Get Abelian Varieties, Theta Functions and the Fourier Transform PDF

By Alexander Polishchuk

ISBN-10: 0511063911

ISBN-13: 9780511063916

ISBN-10: 051154653X

ISBN-13: 9780511546532

ISBN-10: 0521808049

ISBN-13: 9780521808040

This e-book is a latest therapy of the idea 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 begins by way of discussing the classical conception of theta services from the point of view of the illustration concept of the Heisenberg team (in which the standard Fourier rework performs the popular role). He then indicates that during the algebraic method of this thought (originally because of Mumford) the Fourier-Mukai remodel can usually 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 robust curiosity in algebraic geometry.

Show description

Read or Download Abelian Varieties, Theta Functions and the Fourier Transform PDF

Best algebraic geometry books

Download e-book for kindle: Algebraic spaces by Michael Artin

Those notes are in line with lectures given at Yale collage within the spring of 1969. Their item is to teach how algebraic services can be utilized systematically to boost yes notions of algebraic geometry,which are typically handled via rational capabilities through the use of projective equipment. the worldwide constitution that is typical during this context is that of an algebraic space—a area got through gluing jointly sheets of affine schemes through algebraic services.

Get Topological Methods in Algebraic Geometry PDF

In recent times new topological equipment, specially the speculation of sheaves based through J. LERAY, were utilized effectively to algebraic geometry and to the idea of capabilities of numerous complicated variables. H. CARTAN and J. -P. SERRE have proven how primary theorems on holomorphically entire manifolds (STEIN manifolds) might be for­ mulated when it comes to sheaf concept.

Get Introduction to Intersection Theory in Algebraic Geometry PDF

This e-book introduces a few of the major principles of recent intersection conception, lines their origins in classical geometry and sketches a number of normal purposes. It calls for little technical historical past: a lot of the cloth is out there to graduate scholars in arithmetic. A vast survey, the publication touches on many themes, most significantly introducing a strong new strategy built by way of the writer and R.

Download e-book for kindle: Rational Points on Curves over Finite Fields: Theory and by Harald Niederreiter

Rational issues on algebraic curves over finite fields is a key subject for algebraic geometers and coding theorists. right here, the authors relate a big software of such curves, specifically, to the development of low-discrepancy sequences, wanted for numerical equipment in diversified parts. They sum up the theoretical paintings on algebraic curves over finite fields with many rational issues and speak about the purposes of such curves to algebraic coding concept and the development of low-discrepancy sequences.

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

Example text

Namely, we set 1 F (L) = F(L) meas(K /L) 2 (morally, one should think of elements of F (L) as half-measures on K /L). Note that the space F (L) has a canonical Hermitian metric. Namely, for φ1 , φ2 ∈ F (L) the product φ1 φ2 descends to a measure on K /L, which then can be integrated. Definition. Lagrangian subgroups L 1 , L 2 ⊂ K equipped with liftings to H , are called compatible if L 1 + L 2 is a closed subgroup and their lifting homomorphisms to H agree on L 1 ∩ L 2 . Note that for compatible subgroups one has L 1 + L 2 = (L 1 ∩ L 2 )⊥ .

1. 1. Fourier Transform Let K be a locally compact abelian group, K be the Pointriagin dual group. As was already mentioned earlier, the reader will not lose much by assuming that K is either a real vector space or a finite abelian group. For every Haar measure µ on K the Fourier transform is the operator Sµ : L 2 (K ) → L 2 ( K ) given by ˆ = Sµ (φ)(k) ˆ k(k)φ(k)µ. k∈K In this situation there is a unique dual measure µ ˆ on K such that Sµ is unitary with respect to the Hermitian metrics defined using µ and µ.

5. 1) where v ∈ V , γ ∈ . Definition. 1). Its elements are called canonical theta functions for (H, , α). 2) coming from the Hermitian metric on L(H, α −1 ). 1) with the definition of the Fock representation we would like interpret the condition f ∈ T (H, , α) for a holomorphic function f on V as the invariance of f under operators Uα(γ ),γ for all γ ∈ (where is lifted to H using α). Since nonzero elements of T (H, , α) are not square-integrable, to achieve such an interpretation we have to extend operators Uλ,v to a larger space of holomorphic functions.

Download PDF sample

Abelian Varieties, Theta Functions and the Fourier Transform by Alexander Polishchuk


by Richard
4.1

Rated 4.05 of 5 – based on 36 votes