Density of Vanishing of Certain Eigenspaces of Cyclotomic Class Groups

Abstract

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 $d\ge 4$, put $N=(p-1)/d$, and let $U_d=(\mathbb{Z}/d\mathbb{Z}{)}^{\times }$. For a relative density-one set of primes $p\equiv d+1(\mathrm{mod}2d)$, we prove that the odd block ${⨁}_{a\in U_d}{A}_{aN}$ has order at most $p^{\phi (d)/2-1}$, and reflection shows that the even block ${⨁}_{a\in U_d}{A}_{p-aN}$ has $p$-rank at most $\phi (d)/2-1$. For each $d\in \{4,6\}$, both blocks vanish for a relative density-one set of primes $p\equiv d+1(\mathrm{mod}2d)$; 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.

13 pages