On the average number of octahedral modular forms

On the average number of octahedral modular forms

λf (p) = Tr(ρ(Frobp)) χ(p) = det(ρ(Frobp)) for all p not dividing N . Following [4], we define SArtin 1/4,k (N,χ) to be the finite set of primitive weight k cupsidal eigenforms which admit an associated Galois representation. If ρ : Gal(Q̄/Q)→ GL2(C) is a Galois representation, we define Pρ to be the composition of ρ with the natural projection GL2(C)→ PGL2(C). Two-dimensional complex Galois rep...

Lectures on Applications of Modular Forms to Number Theory

Cm Number Fields and Modular Forms

x∂F⊂a⊂OF (Na)k−1. Here the superscript ‘+′ stands for totally positive elements. Siegel further derived from this a simpler proof of his famous theorem that ζF (1− k) is rational for all k ≥ 1. Siegel’s construction is based on the simple observation that a Hilbert modular form becomes an elliptic modular form when restricting diagonally to the upper half plane. Indeed, Hecke constructed and pr...

Modular Forms and Applications in Number Theory

Modular forms are complex analytic objects, but they also have many intimate connections with number theory. This paper introduces some of the basic results on modular forms, and explores some of their uses in number theory.

Adjoint Motives of Modular Forms and the Tamagawa Number Conjecture

This paper concerns the Tamagawa number conjecture of Bloch and Kato [B-K] for adjoint motives of modular forms of weight k ≥ 2. The conjecture relates the value at 0 of the associated L-function to arithmetic invariants of the motive. We prove that it holds up to powers of certain “bad primes.” The strategy for achieving this is essentially due to Wiles [Wi], as completed with Taylor in [T-W]....