By Alexander Polishchuk

This booklet is a latest therapy of the idea of theta services within the context of algebraic geometry. the newness of its strategy lies within the systematic use of the Fourier-Mukai rework. Alexander Polishchuk begins through discussing the classical concept of theta features from the point of view of the illustration concept of the Heisenberg crew (in which the standard Fourier remodel performs the famous role). He then indicates that during the algebraic method of this thought (originally because of Mumford) the Fourier-Mukai rework can frequently be used to simplify the prevailing proofs or to supply thoroughly new proofs of many very important theorems. This incisive quantity is for graduate scholars and researchers with robust curiosity in algebraic geometry.

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1) → U (1) should satisfy α(γ1 + γ2 ) = exp(πi E(γ1 , γ2 ))α(γ1 )α(γ2 ). 2) on the data used in the construction of a line bundle on V / . The corresponding space F( ) can be identiﬁed with the space of L 2 -sections of certain line bundle on V / (see Exercise 1). Henceforward, referring to the above situation, we will say that a lifting homomorphism σα is given by the quadratic map α and will freely use the correspondence α → σα when discussing liftings of a lattice to the Heisenberg group. , if and only if the skew-symmetric form E| × is unimodular.

6, the space of canonical theta functions is 1-dimensional, so α for different choices of a Lagrangian subspace L the elements θ H, ,L should be proportional. In Chapter 5 we will compute these proportionality coefﬁcients and deduce from this the classical functional equation for theta series. In the case when is not necessarily maximal isotropic, we equip the space of canonical theta functions with the structure of a representation of a ﬁnite Heisenberg group associated with , and show that it is irreducible.

1) from A × A to U (1) is a bihomomorphism. 1) induces an isomorphism A A. We call q a (nondegenerate) quadratic form if q is a (nondegenerate) quadratic function on A such that q(−a) = q(a) for any a ∈ A. Now let L 1 , L 2 , L 3 be an admissible triple of Lagrangian subgroups in K . 2) where d1 (x) = (x, x, x), d2 (x12 , x23 , x31 ) = (x12 − x31 , x23 − x12 , x31 − x23 ), d3 (x1 , x2 , x3 ) = x1 + x2 + x3 . 2. 2) has only one potentially nontrivial cohomology group A(L 1 , L 2 , L 3 ) = ker(d3 )/ im(d2 ).