MODULAR FORMS-page74

MODULAR FORMS-page74 - 70 LECTURE 7 THE ALGEBRA OF MODULAR...

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

View Full Document Right Arrow Icon
70 LECTURE 7. THE ALGEBRA OF MODULAR FORMS ( e 0 1 ,...,e 0 n ). Then the pre-image of the standard lattice Z n = Z e 1 + ... + Z e n is an integral lattice L with the distance function Q . Let us define the theta function of the lattice L by setting θ L ( τ ) = X m =0 r L ( m ) q m = X v L q Q ( v ) / 2 , (7.7) where r L ( m ) = # { v L : Q ( v ) = 2 m } . (see Exercise 6.10). Since r L ( m ) (2 m ) n/ 2 (inscribe the cube around the sphere of radius 2 m ), and hence grows only polynomially, we easily see that θ L ( τ ) absolutely converges on any bounded subset of H , and therefore defines a holomorphic form on H . We shall assume that L is unimodular , i.e. the determinant of the matrix ( a ij ) is equal to 1. This definition does not depend on the choice of a basis in L and is equivalent to the property that L is equal to the set of vectors w in R n such that w · v Z for all v L . For example, if L is the standard lattice Z n we see from Lecture 4 that θ
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