By Alexander Polishchuk

This ebook is a contemporary remedy of the speculation of theta features within the context of algebraic geometry. the newness of its strategy lies within the systematic use of the Fourier-Mukai remodel. Alexander Polishchuk starts off through discussing the classical concept of theta services from the perspective of the illustration concept of the Heisenberg staff (in which the standard Fourier remodel performs the widespread role). He then exhibits that during the algebraic method of this idea (originally as a result of Mumford) the Fourier-Mukai rework can frequently be used to simplify the present proofs or to supply thoroughly new proofs of many vital theorems. This incisive quantity is for graduate scholars and researchers with powerful curiosity in algebraic geometry.

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

**Example text**

Since by Chow theorem (see [52]; Chapter 1, Section 3) analytic subvarieties of P N are algebraic, Lefschetz theorem implies that (T ) is an algebraic subvariety of P N . Thus, a choice of polarization on a complex abelian variety gives an algebraic structure on it. The GAGA principle (see [124]) implies that every holomorphic bundle on T is algebraic with respect to this structure. Hence, the algebraic structure does not depend on a choice of polarization. Writing explicitly equations deﬁning the subvariety (T ) ⊂ P N is a beautiful chapter in the theory of theta functions.

Thus, a choice of polarization on a complex abelian variety gives an algebraic structure on it. The GAGA principle (see [124]) implies that every holomorphic bundle on T is algebraic with respect to this structure. Hence, the algebraic structure does not depend on a choice of polarization. Writing explicitly equations deﬁning the subvariety (T ) ⊂ P N is a beautiful chapter in the theory of theta functions. The main idea is to use the group structure on T (plus the fact that line bundles behave like quadratic forms with respect to addition on T ) to derive some universal identities between theta functions.

To get a unitary representation of H we modify the space C(L) as follows. Note that for every φ1 , φ2 ∈ C(L) the function φ1 φ2 is U (1)-invariant, hence it descends to a function on K /L. In particular, for every φ ∈ C(L) we can consider |φ|2 as a function on K (L). Now we set F(L) = φ ∈ C(L)| where dk is a Haar measure4 on K /L, to the hermitian metric given by φ1 , φ2 = K /L |φ|2 dk < ∞ , denotes the completion with respect K /L φ1 φ2 dk. Remark. The representation F(L) is equivalent to the induced representation Ind Hp−1 (L) χ L , where χ L is the 1-dimensional representation of p −1 (L) ⊂ H 4 By the deﬁnition, Haar measure is a translation-invariant measure.