44 citations · 118 across the 9 of their papers we have counts for
10 papers · 1 filter
Geometry-of-numbers methods over global fields II: Coregular representations
Manjul Bhargava, Arul Shankar, Xiaoheng Wang
We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ra…
Squarefree values of polynomial discriminants II
Manjul Bhargava, Arul Shankar, Xiaoheng Wang
We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we…
An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree
Manjul Bhargava, Arul Shankar, Xiaoheng Wang
We prove an improvement on Schmidt's upper bound on the number of number fields of degree and absolute discriminant less than X for . We carry this out by imp…
On the squarefree values of
Gian Cordana Sanjaya, Xiaoheng Wang
In this article, we prove that the density of integers such that is squarefree, when ordered by , equals the conjectured product of th…
Families of elliptic curves ordered by conductor
Ananth N. Shankar, Arul Shankar, Xiaoheng Wang
In this article, we study the family of elliptic curves , having good reduction at and , and whose -invariants are small. Within this set of elliptic curves…
Squarefree values of polynomial discriminants I
Manjul Bhargava, Arul Shankar, Xiaoheng Wang
We determine the density of monic integer polynomials of given degree that have squarefree discriminant; in particular, we prove for the first time that the lower density of…
Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces
Manjul Bhargava, Arul Shankar, Xiaoheng Wang
We develop geometry-of-numbers methods to count orbits in prehomogeneous vector spaces having bounded invariants over any global field. As our primary example, we apply these techn…
Arithmetic invariant theory II
Manjul Bhargava, Benedict H. Gross, Xiaoheng Wang
Let be a field, let be a reductive group, and let be a linear representation of . Let denote the geometric quotient and let …
A positive proportion of locally soluble hyperelliptic curves over have no point over any odd degree extension
Manjul Bhargava, Benedict H. Gross, Xiaoheng Wang
A hyperelliptic curve over is called "locally soluble" if it has a point over every completion of . In this paper, we prove that a positive proportion of hyp…
Average size of the 2-Selmer group of Jacobians of monic even hyperelliptic curves
Arul Shankar, Xiaoheng Wang
In [5], Manjul Bhargava and Benedict Gross considered the family of hyperelliptic curves over $\Q$ having a fixed genus and a marked rational Weierstrass point. They showed that th…