Greenberg's conjecture for real quadratic number fields
Pietro Mercuri, Maurizio Paoluzi, René Schoof
TL;DR
The paper verifies Greenberg's conjecture for $p=3$ across real quadratic fields with discriminants $f<10^5$ by computing the $3$-class groups in the cyclotomic $\mathbf{Z}_3$-extension. It recasts the problem in terms of a Galois module $C(f)=\Lambda/J$ over the Iwasawa algebra $\Lambda=\mathbf{Z}_3[[T]]$ and reduces the question to the finite-quotient behavior $C_n=C(f)/\omega_n C(f)$ (or $\omega'_n$ when $f\equiv1\pmod 3$). An explicit, coefficient-heavy algorithm computes shrinking ideals $J+(\omega_n)$ (or $J+(\omega'_n)$) until stabilization occurs, using upper bounds from cyclotomic units and Gras-type lower bounds; Nakayama’s lemma then yields finiteness of $C(f)$ and hence $C_n=C(f)$ for large $n$. Across $f<10^5$, the authors find $C(f)$ finite in all cases, with most fields giving $C(f)=0$ and a minority giving nontrivial but finite modules, thus providing strong computational support for Greenberg's conjecture in this setting. The results distinguish cases $f\not\equiv 1\pmod 3$ and $f\equiv 1\pmod 3$ and report detailed data, including several exotic $J$-shapes, contributing a comprehensive numerical portrait of $3$-class growth in these towers.
Abstract
We compute the $3$-class groups $A_n$ of the fields $F_n$ in the cyclotomic $\mathbf{Z}_3$-extensions of the real quadratic fields of discriminant $f<100,000$. In all cases the orders of $A_n$ remain bounded as $n$ goes to infinity. This is in agreement with Greenberg's conjecture.