# An Introduction to Algebraic Geometry by K. Ueno

By K. Ueno

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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 ).