By Alexander Polishchuk
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.
Read or Download Abelian Varieties, Theta Functions and the Fourier Transform PDF
Best algebraic geometry books
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.
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.
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.
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.
- Essays in Constructive Mathematics
- Galois Theory of Difference Equations
- Hodge theory and complex algebraic geometry 2
- Algebraic Spaces
- Elementary Number Theory. Primes, Congruences and Secrets
Extra resources for Abelian Varieties, Theta Functions and the Fourier Transform
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.
Abelian Varieties, Theta Functions and the Fourier Transform by Alexander Polishchuk