Greenberg's conjecture for real quadratic number fields

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.