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By Alexander Polishchuk

This publication is a contemporary therapy of the idea of theta services within the context of algebraic geometry. the newness of its technique lies within the systematic use of the Fourier-Mukai remodel. Alexander Polishchuk starts off by means of discussing the classical thought of theta services from the point of view of the illustration thought of the Heisenberg team (in which the standard Fourier rework performs the well-liked role). He then exhibits that during the algebraic method of this conception (originally because of Mumford) the Fourier-Mukai rework can usually be used to simplify the present proofs or to supply thoroughly new proofs of many vital theorems. This incisive quantity is for graduate scholars and researchers with powerful curiosity in algebraic geometry.

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B) Show that the line bundle L = L(H, α0 ) on C/ has degree one. (c) Prove that for fixed τ the function θ(z, τ ) has simple zeroes at z ∈ τ +1 + Z + Zτ and no other zeros. 2 Let ⊂ V be a lattice. Assume that for some nonzero holomorphic function f on V one has f (x + γ ) = c(γ ) · exp π H (x, γ ) + 7. π H (γ , γ ) f (x) 2 for all x ∈ V , γ ∈ . Show that c(γ1 + γ2 ) = c(γ1 )c(γ2 ) exp(πi E(γ1 , γ2 )). Let us fix τ in the upper half-plane and consider the following function of complex variables z 1 , z 2 : F(z 1 , z 2 ) = sign(α(z 1 ) + m) exp(2πi[τ mn + nz 1 + mz 2 ]), (α(z 1 )+m)(α(z 2 )+n)>0 where α(z) = Im(z)/ Im(τ ).

If were not injective, we would have a pair of points v1 , v2 ∈ V such that v2 −v1 ∈ , and a constant λ ∈ C∗ such that for every f ∈ T (3H, , α 3 ) one has f (v2 ) = λ f (v1 ). To get a contradiction it is enough to consider f (v) = θ(v − a)θ(v − b)θ(v + a + b) for various θ ∈ T (H, , α) \ 0. Then the trick is to consider both sides of the identity f (v2 ) = λ f (v1 ) as functions of a and to take the logarithmic derivative of the identity. As a result we get the invariance of the meromor2 + v) 2 + v) under translations.

2). 1. 1) given by α. Recall that the space T (H, , α) of canonical theta functions consists of holomorphic 27 28 Theta Functions I functions f on V , invariant under the action of in (an extension of) Fock representation, where is lifted to H(V ) using σα . Let N ( ) be the normalizer of in H(V ). Then the group G(E, , α) := N ( )/ acts naturally on the space T (H, , α). 1, G(E, , α) is a Heisenberg group. More precisely, it is a central extension of the finite abelian group ⊥ / by U (1), where ⊥ = {v ∈ V : E(v, ) ⊂ Z}.

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