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By Walker R. J.

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This property is easily proved by distinguishing separately points of ~ and cusps, and making use of the formula for the imaginary parts of yz and z. Hence we have an injection and rv\U is an open neighborhood of the projection of v in r\f>*. Case I. z E ~ and r z contains only 1 or - 1. Then the map U ---. F\ U is a homeomorphism. We take the inverse mapping as a chart. 27 § 1. The Rieman Surface I'\~* Case 2. Z E i) and r ~ contains other elements besides 1 or possibly - 1. Let I'~ = r ~ or r z/{± I} according to these two cases.

We need some definitions. We define a function H(n) for integers n first by putting H(n) = 0 ifn < 0 and H(O) = -1/12. If n > 0, we let H(n) be the number of equivalence classes with respect to SL 2 (Z) of positive definite binary quadratic forms ax 2 + bxy + cy2 with discriminant b2 - 4ac = -n, counting forms equivalent to a multiple of x 2 + y2 (resp. x 2 + xy multiplicity t (resp. 1)· If n == 1 or 2 (mod 4) then H(n) = O. We have the following table. 3 n 0 H(n) -TI 1 1 3" + y2) with 4 7 8 11 12 15 16 19 20 23 24 1 1 1 1 3" 4 2 "2 3 I 2 2 "2 3 We also define a polynomial Pk(t, N) (k > 0 even) as the coefficient of Xk- 2 in the power series development of We also have = Pk(t, N) Pk-l - p-k-l _ p-p where p+ p= t and For instance P 2 (t, N) =' 1 and P4 (t, N) = t 2 pp = N - N.

We reproduce the lemma below for the convenience of the reader, and this concludes the proof. "'1, ... ,"'r Lemma. Let be distinct characters they are linearly independent. 0/ an algebra over afield. Then Proof If there exists a shortest non-trivial linear relation, say C1"'1 +C2"'2+"'=0, with c 1, C2 =/- 0, pick To such that'" 1 (To)=/- '" 2(To). We evaluate the relation at both a variable element T, and at ToT. One of the values, say'" 1 (To) cannot be equal to O. We divide by", 1 (To) and subtract the two relations to give a shorter relation, contradiction.

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Algebraic Curves by Walker R. J.

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