Eisenstein congruences in tame families and the class group of a metabelian extension

Source abstract

Ribet's 1976 proof of the converse to Herbrand's theorem created a new paradigm in algebraic number theory by illustrating that unramified abelian Galois extensions of number fields can be constructed using Galois representations associated to Eisenstein congruences, i.e., cuspidal eigenforms that are congruent to Eisenstein series. In recent years, further progress has been made in investigating the quantity of such Eisenstein congruences, showing that when there are many such congruences, additional and finer information can be gleaned about the arithmetic structure of the related extensions. In this vein, we use new results on the existence of Eisenstein congruences in tame families to explain a computational observation about splitting behavior of primes in a certain metabelian number field.