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Rational points on ellipsoids and modular forms

Claire Burrin, Matthias Gröbner

Abstract

The theory of modular forms and spherical harmonic analysis are applied to establish new best bounds towards the counting and equidistribution of rational points on spheres and other higher dimensional ellipsoids, in what may be viewed as a textbook display of the 'unreasonable effectiveness' of modular forms.

Rational points on ellipsoids and modular forms

Abstract

The theory of modular forms and spherical harmonic analysis are applied to establish new best bounds towards the counting and equidistribution of rational points on spheres and other higher dimensional ellipsoids, in what may be viewed as a textbook display of the 'unreasonable effectiveness' of modular forms.
Paper Structure (21 sections, 19 theorems, 139 equations, 3 tables)

This paper contains 21 sections, 19 theorems, 139 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Discussion: Sketch of proofs and further results
  3. Modular forms and $L$-functions
  4. Modular forms for $\Gamma_0(N)$
  5. $L$-functions associated to modular forms
  6. Half-integral weight modular forms
  7. Theta functions
  8. Estimates on Fourier coefficients
  9. Harmonic analysis on the sphere
  10. Spherical functions
  11. Spectral decomposition of $L^2(S^{d})$
  12. Proof of \ref{['Thm2']} (and \ref{['Thm2bis']})
  13. Proof of \ref{['thm:modular decomposition']}
  14. Identities involving divisor sums
  15. Proof of \ref{['thm:modular decomposition']}
  16. ...and 6 more sections

Key Result

Theorem 1

Let $Q$ be a positive definite quadratic form as described above. Let $\Psi\subset \mathcal{E}_Q$ be a convex domain with piecewise smooth boundary, and let $\mu_Q$ denote the normalized Lebesgue measure on $\mathcal{E}_Q$. Then as $n\to\infty$ along the sequence of integers coprime to $N$.

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof : Sketch of proof
  • Theorem 5
  • Theorem 3.1
  • proof
  • Proposition 4.1
  • proof
  • ...and 26 more