By Alexander Polishchuk

This e-book is a latest therapy of the speculation 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 via discussing the classical thought of theta services from the perspective of the illustration thought of the Heisenberg workforce (in which the standard Fourier remodel performs the widespread role). He then indicates that during the algebraic method of this concept (originally as a result of Mumford) the Fourier-Mukai remodel can frequently be used to simplify the prevailing proofs or to supply thoroughly new proofs of many vital theorems. This incisive quantity is for graduate scholars and researchers with robust curiosity in algebraic geometry.

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**Example text**

This can be proven by comparing the integral of ζ (z)dz along the boundary of the parallelogram formed by 1 and τ (slightly shifted) with the residue of this 1-form at the unique pole inside. The following theorem gives a relation between σ (z, τ ) and the theta series on elliptic curve C/ given by θ11 (z, τ ) = exp πi n + n∈Z 1 2 2 τ + 2πi n + 1 2 z+ 1 2 . Note that θ11 (z, τ ) = exp( πiτ + πi(z + 12 ))θ(z + τ +1 , τ ), where θ(z, τ ) is the 4 2 theta series considered in Exercise 5. 9. 3) where η(τ ) is the Dedekind η-function: η(τ ) = exp πiτ 12 ∞ (1 − q n ), n=1 where q = exp(2πiτ ).

4) we can assume that L 2 ⊂ L 1 + L 3 . We will use the presentation A = A(L 1 , L 2 , L 3 ) = L 2 /(L 1 ∩L 2 +L 2 ∩L 3 ), so that q(x2 ) = q L 1 ,L 2 ,L 3 (x2 ) = σ (x1 )−1 σ (x2 )σ (x3 )−1 for x2 ∈ L 2 , where x1 ∈ L 1 and x3 ∈ L 3 are chosen in such a way that x2 = x1 +x3 . Note that since both numbers c(L 1 , L 2 , L 3 ) and γ (q L 1 ,L 2 ,L 3 ) have absolute value 1, it sufﬁces to prove that they differ by a positive constant, so we can be imprecise with our choices of Haar measures when integrating.

Note that this theorem implies that all representations F(L) for different Lagrangian subgroups are equivalent to each other. The unique irreducible representation of H (on which U (1) acts in the standard way) is called Schr¨odinger representation of H . We will not prove this theorem (the particular case when K is ﬁnite is considered in Exercises 3 and 4, the complete proof can be found in [97] or [84]). However, in Chapter 4 we will construct explicitly intertwining operators between representations F(L) for some pairs of Lagrangian subgroups in K .