By Alexander Polishchuk

This e-book is a contemporary remedy of the idea of theta capabilities within the context of algebraic geometry. the newness of its method lies within the systematic use of the Fourier-Mukai remodel. Alexander Polishchuk begins by means of discussing the classical concept of theta features from the perspective of the illustration thought of the Heisenberg workforce (in which the standard Fourier remodel performs the favorite role). He then indicates that during the algebraic method of this concept (originally as a result of Mumford) the Fourier-Mukai rework can usually 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 powerful curiosity in algebraic geometry.

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**Extra resources for Abelian Varieties, Theta Functions and the Fourier Transform**

**Example text**

Theta Series and Weierstrass Sigma Function 39 even n and the sum over odd n, we get 1 3πi 3 τ + ,τ = τ+ (−1)n exp πi 4n 2 + 4n + θ11 2 4 4 4 n + exp πi(2n + 1)2 τ + 2πi n + n 1 2 2τ + 1 2 3πi (τ + 1) . 4 It remains to note that the ﬁrst sum is zero (as seen by substituting n → −n − 1). 3). 4) (1 − q n )2 n=1 where in the right-hand side we use multiplicative variables q = exp(2πiτ ), 1 u = exp(2πi z) (and where u 2 = exp(πi z)). This identity in turn is proven as follows. It is easy to see that ratio of the left-hand and right-hand sides is periodic in z with respect to .

Show that the representation F( ) can be identiﬁed with the natural representation of H(V ) on the space of square-integrable sections of L. 2. In the situation of the previous exercise show that c1 (L) ∈ H 2 (T, Z) is given by the skew-symmetric form E| × . ] 3. , such that the corresponding group K is ﬁnite). (a) Prove that there exists a Lagrangian subgroup I ⊂ K . (b) Let W be a representation of H such that U (1) acts by the identity character. Consider the decomposition W = ⊕χ ∈ I Wχ 4. in isotipic components with respect to the I -action.

4 for H i (T, L(H, α)). The ﬁrst step in this direction was recently done by I. Zharkov (see [138]) who constructed a canonical cohomology class in H i ( , H 0 (V, O)), where H 0 (V, O) is the space of holomorphic functions on V . There remains a question, whether there exists an H(V )-submodule F−∞ ⊂ H 0 (V, O) such that the above class lies in H i ( , F−∞ ) and such that the projectivization of F−∞ does not depend on a choice of complex structure. Exercises 1. Let (V, E) be a symplectic vector space and ⊂ V be an isotropic lattice equipped with a lifting to the Heisenberg group H(V ).