Operads and hecke operators on modular forms

Abstract

Let Mk(Γ) be the collection of modular forms over C of weight k with respect to a congruence subgroup Γ, it is well-known double cosets ΓgΓact on it as linear maps. Those operators are known as Hecke operators. In this paper, we first show that similar double cosets ΓngΔΓ give multi-linear maps Mk1(Γ) x ⋅ ⋅ ⋅ x Mkn(Γ) → Mk1+⋅⋅⋅+kn(Γ), and we show these operators form an algebraic structure called operad . Then we define a Galois action on this operad which is compatible with the Galois action on modular forms. By taking the Galois orbit, we find a suboperad which acts on the integral modular forms.

Date of Award	2011
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology