6. Theta Series

Question 1

Define the Fourier transform f ̂ (y) of some function f(x)

Answer 1

f ̂ (y) = INT -∞,∞ f(x)e^2πixy dx

Question 2

Define the Poisson summation formula

Answer 2

The periodisations F(x) = SUM_n f(x+n) and F ̂ (y) = SUM_n f ̂ (y+n) are both absolutely and uniformly convergent to the function F(n) = F ̂ (n)

Question 3

Define the Jacobi theta series θ(τ)

Answer 3

θ(τ) = SUM_n e^2πin^2 τ = 1 +2 SUM 1,∞ q^n^2

Question 3

Define the theta transformation formula

Answer 3

θ(-1/4τ) = sqrt(-2iτ)θ(τ)

Question 4

Define the Jacobi theta series as a modular form

Answer 4

A modular form of weight 1/2 for the congruence subgroup Γ_0(4)

Question 5

Give the mth power of the Jacobi-theta series

Answer 5

θ^m(τ) = SUM n,∞ r_m (n)q^n

Question 6

Define r_m in the mth power of the jacobi-theta series

Answer 6

r_m(n) = #{(x1, x2, … , xm) in Z^m; SUM 1,m x^2 = n}

The number of ways you can write integer n as the sum of m squares.

Question 7

What do we find for the Jacobi theta series θ^m(τ) where m divisible by 4

Answer 7

θ^m(τ) ∈ M_m/2 (Γ_0(4))