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By David Mumford (auth.)

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Extra resources for Tata Lectures on Theta I

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The set of modular forms of weight k & level N is a vector space and is denoted by Mod (N) . k Thus any f c Mod strip; and the circles of radii d P. (N) . q (i. • touching the real axis at the rational points p/q) are called the horocircles for f. See Fig. l) acts on the "rational boundary points" ~U(m}ofH and that if f«a 1'+b)/(c 1'+dl) = (c T+d) k f( 1') , a( l +b then the bound at p/qc~U(ro1is equivalent to the bound at c(p q) + d Cthe bound at 00 being the condition (b) (il : I f(T )1 ~ c if 1m 'I' > d).

Exp (-2niy(ad-bc) + nic ( , d) C 1'+ ! (y, 1') 2 - nia 1') = exp (-2niy - ni d(a 2 1' (C1'+d) _ cIa 1'+b)2)] c 1'+ . (a 1'd - 2abc l' - b c») c 1'+d 30 But a 2 '1'd - 2 abc 'I' _b 2 c • a(ad-bc)'I' -ab(c'l'+d) + b(ad-bc) = (a 'I' +b) - ab(c 'I' +d). Now using ab is even, we get what we want. If we now recall the characteri$ation of I(y, 1") as a function of y as in S I, namely, '(y, 'I' ') is the unique function (upto scalars) invariant under"or' when> ... (a-r+b)/(c1'+d)), we find t(y, 1') is one such.

The details a re lengthy (and hence omitted)but straight forward (the usual properties of the Jacobi symbol, e, g •• reciprocity, must be used), It is, however, a priori clear that the method must give a function equation of type (F 1) for some 8th root , of 1. S 8. The Heat equation again. The transformation formula for "(z, '1') allows us to see very explicitly what happens to the real valued function "(x, it), studied in S 2, when t---;:' O. In fact, (F 3) says: .! 2 '(xl it, i/t) = t 2 exp (TT x It) "(x, it) hence '(x,it) "t _J.

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