MODULAR FORMS-page67

MODULAR FORMS-page67 - 63 (iii) Show that L ( ) is almost...

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

View Full Document Right Arrow Icon
63 (iii) Show that θ L ( τ ) is “almost” modular form for the group Γ 0 ( N ) = { α β γ δ « SL(2 , Z ) : N | c } , i.e. θ L ( ατ + β γτ + δ ) = ( γτ + δ ) k/ 2 χ ( d ) θ L ( τ ) , α β γ δ « Γ 0 ( N ) , where χ ( d ) = ( ( - 1) k 2 D d ) is the quadratic residue symbol. (iv) Prove that θ L ( τ ) is a modular form for Γ 0 (2) whenever D = 1 and k 0 mod 4. 6.11 Let Φ( z ; τ ) be a function in z and τ satisfying the assumptions of Theorem 6.2 (such a function is called a Jacobi form of weight m and index 0 with respect to the group Γ). Show that (i) ( z ; τ ) is a Jacobi form of weight 2 and index 0 with respect to Γ(1); (ii) σ ( z ;1 ) is a Jacobi form of weight 1 with respect to Γ(1). 6.12 Let n be a positive integer greater than 2. Consider the map of a complex torus E τ \ { 0 } → C n given by the formula z (1 ,℘ ( z ) ,...,℘ ( z ) n - 1 2 ,℘ ( z ) 0 ,℘ ( z ) ( z ) 0 ,...,℘ ( z ) n - 3 2 ( z ) 0 ) if n is odd and z (1 ,℘ ( z ) ,...,℘ ( z ) n 2 ,℘ ( z ) 0 ,℘ ( z ) ( z ) 0 ,...,℘
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