63 citations · 557 across the 46 of their papers we have counts for
10 papers · 2 filters
The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
Manjul Bhargava, Arul Shankar
In this article, we prove that the average rank of elliptic curves over , when ordered by height, is less than (in fact, less than ). As a consequence of our…
The average number of elements in the 4-Selmer groups of elliptic curves is 7
Manjul Bhargava, Arul Shankar
We prove that when all elliptic curves over are ordered by height, the average size of their 4-Selmer groups is equal to 7. As a consequence, we show that a positive p…
On the number of integral binary -ic forms having bounded Julia invariant
Manjul Bhargava, Andrew Yang
In 1848, Hermite introduced a reduction theory for binary forms of degree which was developed more fully in the seminal 1917 treatise of Julia. This canonical method of reducti…
The proportion of plane cubic curves over that everywhere locally have a point
Manjul Bhargava, John Cremona, Tom Fisher
We show that the proportion of plane cubic curves over that have a -rational point is a rational function in , where the rational function is inde…
What is the probability that a random integral quadratic form in variables is isotropic?
Manjul Bhargava, John Cremona, Tom Fisher
We show that the density of quadratic forms in variables over that are isotropic is a rational function in , where the rational function is independent of $p…
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 …