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arXiv · 1111.5592

Modular forms with large coefficient fields via congruences

Luis Dieulefait·Jorge Jimenez Urroz·Kenneth Ribet
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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.

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Luis Dieulefait, Jorge Jimenez Urroz, Kenneth Ribet. 2011-11-23. Modular forms with large coefficient fields via congruences. https://arxiv.org/abs/1111.5592

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