Density of Vanishing of Certain Eigenspaces of Cyclotomic Class Groups

RESUMO

For an odd prime $p$, let $A_j$ be the $\omega^j$-eigenspace of the $p$-primary class group of $\mathbb Q(\zeta_p)$. Fix an even integer $d\ge4$, put $N=(p-1)/d$, and let $U_d=(\mathbb Z/d\mathbb Z)^\times$. For a relative density-one set of primes $p\equiv d+1\pmod{2d}$, we prove that the odd block $\bigoplus_{a\in U_d}A_{aN}$ has order at most $p^{\varphi(d)/2-1}$, and reflection shows that the even block $\bigoplus_{a\in U_d}A_{p-aN}$ has $p$-rank at most $\varphi(d)/2-1$. For each $d\in\{4,6\}$, both blocks vanish for a relative density-one set of primes $p\equiv d+1\pmod{2d}$; in particular, the even components vanish in accordance with Vandiver's conjecture. For each $d\in\{8,10,12\}$, the odd block has order at most $p$ for a relative density-one set of primes in the same progression, so every summand $A_{aN}$, $a\in U_d$, is cyclic, in accordance with Iwasawa's cyclicity conjecture. The proof uses an atomless limiting law for products of Dirichlet $L$-values, integrality of generalized Bernoulli norms, the relative class-number formula and reflection. A separate exact computation proves $A_{34}=0$ for every odd prime $p$; a single-file PARI/GP program reproduces the calculation.