Class groups and local indecomposability for non-CM forms
Number Theory
2023-10-03 v3
Abstract
In the late 1990's, R. Coleman and R. Greenberg (independently) asked for a global property characterizing those -ordinary cuspidal eigenforms whose associated Galois representation becomes decomposable upon restriction to a decomposition group at . It is expected that such -ordinary eigenforms are precisely those with complex multiplication. In this paper, we study Coleman-Greenberg's question using Galois deformation theory. In particular, for -ordinary eigenforms which are congruent to one with complex multiplication, we prove that the conjectured answer follows from the -indivisibility of a certain class group.
Cite
@article{arxiv.1807.02499,
title = {Class groups and local indecomposability for non-CM forms},
author = {Francesc Castella and Carl Wang-Erickson and Haruzo Hida},
journal= {arXiv preprint arXiv:1807.02499},
year = {2023}
}
Comments
40 pages, with a 11-page appendix by Haruzo Hida. v3: improvements to exposition, minor corrections