math.NT — Number Theory

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

Contributed by Anwesh Ray

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