Modular forms with large coefficient fields via congruences
Number Theory
2011-11-24 v1
Abstract
In this paper we apply results from the theory of congruences of modular forms (control of reducible primes, level-lowering), the modularity of elliptic curves and Q-curves, and a couple of Frey curves of Fermat-Goldbach type, to show the existence of newforms of weight 2 and trivial nebentypus with coefficient fields of arbitrarily large degree and square-free or almost square-free level. More precisely, we prove that for any given numbers t and B, there exists a newform f of weight 2 and trivial nebentypus whose level N is square-free (almost square-free), N has exactly t prime divisors (t odd prime divisors and a small power of 2 dividing it, respectively), and the degree of the field of coefficients of f is greater than B.
Cite
@article{arxiv.1111.5592,
title = {Modular forms with large coefficient fields via congruences},
author = {Luis Dieulefait and Jorge Jimenez Urroz and Kenneth Ribet},
journal= {arXiv preprint arXiv:1111.5592},
year = {2011}
}