By Alexander Polishchuk
This e-book is a contemporary remedy of the idea of theta capabilities within the context of algebraic geometry. the newness of its method lies within the systematic use of the Fourier-Mukai remodel. Alexander Polishchuk begins by means of discussing the classical concept of theta features from the perspective of the illustration thought of the Heisenberg workforce (in which the standard Fourier remodel performs the favorite role). He then indicates 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.
Read Online or Download Abelian Varieties, Theta Functions and the Fourier Transform PDF
Best algebraic geometry books
Conics and Cubics is an obtainable advent to algebraic curves. Its specialize in curves of measure at such a lot 3 retains effects tangible and proofs obvious. Theorems stick to clearly from highschool algebra and key rules: homogenous coordinates and intersection multiplicities.
By classifying irreducible cubics over the true numbers and proving that their issues shape Abelian teams, the e-book provides readers easy accessibility to the learn of elliptic curves. It contains a uncomplicated facts of Bezout's Theorem at the variety of intersections of 2 curves.
The booklet is a textual content for a one-semester path on algebraic curves for junior-senior arithmetic majors. the one prerequisite is first-year calculus.
The new version introduces the deeper examine of curves via parametrization through energy sequence. makes use of of parametrizations are provided: counting a number of intersections of curves and proving the duality of curves and their envelopes.
About the 1st edition:
"The booklet. .. belongs within the admirable culture of laying the principles of a tough and in all probability summary topic by way of concrete and obtainable examples. "
- Peter Giblin, MathSciNet
Within the spring of 1976, George Andrews of Pennsylvania nation college visited the library at Trinity university, Cambridge, to ascertain the papers of the overdue G. N. Watson. between those papers, Andrews stumbled on a sheaf of 138 pages within the handwriting of Srinivasa Ramanujan. This manuscript used to be quickly specified, "Ramanujan's misplaced pc.
Written for graduate scholars and researchers alike, this set of lectures offers a established advent to the idea that of equidistribution in quantity thought. this idea is of becoming value in lots of components, together with cryptography, zeros of L-functions, Heegner issues, leading quantity idea, the speculation of quadratic varieties, and the mathematics features of quantum chaos.
This quantity comprises the complaints of the convention on Interactions of Classical and Numerical Algebraic Geometry, held may well 22-24, 2008, on the collage of Notre Dame, in honor of the achievements of Professor Andrew J. Sommese. whereas classical algebraic geometry has been studied for centuries, numerical algebraic geometry has just recently been built.
Extra resources for Abelian Varieties, Theta Functions and the Fourier Transform
Theta Series and Weierstrass Sigma Function 39 even n and the sum over odd n, we get 1 3πi 3 τ + ,τ = τ+ (−1)n exp πi 4n 2 + 4n + θ11 2 4 4 4 n + exp πi(2n + 1)2 τ + 2πi n + n 1 2 2τ + 1 2 3πi (τ + 1) . 4 It remains to note that the ﬁrst sum is zero (as seen by substituting n → −n − 1). 3). 4) (1 − q n )2 n=1 where in the right-hand side we use multiplicative variables q = exp(2πiτ ), 1 u = exp(2πi z) (and where u 2 = exp(πi z)). This identity in turn is proven as follows. It is easy to see that ratio of the left-hand and right-hand sides is periodic in z with respect to .
Show that the representation F( ) can be identiﬁed with the natural representation of H(V ) on the space of square-integrable sections of L. 2. In the situation of the previous exercise show that c1 (L) ∈ H 2 (T, Z) is given by the skew-symmetric form E| × . ] 3. , such that the corresponding group K is ﬁnite). (a) Prove that there exists a Lagrangian subgroup I ⊂ K . (b) Let W be a representation of H such that U (1) acts by the identity character. Consider the decomposition W = ⊕χ ∈ I Wχ 4. in isotipic components with respect to the I -action.
4 for H i (T, L(H, α)). The ﬁrst step in this direction was recently done by I. Zharkov (see ) who constructed a canonical cohomology class in H i ( , H 0 (V, O)), where H 0 (V, O) is the space of holomorphic functions on V . There remains a question, whether there exists an H(V )-submodule F−∞ ⊂ H 0 (V, O) such that the above class lies in H i ( , F−∞ ) and such that the projectivization of F−∞ does not depend on a choice of complex structure. Exercises 1. Let (V, E) be a symplectic vector space and ⊂ V be an isotropic lattice equipped with a lifting to the Heisenberg group H(V ).