Modular forms applied to the computational inverse Galois problem

Abstract

For each of the groups PSL 2 (F 25 ), PSL 2 (F 32 ), PSL 2 (F 49 ), PGL 2 (F 25 ), and PGL 2 (F 27 ), we display the first explicitly known polynomials over Q having that group as Galois group. Each polynomial is related to a Galois representation associated to a modular form. We indicate how computations with modular Galois representations were used to obtain these polynomials. For each polynomial, we also indicate how to use Serre's conjectures to determine the modular form giving rise to the related Galois representation.

Computing modular Galois representations

“…It would be interesting to compute explicitly these Galois representations ρ f,l for several reasons: first, simply for the sake of the Galois representation itself, next, because the number field L will often 1 be an explicit solution to the inverse Galois problem for GL 2 (F ℓ ) (cf [Bos07a], [Bos11]) with controlled ramification, and even for the Gross problem, and, last but not least, because it gives a fast way of computing the q-expansion coefficients a p of f modulo l. Letting ℓ vary, we thus obtain a Schoof-like algorithm (cf [Sch95]) to compute q-expansions of newforms, as bounds on the coefficients a p are well-known.…”