63 citations · 557 across the 46 of their papers we have counts for
6 papers · 1 filter
A majority of elliptic curves over satisfy the Birch and Swinnerton-Dyer conjecture
Manjul Bhargava, Christopher Skinner, Wei Zhang
We prove that a majority (in fact, ) of all elliptic curves over , when ordered by height, satisfy the Birch and Swinnerton-Dyer rank conjecture.
On the mean number of 2-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields
Manjul Bhargava, Ila Varma
Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of 2-torsion elements in the class groups and narrow class groups…
A positive proportion of plane cubics fail the Hasse principle
Manjul Bhargava
When all ternary cubic forms over are ordered by the heights of their coefficients, we show that a positive proportion of them fail the Hasse principle, i.e., they have…
The geometric sieve and the density of squarefree values of invariant polynomials
Manjul Bhargava
We develop a method for determining the density of squarefree values taken by certain multivariate integer polynomials that are invariants for the action of an algebraic group on a…
The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders
Manjul Bhargava, Ila Varma
We determine the mean number of 3-torsion elements in the class groups of quadratic orders, where the quadratic orders are ordered by their absolute discriminants. Moreover, for a…
A positive proportion of elliptic curves over have rank one
Manjul Bhargava, Christopher Skinner
We prove that, when all elliptic curves over are ordered by naive height, a positive proportion have both algebraic and analytic rank one. It follows that the average…