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Extra info for Elliptic Curves: Notes from Postgraduate Lectures Given in Lausanne 1971/72

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D. 5) Corollary. The holomorphic functions for any 0(= (ca b) G2k ( ex ('1:')) = ( c 1: GZk(~) satisfy + d) 2kG 2k ( 1:) , ( k >' l) d e:SLZ(Zl), and tend to a finite limit when 1: tends to infinity on the imaginary axis. This is just a reformulation of the homogeneity properties, in view of the proposition. ) Thus, if 0( = (ca ~), J (0<, "t') ( a1:' + C'r + b)' _ d (ac1:' + ad - ~ =7. ac~ - , (DCESL ( 7l)). 2 cb)/(c~ + d) 2 (C1: + d)-2 hope that this function will never be confused with the modular ~e invariant J = g~/ f:.

In any 3m by definition. Another way of saying the same thing would be to consider the graded algebras [[X], d' being defined as the double of the usual degree, and a: [y] , d" being defined as the triple of the usual degree, and then [[X,Y] = [[X]0 [[Y]with the degree d = d'~ d". The Poincare series of ([ [X] ,d') is obviously 1 + T2 + r 4 + r 6 + = 1/(1 - r 2) because this algebra has one generator Xk in degree d' = 2k and no non-zero element of odd degree. Similarly the Poincare series of (a: [Y] ,d tt ) is (1 - T3 ) -1.

0 whenever £ - £' whenever - ~ 0 In particular if d is the divisor of a theta function 9, we see that div(9) ~ 0 exactly when 9 is holomorphic. Although the divisor of the zero function is not defined, we make the formal convention that div(O) ~ £ for all £ € Div(E). Then we define L(~) for any divisor £ ~ = {f L-elliptic: div(f) ~ -£ i Div(E). Due to the preceding convention 0 E:L(£) , and it is easily seen that L(~) is a complex vector space (the nota- tion L comes from the appellation linear system, sometimes used for this space, or for the set of divisors of the form div(f) for f E: L (£) ).

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