Elementary local representation densities at all primes via lifting recursions

Abstract

Let $p$ be a prime and let $L$ be a quadratic ${\mathbb{Z}}_p$-lattice with quadratic form $Q$. For $t\ne 0$ the local representation density $\hat{I}{\pm }_p(t;L)$ is the stable normalised growth of the congruence counts of solutions to $Q(v)\equiv t(\mathrm{mod}{p}^m)$. 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 $2^{a}H\epsilon$), together with the anisotropic ternary lattice $L_3=⟨2{⟩}^{\oplus 3}$. At the dyadic prime the usual Jacobian/Hensel lifting mechanism breaks down in the bilinear-lattice convention $Q(v)=⟨v,v⟩$. 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 $\hat{I}{\pm }_2(t;L_3)$) and a prime-uniform formula for the densities of the scaled hyperbolic planes $p^e{H}_0$ in the standard normalisation $q=⟨\cdot ,\cdot ⟩/2$.

24 pages. Ancillary files include Lean 4 formalization and Python verification scripts