Elementary local representation densities at all primes via lifting recursions

TL;DR

This paper explicitly computes local representation densities for quadratic lattices over p-adic integers, including challenging cases at p=2, using lifting recursions and involutions, with applications to counting solutions to quadratic congruences.

Contribution

It introduces a new explicit half-lift involution for diagonal sums of squares, enabling stable lifting recursions at dyadic primes, and provides uniform formulas for local densities and congruence counts.

Findings

Explicit formulas for local densities of hyperbolic planes over all primes.

Stable lifting recursion with factor 2^{d-1} at p=2.

Closed-form solutions for three-squares congruence counts.

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 $\alpha_p(t;L)$ 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 $\alpha_2(t;L_3)$) and a prime-uniform formula for the densities of the scaled hyperbolic planes $p^eH_0$ in the standard normalisation $q=\langle\cdot,\cdot\rangle/2$.