The local structure of refined class groups in cyclotomic Iwasawa theory

Abstract

We give an explicit local description of ray class refinements in cyclotomic Iwasawa theory. For distinct primes $p\geq5$ and $\ell$, and even $2\leq k\leq p-3$, we prove that the $ω^{1-k}$-component of the $p$-primary ray class group of $\mathbb{Q}(ζ_p)$, with squarefree modulus supported above $\ell$, is nonzero if and only if $p\mid\ell(1-\ell^k)B_k$. In the cyclotomic tower, we determine the local modules and their correction modulo the diagonal global Tate module, obtaining an explicit determinant formula. Their Frobenius Euler factors, combined with the Mazur--Wiles theorem, yield a ray class form of the Iwasawa main conjecture for nonexceptional odd characters. We also obtain exact class number valuation comparisons at every layer, including those before stabilization, recover Finat's formula for $\ell=2$, and deduce refined $μ$-vanishing from Ferrero--Washington. The cyclotomic character explains the Wieferich exception, while existing Eisenstein congruence theorems connect the divisibility criteria with prime-level Ramanujan-style congruences.