By Alexander Polishchuk
This e-book is a latest therapy of the speculation 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 starts off via discussing the classical thought of theta services from the perspective of the illustration thought of the Heisenberg workforce (in which the standard Fourier remodel performs the widespread role). He then indicates that during the algebraic method of this concept (originally as a result of Mumford) the Fourier-Mukai remodel can frequently be used to simplify the prevailing 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 Online or Download Abelian Varieties, Theta Functions and the Fourier Transform PDF
Best algebraic geometry books
The elemental challenge of deformation concept in algebraic geometry includes gazing a small deformation of 1 member of a family members of gadgets, akin to kinds, or subschemes in a set house, or vector bundles on a set scheme. during this new e-book, Robin Hartshorne experiences first what occurs over small infinitesimal deformations, after which steadily builds as much as extra worldwide occasions, utilizing tools pioneered by way of Kodaira and Spencer within the advanced analytic case, and tailored and multiplied in algebraic geometry by means of Grothendieck.
The most issues of this publication are to set up the triple formulation with none hypotheses at the genericity of the morphism, and to advance a thought of entire quadruple issues, that's a primary step in the direction of proving the quadruple aspect formulation lower than much less restrictive hypotheses. This publication may be of curiosity to graduate scholars and researchers within the box of algebraic geometry.
- The semi-simple zeta function of quaternionic Shimura varieties
- Plane Algebraic Curves (Student Mathematical Library, Volume 15)
- Recurrence in Topological Dynamics: Furstenberg Families and Ellis Actions
- Quadratic and hermitian forms over rings
- Computational Algebraic Geometry (Progress in Mathematics)
- Positivity in algebraic geometry
Extra resources for Abelian Varieties, Theta Functions and the Fourier Transform
This can be proven by comparing the integral of ζ (z)dz along the boundary of the parallelogram formed by 1 and τ (slightly shifted) with the residue of this 1-form at the unique pole inside. The following theorem gives a relation between σ (z, τ ) and the theta series on elliptic curve C/ given by θ11 (z, τ ) = exp πi n + n∈Z 1 2 2 τ + 2πi n + 1 2 z+ 1 2 . Note that θ11 (z, τ ) = exp( πiτ + πi(z + 12 ))θ(z + τ +1 , τ ), where θ(z, τ ) is the 4 2 theta series considered in Exercise 5. 9. 3) where η(τ ) is the Dedekind η-function: η(τ ) = exp πiτ 12 ∞ (1 − q n ), n=1 where q = exp(2πiτ ).
4) we can assume that L 2 ⊂ L 1 + L 3 . We will use the presentation A = A(L 1 , L 2 , L 3 ) = L 2 /(L 1 ∩L 2 +L 2 ∩L 3 ), so that q(x2 ) = q L 1 ,L 2 ,L 3 (x2 ) = σ (x1 )−1 σ (x2 )σ (x3 )−1 for x2 ∈ L 2 , where x1 ∈ L 1 and x3 ∈ L 3 are chosen in such a way that x2 = x1 +x3 . Note that since both numbers c(L 1 , L 2 , L 3 ) and γ (q L 1 ,L 2 ,L 3 ) have absolute value 1, it sufﬁces to prove that they differ by a positive constant, so we can be imprecise with our choices of Haar measures when integrating.
Note that this theorem implies that all representations F(L) for different Lagrangian subgroups are equivalent to each other. The unique irreducible representation of H (on which U (1) acts in the standard way) is called Schr¨odinger representation of H . We will not prove this theorem (the particular case when K is ﬁnite is considered in Exercises 3 and 4, the complete proof can be found in  or ). However, in Chapter 4 we will construct explicitly intertwining operators between representations F(L) for some pairs of Lagrangian subgroups in K .