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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
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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.

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