On indefinite modular forms of weight one

Modular Forms of Weight One

The decomposition into an Euler product and the functional equation of the Dirichlet series associated by Hecke to modular forms of weight one suggests that these correspond to Artin L-functions of degree 2 over Q, otherwise known as Galois representations Gal(Q/Q)→ GL2(C). It is this correspondence, conjectured by Langlands, which we establish here. The first three sections are preliminary. Th...

Mock-modular Forms of Weight One

The object of this paper is to initiate a study of the Fourier coefficients of a weight one mock-modular form and relate them to the complex Galois representations associated to the form’s shadow, which is a weight one newform. In this paper, our focus will be on weight one dihedral newforms of prime level p ≡ 3 (mod 4). In this case we give properties of the Fourier coefficients themselves tha...

On “good” Half-integral Weight Modular Forms

If k is a positive integer, let Sk(N) denote the space of cusp forms of weight k on Γ1(N), and let S k (N) denote the subspace of Sk(N) spanned by those forms having complex multiplication (see [Ri]). For a non-negative integer k and any positive integer N ≡ 0 (mod 4), let Mk+ 2 (N) (resp. Sk+ 2 (N)) denote the space of modular forms (resp. cusp forms) of half-integral weight k + 12 on Γ1(N). S...

Level-one elliptic modular forms

[1] Traditional terminology is that f → f |2kγ is the slash operator, although this name fails to suggest any meaning other than reference to the notation itself. In fact, obviously f(z)→ f(γz)(cz + d)−2k is a left translation operator, albeit complicated by the automorphy factor. That is, this is a right action of SL2(Z) on functions on H, while the group action of SL2(Z) on H is written on th...

MAT1200: Modular Forms of Half Integral Weight