Abstract
We use the theory of congruences between modular forms to prove the existence of newforms with square-free level having a fixed number of prime factors such that the degree of their coefficient fields is arbitrarily large. We also prove a similar result for certain almost square-free levels.
Highlights
1 Background In this paper we will exploit the theory of congruences between modular forms to deduce the existence of newforms with levels of certain specific types having arbitrarily large coefficient fields
If the level is allowed to be divisible by a large power n of a fixed prime, or by the cube of a large prime p, the coefficient fields of all newforms of this level will grow with n due to results of Hiroshi Saito showing that the maximal real subfield of certain cyclotomic field whosedegree grows with n will be contained in these fields of coefficients
For t ≥ 3 we follow a completely different approach, namely, we exploit congruences involving certain elliptic curves whose construction is on the one hand related to Chen’s celebrated results on Goldbach’s conjecture and on the other hand inspired by Frey curves as in the proof of Fermat’s Last Theorem
Summary
In this paper we will exploit the theory of congruences between modular forms to deduce the existence of newforms (in particular, cuspidal Hecke eigenforms) with levels of certain specific types having arbitrarily large coefficient fields. For t ≥ 3 we follow a completely different approach, namely, we exploit congruences involving certain elliptic curves whose construction is on the one hand related to Chen’s celebrated results on (a partial answer to) Goldbach’s conjecture (cf [2]) and on the other hand inspired by Frey curves as in the proof of Fermat’s Last Theorem (the diophantine problem that we will consider will be a sort of Fermat-Goldbach mixed problem). There exist α ∈ {5, 8} and t different odd primes p1, p2, ...., pt such that if we call N the product of these t primes, in the space of cuspforms of weight 2, level 2αN and trivial nebentypus there exists a newform f with field of coefficients Qf satisfying: Let us stress that the results on prime values of x4 + y2 /c, in Theorem 2 and its corollary, are interesting in its own right, independently of the application to finding newforms with large coefficient fields. It is the case that some of the computations done in [8] do not apply to this case in a straightforward manner and, they must be done with the required level of generality in the variable c
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