Bernoulli numbers and the integer group determinants of $C_p \times C_p$

Abstract

For a prime $p\ge7$, let $S(C_p\times C_p)$ be the set of integer group determinants and $e(p)$ the least exponent for which $p^{e(p)}\mathbf Z\subseteq S(C_p\times C_p)$. Let $r_p$ be the least integer $r\ge1$ satisfying $(2r+1)i>r(p-1)$ for every irregular pair $(p,2i)$ with $r\mid i$. We prove the unconditional inclusion $p^{p+3+2r_p}\mathbf Z\subseteq S(C_p\times C_p)$. Equality $e(p)=p+3+2r_p$ holds when Vandiver's conjecture holds for $p$ and $r_p\le(p-5)/2$, hence for every prime $7\le p<2^{31}$. The first exception to $e(p)=p+5$ is $p=103$, where $e(103)=116$. A polynomial obstruction from Bernoulli numbers supplies the lower bound, and circular units give the construction. For regular primes, we characterize the entire determinant set through explicit affine systems over $\mathbf F_p$ and principality conditions in $\mathbf Z[ζ_p]$. For regular $p\ge11$, prime cofactors cannot occur at valuation $p+3$; at valuation $p+4$ they have natural density $(p+1)/(hp^{(p-3)/2})$, where $h$ is the class number of $\mathbf Q(ζ_p)$. We also show that $p^{p+5}$ is the least absolute value of a nonzero $p$-divisible determinant for these regular primes.