8. Hecke Theory

Question 1

Define a Hecke operator

Answer 1

The operator T_p which, when acting on a function f ∈ M_k(Γ), we find T_p f ∈M_k(Γ).

Equivalent for cusp form

Question 2

What conditions must the Hecke operator satisfy

Answer 2

Those for a modular form: i.e. holomorphicity and modularity.

Question 3

How can we relate the q-coefficients for f and Tf

Answer 3

b_n = a_pn +p^k-1 a_n/p

Question 4

Define an eigenvector and eigenvalue for the Hecke operator

Answer 4

T_p f = λ_p f

where a_p = λ_p a_1

Question 5

Give the eigenequation for the Hecke operator and Eisenstein series

Answer 5

Eigenfunction G_k
Eigenvalue σ_k-1 (p) = p^k-1 +1

Question 6

What are the commutators of Hecke operators for primes p and q?

Answer 6

All commute

Question 7

Given we can write every integer as a unique product of primes to different powers, what is the corresponding T_m

Answer 7

The product of all the T_p^l which make up m

Hence, the Heck operator is multiplicative for coprime values

Question 8

Define an eigenform

Answer 8

A modular form which is an eigenfunction for all T_m