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

Manjul Bhargava, Arul Shankar, Xiaoheng Wang

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 $F$.

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 , the density of discriminants of field extensions of degree at most 5 over .

Paper Structure

This paper contains 28 sections, 32 theorems, 117 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. The local masses $m_{\mathfrak p}$
  5. Tamagawa number of a reductive group over $F$
  6. Parametrization of quadratic, cubic, quartic, and quintic extensions of a fixed global field
  7. Counting $S$-integral orbits over global fields
  8. Reduction Theory
  9. Averaging
  10. The number of generic elements in the cusp is negligible
  11. The main term
  12. The number of nongeneric elements in the main body is negligible
  13. Conclusion
  14. Congruence conditions and a squarefree sieve
  15. Application to field counting
  16. ...and 13 more sections

Key Result

Theorem 1

Let $F$ be a global number or function field of any characteristic. For any $n=2$, $3$, $4$ or $5$, we let $N_n(F,X)$ denote the number of isomorphism classes of degree-$n$ field extensions $L$ of $F$ weighted by $(\#{\rm Aut}(L/F))^{-1}$, whose normal closure over $F$ has full Galois group $S_n$ su When $F$ is a function field, $X$ only runs through the possible norms of relative discriminants in

Theorems & Definitions (42)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Corollary 7
  • Theorem 8
  • Theorem 9
  • Theorem 3.1
  • ...and 32 more