By Alexander Polishchuk

This ebook is a contemporary therapy of the idea of theta services within the context of algebraic geometry. the newness of its procedure lies within the systematic use of the Fourier-Mukai remodel. Alexander Polishchuk begins through discussing the classical thought of theta capabilities from the perspective of the illustration conception of the Heisenberg team (in which the standard Fourier rework performs the renowned role). He then indicates that during the algebraic method of this idea (originally as a result of Mumford) the Fourier-Mukai rework 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 powerful curiosity in algebraic geometry.

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

**Similar algebraic geometry books**

**Algebraic geometry 3. Further study of schemes**

Algebraic geometry performs a huge position in different branches of technology and know-how. this is often the final of 3 volumes by means of Kenji Ueno algebraic geometry. This, in including Algebraic Geometry 1 and Algebraic Geometry 2, makes a great textbook for a path in algebraic geometry. during this quantity, the writer is going past introductory notions and offers the speculation of schemes and sheaves with the target of learning the houses beneficial for the whole improvement of recent algebraic geometry.

**Equidistribution in Number Theory: An Introduction **

Written for graduate scholars and researchers alike, this set of lectures offers a established creation to the idea that of equidistribution in quantity conception. this idea is of turning out to be value in lots of components, together with cryptography, zeros of L-functions, Heegner issues, leading quantity thought, the idea of quadratic kinds, and the mathematics points of quantum chaos.

**Lectures on Resolution of Singularities**

Solution of singularities is a robust and often used instrument in algebraic geometry. during this e-book, J? nos Koll? r presents a entire remedy of the attribute zero case. He describes greater than a dozen proofs for curves, many in line with the unique papers of Newton, Riemann, and Noether. Koll?

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

**Sample text**

2). Let T be a complex torus, e1 , . . , e2n be the basis of the lattice H 1 (T, Z), ∗ e1∗ , . . , e2n be the dual basis of H 1 (T ∨ , Z), where T ∨ is the dual torus. Show that the ﬁrst Chern class of the Poincar´e bundle on T × T ∨ is given by 2n c1 (P) = i=1 ei ∧ ei∗ . 2 Representations of Heisenberg Groups I This chapter in an introduction to the representation theory of Heisenberg groups. This theory will be our principal tool in the study of theta functions. Throughout this chapter V is a real vector space with a ﬁxed symplectic form E.

With every non-degenerate quadratic function q on a ﬁnite abelian group A we associate the Gauss sum γ (q) = |A|− 2 1 q(a). a∈A Here are some simple properties of γ (q). 6. (i) Let A = A1 ⊕ A2 , q = q1 ⊕ q2 , where qi is a nondegenerate quadratic function on Ai (i = 1, 2). By the deﬁnition, this means that q(a1 , a2 ) = q1 (a1 )q2 (a2 ). Then γ (q1 ⊕ q2 ) = γ (q1 ) · γ (q2 ). (ii) Assume that there is a subgroup I ⊂ A such that q| I = 1 and |I | = 1 |A| 2 . Then γ (q) = 1. (iii) One has γ (q −1 ) = γ (q)−1 .

2) has only one potentially nontrivial cohomology group A(L 1 , L 2 , L 3 ) = ker(d3 )/ im(d2 ). One has A(L 1 , L 2 , L 3 ) (L 1 + L 2 ) ∩ L 3 /(L 1 ∩ L 3 + L 2 ∩ L 3 ). 3. Quadratic Function Associated with an Admissible Triple 45 The proof is left to the reader. Let us deﬁne a function q : ker(d3 ) → U (1) by setting q(x1 , x2 , x3 ) = σ (x1 )σ (x2 )σ (x3 ), where σ : L i → H , i = 1, 2, 3, are lifting homomorphisms. Since x1 + x2 + x3 = 0, the expression in the RHS belongs to U (1) ⊂ H . 3.