## Algebraic Topology by Edwin H. Spanier

By Edwin H. Spanier

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A, b, c, d € 7Lt ad-be = +1 a and assume that ab,cd are even. Multiplying by -1 if necessary, we assume that c > 0. Consider the function d((cT + d) y, T). Clearly, when y is replaced by y + l , the function is unchanged except for an exponential factor. It is not 29 o hard to rig up an exponential factor of the type exp (Ay ) which c o r r e c t s 0 ( ( c f + d ) y, T) to a p e r i o d i c function for y l—> y + l . In fact, let Y(y, f ) = e x p (Trie ( c T + d)y 2 ) * ( ( c T +d)y, T). B: a factor exp (TTicd) a p p e a r s , s o w e u s e H o w e v e r , the p e r i o d i c behaviour of 0 for cd even in the v e r i f i c a t i o n ) .

2. , * , 2 0 ) / * 2 k (0, T). 2 2 2 Finally, there can be no further relations between & , f , f because the only polynomials that vanish on the conic x x 2 O - x 2 = x- + x 2 are multiples of 2 - x . , as required. £i 1 % 11. £ as an automorphic form in 2 variables. So far we have concentrated on the behaviour of 0 ( z , T ) as a function of z for fixed T , and as a function of T for z = 0. Let us now put all this together and consider £ as a function of both variables. First of all, it is easy to see that the functional equations on £ , plus its limiting behaviour as Im T Proposition 1 1 .

Thus K • Mr Mod « © Mod, k k«ZZ+ is a graded ring, called the ring of modular forms of level N. 2. £ 2 (0, T), * 2 (0, T) and * (0, T) are modular forms of weight 1 & level 4. Proof. To start with, condition (a) for 0 2 (0, T) amounts to saying that C , QO the 8th root of 1, in the functional equation ( F ^ is 1 1 when (* j j ) « r ™ is immediate from the description of C (in fact, we only need c even and d = 1 (mod 4)). We can also verify immediately the bound (b)(ii) at co for S 40 2 £ (0, T).