Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces

Abstract

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 techniques to determine, for any base global field $F$, the density of discriminants of field extensions of degree at most 5 over $F$.

Main results have been extended to all global fields, including those of small positive characteristic. Several new results are also proven, including applications to relative class numbers of quadratic and cubic extensions, unramified nonabelian extensions of quadratic extensions, and the expected number of points in the fibers of degree 4 and 5 covers of curves over a finite field. 38 pages

References in corpus (6)

• The geometric sieve and the density of squarefree values of invariant polynomials
• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
• Most hyperelliptic curves over Q have no rational points
• The average number of elements in the 4-Selmer groups of elliptic curves is 7
• A positive proportion of plane cubics fail the Hasse principle
• A majority of elliptic curves over $\mathbb{Q}$ satisfy the Birch and Swinnerton-Dyer conjecture

Cited by in corpus (6)

• The average size of the 3-isogeny Selmer groups of elliptic curves $y^2=x^3+k$
• Bounds for the $\ell$-torsion in class groups
• Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle's conjecture for function fields
• Ranks of abelian varieties in cyclotomic twist families
• Upper bound on the number of extensions of a given number field
• Counting extensions of number fields with Frobenius Galois group