By Jan H. Bruinier

ISBN-10: 3540433201

ISBN-13: 9783540433200

Round 1994 R. Borcherds found a brand new kind of meromorphic modular shape at the orthogonal team $O(2,n)$. those "Borcherds items" have limitless product expansions analogous to the Dedekind eta-function. They come up as multiplicative liftings of elliptic modular types on $(SL)_2(R)$. the truth that the zeros and poles of Borcherds items are explicitly given by way of Heegner divisors makes them fascinating for geometric and mathematics purposes. within the current textual content the Borcherds' development is prolonged to Maass wave types and is used to review the Chern sessions of Heegner divisors. A communicate theorem for the lifting is proved.

**Read or Download Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors PDF**

**Best abstract books**

**Download PDF by Alejandro Adem: Cohomology of finite groups**

Adem A. , Milgram R. J. Cohomology of finite teams (Springer, 1994)(ISBN 354057025X)

**Download PDF by F.E.A. Johnson: Syzygies and Homotopy Theory**

An important invariant of a topological house is its basic team. while this is often trivial, the ensuing homotopy thought is definitely researched and usual. within the basic case, in spite of the fact that, homotopy conception over nontrivial basic teams is far extra complex and much much less good understood. Syzygies and Homotopy concept explores the matter of nonsimply attached homotopy within the first nontrivial circumstances and offers, for the 1st time, a scientific rehabilitation of Hilbert's approach to syzygies within the context of non-simply hooked up homotopy thought.

- Abstract Painting. 50 Years of Accomplishment From Kandinsky to Jackson Pollock
- Homology of Locally Semialgebraic Spaces
- The Skeleton Key of Mathematics
- general topology. muller
- Harmonic Measure (New Mathematical Monographs)

**Extra resources for Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors **

**Sample text**

Here condition (iii) means that f has a Fourier expansion of the form c(γ, n)eγ (nτ ). f (τ ) = γ∈L /L n∈Z−q(γ) n≥0 Moreover, if all c(γ, n) with n = 0 vanish, then f is called a cusp form. The C-vector space of modular forms of weight κ with respect to ∗L and Mp2 (Z) is denoted by Mκ,L , the subspace of cusp forms by Sκ,L . It is easily seen that Mκ,L is ﬁnite dimensional. 3. Fix a positive integer t. Let L be the 1-dimensional lattice Z, equipped with the positive deﬁnite quadratic form q(a) = ta2 .

We ﬁnd that θL+γ (τ, v) is given by 1 2yzv2 e (τ q((λK + cz )w ) + τ¯q((λK + cz )w⊥ )) c,d∈Z λK ∈K+p(γ−cz ) c≡(γ,z) (N ) ×e − |(λK + cz , z)τ + d|2 d(λK + cz + f (c, γ)z, zv − zv⊥ ) − 4iyzv2 2zv2 , where we have used zw = zw⊥ = 0 and q(z) = 0. 1) of ΘK (τ, w; dµ, −cµ), we see that we have to prove − d(λK + cz + f (c, γ)z, zv − zv⊥ ) ≡ −(λK − cµ/2, dµ) − d(γ, z ) + cdq(z ) 2zv2 modulo 1. Since (z, zv − zv⊥ )/2zv2 = 1, we ﬁnd −d (f (c, γ)z, zv − zv⊥ ) = −df (c, γ) 2zv2 (γ, z) − c dB = −dbγ + N 2 = −d(γ, z ) + d(γ, z)z + From (ζ, z ) ∈ Z it follows that B ≡ −N z −d 2 (γ, z) − c dB.

We write L = {x ∈ L ⊗ Q; (x, y) ∈ Z for all y ∈ L} for the dual lattice. Then the quotient L /L is a ﬁnite Abelian group, the socalled discriminant group. The modulo 1 reduction of q(x) is a Q/Z-valued quadratic form on L /L whose associated bilinear form is the modulo 1 reduction of the bilinear form (·, ·) on L . Recall that there is a unitary representation L of Mp2 (Z) on the group algebra C[L /L]. If we denote the standard basis of C[L /L] by (eγ )γ∈L /L , then L can be deﬁned by the action of the generators S, T ∈ Mp2 (Z) as follows (cp.

### Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors by Jan H. Bruinier

by John

4.3