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On indefinite integral ternary quadratic forms

Alexander Gamburd, Amit Ghosh, Peter Sarnak, Junho Peter Whang

Abstract

We resolve two problems pertaining to indefinite integral ternary quadratic forms, one highlighted by Margulis and the other initiated by Serre, both from 1990. To do so we develop tools for dealing with high ramification in problems involving sums over classes of such forms weighted by their diophantine invariants.

On indefinite integral ternary quadratic forms

Abstract

We resolve two problems pertaining to indefinite integral ternary quadratic forms, one highlighted by Margulis and the other initiated by Serre, both from 1990. To do so we develop tools for dealing with high ramification in problems involving sums over classes of such forms weighted by their diophantine invariants.
Paper Structure (22 sections, 55 theorems, 119 equations, 2 tables)

This paper contains 22 sections, 55 theorems, 119 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Part 1 : Spectrum of anisotropic forms
  3. Watson's papers revisited
  4. Upper and lower bounds for MAR(X)
  5. Genus count asymptotics
  6. Preliminaries for proofs of the Theorems and (B)-(D)
  7. Spinor genera
  8. Sums over squarefree numbers
  9. Packets
  10. Class number asymptotics
  11. Local representations
  12. Asymptotic formula for MAR(X)
  13. Part 2 : Isotropic forms
  14. Isotropic Siegel asymptotics
  15. Alternative proof of Prop. \ref{['isotropic-siegel-2']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

There is a positive constant $\gamma$ such that

Theorems & Definitions (88)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • Lemma 2.6
  • ...and 78 more