MODULAR FORMS-page57

MODULAR FORMS-page57 - 53 6.3 Let us give some examples....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
53 6.3 Let us give some examples. Example 6.1 . Let Δ( τ ) = η ( τ ) 24 . It is called the discriminant function. We know that Δ( τ ) satisfies (6.1) with k = 6 with respect to the group Γ = SL(2 , Z ). By (4.9) Δ( τ ) = 1 (2 π ) 8 ϑ 0 1 2 1 2 8 . Since Δ( τ ) = q Y m =1 (1 - q m ) 24 we see that the Fourier expansion of Δ( τ ) contains only positive powers of q . This shows that Δ( τ ) is a cusp form of weight 6. Example 6.2 . The function ϑ 00 ( τ ) has the Fourier expansion P q m 2 / 2 . It is convergent at q = 0. So ϑ 4 k 00 is a modular form of weight k . It is not a cusp form. Let us give more examples of modular forms. This time we use the groups other than SL(2 , Z ). For each N let us introduce the principal congruence subgroup of SL(2 , Z ) of level N Γ( N ) = { M = α β γ δ « SL(2 , Z ) : M I mod N } . Notice that the map SL(2 , Z ) SL(2 , Z /N Z ) , α β γ δ « ¯ α ¯ β ¯ γ ¯ δ « is a homomorphism of groups. Being the kernel of this homomorphism, Γ( N ) is a normal subgroup of Γ(1) = SL(2 , Z ). I think it is time to name the group Γ(1). It is called the full modular group . We have Lemma 6.2. The group
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.

Ask a homework question - tutors are online