Integer group determinants for the elementary abelian group of order 49

Abstract

Let $G=C_7\times C_7$, and let $S(G)$ denote the set of integer values of its group determinant. We determine $S(G)$ by classifying the values divisible by $7$; the coprime values are already known. Every nonzero divisible value has valuation at least $10$, and every multiple of $7^{12}$ occurs. At valuations $10$ and $11$, we give necessary and sufficient conditions on the cofactor in terms of two additive invariants of ideals in $\mathbb{Z}[\zeta_7]$. The first condition is a bounded signed sum of prime-ideal invariants. The second requires a prime ideal with nonzero invariant pair whose norm divides the cofactor. Neither cofactor set is a union of congruence classes modulo any positive integer. The least positive divisible value is $43\cdot7^{10}$, and the least positive value of valuation $11$ is $8\cdot7^{11}$. The proof combines integral reconstruction from character values with a calculation of the global-unit image modulo $7$. We conclude by identifying the additional local conditions and realization problems that arise at primes at least $11$.