_2$ and $G$, then we say that $…
View article: Levels of distribution for sieve problems in prehomogeneous vector spaces
Levels of distribution for sieve problems in prehomogeneous vector spaces Open
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 fields, and we carried out these computations in a variety of ca…
View article: The number of ramified primes in number fields of small degree
The number of ramified primes in number fields of small degree Open
In this paper we investigate the distribution of the number of primes which ramify in number fields of degree . In analogy with the classical Erdős-Kac theorem, we prove for -extensions that the number of such primes is normally distribute…
View article: Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves
Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves Open
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 $O_ε(|{\rm Disc}(K)|^{1/2+ε})$ by Brauer--Siegel). This yields cor…
View article: Orbital exponential sums for prehomogeneous vector spaces
Orbital exponential sums for prehomogeneous vector spaces Open
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 explicitly and efficiently evaluating the Fourier transform o…