Paper

{
  "raw_xml": "<entry>\n    <id>http://arxiv.org/abs/2602.21070v1</id>\n    <title>Elementary local representation densities at all primes via lifting recursions</title>\n    <updated>2026-02-24T16:32:07Z</updated>\n    <link href='https://arxiv.org/abs/2602.21070v1' rel='alternate' type='text/html'/>\n    <link href='https://arxiv.org/pdf/2602.21070v1' rel='related' title='pdf' type='application/pdf'/>\n    <summary>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 $q=\\langle\\cdot,\\cdot\\rangle/2$.</summary>\n    <category scheme='http://arxiv.org/schemas/atom' term='math.NT'/>\n    <published>2026-02-24T16:32:07Z</published>\n    <arxiv:comment>24 pages. Ancillary files include Lean 4 formalization and Python verification scripts</arxiv:comment>\n    <arxiv:primary_category term='math.NT'/>\n    <author>\n      <name>Samuel Griffiths</name>\n    </author>\n  </entry>"
}