Iwasawa-Type Spectral Resultant Growth Laws for Grover Walks on Graph Towers

Abstract

Let $X_0\leftarrow X_1\leftarrow \cdots$ be a ${\mathbb{Z}}_p^d$-tower of finite graphs, and let $U_n$ be the Grover transition matrix on $X_n$. We study Iwasawa-type $p$-adic growth laws for the polynomial spectral quantities \[ \det P(U_n), \] where $P(A)$ is a monic polynomial. The basic object is the spectral resultant \[ \mathcal R_{X,P}(T)=\operatorname{Res}_A(\mathcal F_X(A,T),P(A)), \] where ${\mathcal{F}}_X(A,T)$ is the universal Grover--Ihara spectral polynomial of the tower. In the integral setting, this resultant generates the zeroth Fitting ideal of a natural finite module over the Iwasawa algebra; when the resultant is nonzero, this module is torsion. The polynomial $P$ packages prescribed spectral values into a single spectral packet. If $P$ is coprime to the Bass factor $A^2-1$ and ${\mathcal{R}}_{X,P}$ does not vanish at torsion characters, then $\det P(U_n)$ is nonzero for all $n$ and we prove a Cuoco--Monsky type leading asymptotic formula for $v_p(\det P(U_n))$. The leading terms are given explicitly by the $\mu$- and $\lambda$-invariants of ${\mathcal{R}}_{X,P}$, with a separate correction coming from the Bass factor. For $P(A)=A-a$, with $a\ne \pm 1$ and $a$ not an eigenvalue at any level, this recovers the leading invariants in the fixed non-eigenvalue formula for Grover characteristic polynomials. We also prove an equivariant factorization of spectral resultants for finite connected $p$-group covers. As a consequence, we obtain an unramified equivariant Kida formula under explicit integrality and nonzero-resultant assumptions. Finally, when $\gcd (P,A^2-1)=1$, we show that torsion zeros of ${\mathcal{R}}_{X,P}$ correspond exactly to occurrences of roots of $P$ as Grover eigenvalues at finite levels. The examples include the $K_3$-tower, non-abelian Heisenberg $5$-group covers, and an explicit torsion-zero spectral packet.