Eisenstein congruences in tame families and the class group of a metabelian extension
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.
Related
- Degeneration of monodromy in the theory of Eisenstein congruences for classical groups
- Class polynomial obstructions to canonical congruences for the smallest parts function
- Matsumura's extension problem for pluricanonical forms in Kähler families I: the smooth and essentially Moishezon cases
- Infinite families of pairs of real quadratic fields whose class numbers are divisible by a given integer
- A dispersive extension result for a class of nonlocal nonlinearities
BibTeX
@misc{hsu2026eisenstein,
title = {{Eisenstein congruences in tame families and the class group of a metabelian extension}},
author = {Catherine Hsu and Alice Pozzi and Preston Wake and Carl Wang-Erickson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03401},
eprint = {2610.03401},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03401},
}
RIS
TY - UNPB
TI - Eisenstein congruences in tame families and the class group of a metabelian extension
AU - Catherine Hsu
AU - Alice Pozzi
AU - Preston Wake
AU - Carl Wang-Erickson
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03401
UR - https://arxiv.org/abs/2610.03401
AB - 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.
ER -