By Alexander Polishchuk

The purpose of this publication is to give a latest therapy 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 rework. the writer begins by means of discussing the classical conception of theta services from the viewpoint of the illustration idea of the Heisenberg workforce (in which the standard Fourier rework performs the popular role). He then indicates that during the algebraic method of this thought, the Fourier–Mukai rework can usually be used to simplify the present proofs or to supply thoroughly new proofs of many very important theorems. Graduate scholars and researchers with powerful curiosity in algebraic geometry will locate a lot of curiosity during this quantity.

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**Extra resources for Abelian varieties and the Fourier transform**

**Sample text**

Since E has rank 2 over k, we have an embedding of groups: S(k) AutE (E) → GL2 (k) Autk (E) We will consider the tensor product π ⊗ χ as an irreducible representation of the group GL2 (k) × S(k), and wish to restrict this representation to the diagonally embedded subgroup S(k). The central local problem is to compute the space of coinvariants HomS(k) (π ⊗ χ, C). If this is nonzero, we must have ω · Res(χ) = 1 (∗) as a character of k ∗ . Indeed, ω · Res(χ) gives the action of k ∗ ⊂ E ∗ on all vectors in π ⊗ χ.

We can, therefore, estimate the Error by breaking it into smaller sums as follows: |Error| ≤ 4 k

A (noncuspidal) point on the curve X0 (N ), over a field k of characteristic prime to N , is given by a pair (E, F ) of elliptic curves over k and a cyclic N isogeny φ : E → F , also defined over k. We represent the point x by the diagram (E ✲ F ); φ two diagrams represent the same point if they are isomorphic over a separable closure of k. The ring End(x) associated to the point x is the subring of pairs (α, β) in End(E)×End(F ) which are defined over k and give a commutative square E φ α ❄ E φ ✲ F β ❄ ✲ F.

### Abelian varieties and the Fourier transform by Alexander Polishchuk

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