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Elementary local representation densities at all primes via lifting recursions

Samuel Griffiths

Abstract

Let $p$ be a prime and let $L$ be a quadratic $\mathbb{Z}_p$-lattice with quadratic form $Q$. For $t\neq 0$ the local representation density $α_p(t;L)$ is the stable normalised growth of the congruence counts of solutions to $Q(v)\equiv t\pmod{p^m}$. We compute these counts and densities explicitly for the hyperbolic plane $H_0$ over $\mathbb{Z}p$, uniformly in $p$, and at $p=2$ for the basic dyadic blocks (rank-$1$ Type I blocks and the even binary planes $2^aH\varepsilon$), together with the anisotropic ternary lattice $L_3=\langle 2\rangle^{\oplus 3}$. At the dyadic prime the usual Jacobian/Hensel lifting mechanism breaks down in the bilinear-lattice convention $Q(v)=\langle v,v\rangle$. The main new input is an explicit half-lift involution for diagonal sums of squares, which yields a stable lifting recursion with factor $2^{d-1}$ under the primitivity hypothesis $4\nmid a$. As applications we obtain closed forms for the three-squares congruence counts (hence $α_2(t;L_3)$) and a prime-uniform formula for the densities of the scaled hyperbolic planes $p^eH_0$ in the standard normalisation $q=\langle\cdot,\cdot\rangle/2$.

Elementary local representation densities at all primes via lifting recursions

Abstract

Let be a prime and let be a quadratic -lattice with quadratic form . For the local representation density is the stable normalised growth of the congruence counts of solutions to . We compute these counts and densities explicitly for the hyperbolic plane over , uniformly in , and at for the basic dyadic blocks (rank- Type I blocks and the even binary planes ), together with the anisotropic ternary lattice . At the dyadic prime the usual Jacobian/Hensel lifting mechanism breaks down in the bilinear-lattice convention . The main new input is an explicit half-lift involution for diagonal sums of squares, which yields a stable lifting recursion with factor under the primitivity hypothesis . As applications we obtain closed forms for the three-squares congruence counts (hence ) and a prime-uniform formula for the densities of the scaled hyperbolic planes in the standard normalisation .
Paper Structure (17 sections, 31 theorems, 112 equations, 1 table)

This paper contains 17 sections, 31 theorems, 112 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries and dyadic Jordan blocks
  4. Representation counts and densities
  5. Dyadic Jordan blocks
  6. The $q=Q/2$ convention on even dyadic lattices
  7. Congruence counts for the hyperbolic plane
  8. The anisotropic even plane
  9. The half-lift principle and the anisotropic ternary block
  10. A general half-lift principle
  11. Specialisation to the ternary block $L_3$
  12. Explicit density formulas
  13. Hyperbolic plane (all primes)
  14. Type II anisotropic block
  15. Type I (rank one) blocks
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $d\ge 1$ and set For $n\ge 1$ and $a\in\mathbb{Z}$ let Assume $n\ge 3$ and $4\nmid a$. Then Equivalently, among the residue classes modulo $2^n$ satisfying $Q_d\equiv a$, exactly half admit lifts to level $2^{n+1}$; moreover, whenever a class lifts, all$2^d$ lifts in its fibre are solutions (cf. Lemma lem:fibre-invariance).

Theorems & Definitions (72)

  • Theorem 1.1: Half-lift principle for sums of squares
  • Remark 1.2: Scope of the half-lift involution
  • Theorem 1.3: Prime-uniform density for the hyperbolic plane
  • Definition 2.1: Basic dyadic blocks
  • Remark 2.2
  • Proposition 2.3: Jordan decomposition
  • Remark 2.4
  • Lemma 2.5: Dictionary with the $q=\langle\cdot,\cdot\rangle/2$ convention
  • proof
  • Remark 2.6: On the $Q$--vs. $q$ normalisation at $p=2$
  • ...and 62 more