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Unformatted text preview: 101 show that each matrix A from A N is contained in the lefthandside. This follows from the well known fact that each integral matrix can be transformed by integral row and column transformations to the unique matrix of the form ` n n , where n  n . The last assertion can be checked by using elementary number theory. When N = p is prime, we obviously have # A p = p + 1. Now, if N is not prime we have # A N = ( N ) = N Y p  N (1 + p 1 ) = ,N . This can be proved by using the multiplicative property of the function ( n ) and the formula ( N ) = X d  N d ( d, N d ) (( d, N d )) , where is the Euler function. Lemma 10.3 . Let f ( ) be a modular function with respect to (1) which is holomor phic on H and admits the Fourier expansion f = P n = r c n q n . Then f is a polyno mial in j ( ) with coefficients in the subring of C generated by the Fourier coefficients c ,...,c r ....
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 Fall '09
 ONTONKONG
 Transformations

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