Abstract. In this article new cases of the inverse Galois problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL 2 (F p n ) occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their residual Galois representations are as large as a priori possible. Both results essentially use Khare's and Wintenberger's notion of good-dihedral primes. Particular care is taken in order to exclude nontrivial inner twists.
We study modular Galois representations mod p m . We show that there are three progressively weaker notions of modularity for a Galois representation mod p m : We have named these "strongly", "weakly", and "dc-weakly" modular. Here, "dc" stands for "divided congruence" in the sense of Katz and Hida. These notions of modularity are relative to a fixed level M . Using results of Hida we display a level-lowering result ("strippingof-powers of p away from the level"): A mod p m strongly modular representation of some level Np r is always dc-weakly modular of level N (here, N is a natural number not divisible by p). We also study eigenforms mod p m corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod p m to any "dc-weak" eigenform, and hence to any eigenform mod p m in any of the three senses. We show that the three notions of modularity coincide when m = 1 (as well as in other particular cases), but not in general.
Suppose that φ(f, g) = 0. Then we have 0 = θ(Af + F (g)) = θ(Af ), hence 0 = θ(f ). But then f = 0, as the weight of f is not divisible by p. It follows that F (g) = 0, hence that g = 0.
4.5Now that we know how to characterize the image ofKatz , it becomes time to investigate how to compute this ambient vector space. We want to avoid the problems related to the lifting of elements of S p (Γ 1 (N), ε, F) ′ Katz to characteristic zero with a given character (see [22, §1] and [22, Prop. 1.10], they have to do with what is called Carayol's Lemma). So we describeKatz of characteristic zero forms of weight p with no prescribed character.Proposition 4.6 Let p be a prime number, and N ≥ 1 an integer not divisible by p. Suppose that N = 1 or that p ≥ 5.
In this article we consider mod p modular Galois representations which are unramified at p such that the Frobenius element at p acts through a scalar matrix. The principal result states that the multiplicity of any such representation is bigger than 1.
Let f be a cusp form of weight k + 1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato-Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp 2 )} p where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it for the family {a(tn 2 )} n where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato-Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of Dedekind-Dirichlet density.
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