Modular forms with large coefficient fields via congruences

TL;DR

This paper demonstrates the existence of modular forms of weight 2 with arbitrarily large coefficient fields and specified level properties, using congruences, elliptic curves, and Frey curves.

Contribution

It introduces a method to construct newforms with large coefficient fields and controlled levels, expanding understanding of modular forms' coefficient fields.

Findings

Existence of newforms with arbitrarily large coefficient fields.

Construction of forms with specified prime divisors in level.

Application of congruences and elliptic curve theory to modular form construction.

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.