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Subrings of Zn\mathbb{Z}^n of prime-power index and modular forms

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Contributed by Anwesh Ray ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

Let fn(m)f_n(m) denote the number of unital subrings of index mm in Zn\mathbb{Z}^n, with coordinatewise multiplication, and let gn(pe)g_n(p^e) count those whose reduction modulo pp is the diagonal copy of Fp\mathbb{F}_p. Voll's uniformity question asks whether the local subring zeta functions can be described by a single rational function in pp and p−sp^{-s}. Polynomial formulas for fn(pe)f_n(p^e) are known for e≤9e\le9. We prove that neither f10(p17)f_{10}(p^{17}) nor g10(p17)g_{10}(p^{17}) agrees with a polynomial in pp for all sufficiently large primes. We introduce new geometric and cohomological methods for this problem, moving beyond the earlier combinatorial calculations. We compare the counts over Zp\mathbb{Z}_p and Fp[[t]]\mathbb{F}_p[[t]] and, for all sufficiently large primes, express them as point counts on reductions of varieties over Q\mathbb{Q}. The Grothendieck--Lefschetz trace formula relates these counts to compactly supported ℓ\ell-adic \'etale cohomology. We identify a contribution from Tate twists of the two-dimensional Galois representation attached to the weight-twelve cusp form Δ\Delta. We show that this modular contribution does not cancel when all parts of the count are combined. Consequently, there is no polynomial P∈Q[X]P\in\mathbb{Q}[X] such that f10(p17)=P(p)f_{10}(p^{17})=P(p) for all but finitely many primes pp. This gives a negative answer to Voll's uniformity question even when finitely many exceptional primes are allowed.

Provenance statement

The author used ChatGPT 6.1 Sol as an exploratory tool to study related papers and to assist in correcting mistakes in the article. It was not used to assist in writing the paper or in proving the results.

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Version history

  1. v1 · 67 pagesSubmitted by Anwesh RayInitial depositCurrentNov 07, 2026