Download Tata Lectures on Theta III by David Mumford, M. Nori, P. Norman PDF

By David Mumford, M. Nori, P. Norman

This quantity is the 3rd of three in a sequence surveying the idea of theta services which play a important function within the fields of advanced research, algebraic geometry, quantity idea and such a lot lately particle physics. in line with lectures given by way of the writer on the Tata Institute of basic examine in Bombay, those volumes represent a scientific exposition of theta features, starting with their ancient roots as analytic capabilities in a single variable (Volume I), referring to the various appealing methods they are often used to explain moduli areas (Volume II), and culminating in a methodical comparability of theta services in research, algebraic geometry, and illustration idea (Volume III). Researchers and graduate scholars in arithmetic and physics will locate those volumes to be worthwhile additions to their libraries.

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28 TATA LECTURES ON THETA III Recall that in Ch. ^ X2 (T) = e""''^' (^^1+^2) . ^(j^xi + X2, T ) . In terms of theta-functions with characteristic, Xi X2 iT) = ^2 (0;T). We will see that in many ways ^ " is the most natural and most important variant oft?. In particular, Cor. e.. 5. I The space of functions on R^^ ^ XI—^(f>f{x) = (t/(i,^)/,ez), some f G Tioo (The space ofC^ - functions g{x), x eW^\ I such that V . I g{x + n) = e,(f )e(x, ^)g{x), all nel^' J Hence the space on the right is contained in L^(R^^//Z^^).

L so that p{crn(x)£) = p(£) + X. The quotient of n^L by this action is precisely the line bundle L on X. 12. 9, we have j(fXW^^{x) for all X e T{Y) and h{a^'{x), = a'^{V{f){x)) (T'-2(X)) = a^'^^^{x) for all x G T{X). 12 follow by just plain checking. 13. The three subgroups ofQ{L) defined below are iden- tical: (A) the normaliser of a^T{X), (B) the subgroup of{xn,(t>n) with xi G ^"^(L), and (C) the elements g ofQ(L) which have a representative of the type {xn^n) with

Next we consider the sections r{XT,Lj^) of the basic line bundle L over XT^L' We easily see that r(Y t \ c:^ f Sp^c^ of cr(L) — invariant sections of the 1 I A T , L , •-; - I ^^—^j bundle C X C^ on C^ J ^ f Space of entire functions f{z) on C^ invariant by 1 "" \ (T{L) in the Fock representation W^(C^,T) J ' FINITE HEISENBERG GROUPS 39 In fact these invariant functions must belong to the space 7^^(C^,T)_oo, as one checks as follows: a) Define the norm AT : C X C^ — . R+ by b) a small calculation shows that N(a,z) = N(xph{(^,z)), all h G Heis{2g, R), where il)h{(^,z) = {aX'^exp 7ri^yi{2z-y),z -y).

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