6 papers · 1 filter
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…
The existence of infinitely many cubic fields with class group of exact 2-rank 1
Manjul Bhargava, Arul Shankar, Artane Siad +1
We show that infinitely many cubic fields have class group of 2-rank 1.
The second moment of the size of the -class group of monogenized cubic fields
Manjul Bhargava, Arul Shankar, Ashvin Swaminathan
We prove that when totally real (resp., complex) monogenized cubic number fields are ordered by height, the second moment of the size of the -class group is at most (resp.,…
A proof of van der Waerden's Conjecture on random Galois groups of polynomials
Manjul Bhargava
Of the monic integer polynomials with , how many have associated Galois group that is not the full sy…
Integers expressible as the sum of two rational cubes
Levent Alpöge, Manjul Bhargava, Ari Shnidman
We prove that a positive proportion of integers are expressible as the sum of two rational cubes, and a positive proportion are not so expressible, thus proving a conjecture of Dav…
Galois groups of random integer polynomials and van der Waerden's Conjecture
Manjul Bhargava
Of the monic integer polynomials with , how many have associated Galois group that is not the full sy…