Read e-book online Abelian Varieties, Theta Functions and the Fourier Transform PDF

By Alexander Polishchuk

ISBN-10: 0521808049

ISBN-13: 9780521808040

This booklet is a latest remedy of the idea of theta features within the context of algebraic geometry. the newness of its procedure lies within the systematic use of the Fourier-Mukai rework. Alexander Polishchuk starts off by way of discussing the classical concept of theta services from the perspective of the illustration thought of the Heisenberg workforce (in which the standard Fourier remodel performs the well-liked role). He then exhibits that during the algebraic method of this concept (originally as a result of Mumford) the Fourier-Mukai rework can usually be used to simplify the prevailing proofs or to supply thoroughly new proofs of many very important theorems. This incisive quantity is for graduate scholars and researchers with powerful curiosity in algebraic geometry.

Show description

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

Best algebraic geometry books

Get Invariant Factors, Julia Equivalences and the (Abstract) PDF

This publication is principally dedicated to the combinatorics of quadratic holomorphic dynamics. The conceptual kernel is a self-contained summary counterpart of attached quadratic Julia units that's equipped on Thurston's notion of a quadratic invariant lamination and on symbolic descriptions of the angle-doubling map.

Equidistribution in Number Theory: An Introduction - download pdf or read online

Written for graduate scholars and researchers alike, this set of lectures offers a dependent creation to the concept that of equidistribution in quantity conception. this idea is of starting to be value in lots of components, together with cryptography, zeros of L-functions, Heegner issues, best quantity concept, the speculation of quadratic varieties, and the mathematics points of quantum chaos.

Masahiro Shiota's Geometry of Subanalytic and Semialgebraic Sets PDF

Actual analytic units in Euclidean area (Le. , units outlined in the community at every one aspect of Euclidean area via the vanishing of an analytic functionality) have been first investigated within the 1950's via H. Cartan [Car], H. Whitney [WI-3], F. Bruhat [W-B] and others. Their technique was once to derive information regarding actual analytic units from homes in their complexifications.

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

Example text

Note that the 1-dimensional subspace T (H, L , α) ⊂ T (H, , α) coincides with the space of I -invariants in T (H, , α), where I = ⊥ ∩ L/ ∩ L is a maximal isotropic subgroup in ⊥ / , lifted to the finite Heisenberg group G(E, , α) trivially. Thus, we obtain the following corollary. 6. The theta series θ H, ,L is a generator of the 1-dimensional I subspace T (H, , α) ⊂ T (H, , α). 4) is described in the following proposition. 7. 2). (ii) Let ⊂ be a sublattice. Then α θ H, ,L α α(γ )−1 U(1,γ ) θ H, = γ ∈ /( + ∩L) ,L α U(α(γ ),γ ) θ H, = γ ∈ /( + ∩L) ,L .

Indeed, we have ⊥ and l(v) = π H (δ, γ ) = l(γ ) + 2πim(γ ) for some homomorphism m : m : V → R we get → Z. Extending m to an R-linear map π H (δ, v) − l(v) = 2πim(v) for every v ∈ V . It follows that π H (v, δ) − l(v) = 2πi E(v, δ) + 2πim(v). But the LHS is C-linear and the RHS takes values in i R. It follows that both sides are zero which implies our claim. 2) θ(v + δ) = A exp π H (v, δ) + H (δ, δ) θ(v). 2 Set = + Zδ. We have seen that ⊂ ⊥ and the assumption δ ∈ implies that is stricly bigger than .

Here PJr denotes the right-invariant distribution of subspaces on H(V ), which is equal to 0 ⊕ PJ at the point (1, 0). The unitary structure on F − (J ) is given by φ1 , φ2 = φ1 φ2 dv V (F − (J ) is complete with respect to this metric). The second model is a slight modification of the first one. Its importance is due to the fact that it uses holomorphic functions on V with respect to the complex structure J . Definition. 1) 2 where (λ, v) ∈ H(V ), v ∈ V . The Hermitian form on Fock(V, J ) is given by f (v)g(v) exp(−π H (v, v))dv f, g = V (the space Fock(V, J ) is complete with respect to the corresponding norm).

Download PDF sample

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

by Paul

Rated 4.79 of 5 – based on 33 votes