Jan Hendrik Bruinier

On two geometric theta lifts

The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. In this paper we establish for the orthogonal group O(p,2) an adjointness result between Borcherdss singular theta lift and the Kudla-Millson lift. We extend this result to arbitrary signature by introducing a new singular theta lift for O(p,q). On the geometric side, this lift can be interpreted as a differential character, in the sense of Cheeger and Simons, for the cycles under consideration.

Borcherds products on O(2,l) and Chern classes of Heegner divisors

Introduction.- Vector valued modular forms for the metaplectic group. The Weil representation. Poincare series and Einstein series. Non-holomorphic Poincare series of negative weight.- The regularized theta lift. Siegel theta functions. The theta integral. Unfolding against F. Unfolding against theta.- The Fourier theta lift. Lorentzian lattices. Lattices of signature (2,l). Modular forms on orthogonal groups. Borcherds products.- Some Riemann geometry on O(2,l). The invariant Laplacian. Reduction theory and L^p-estimates. Modular forms with zeros and poles on Heegner divisors.- Chern classes of Heegner divisors. A lifting into cohomology. Modular forms with zeros and poles on Heegner divisors II.

Traces of CM values of modular functions

Abstract Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps. Using the theta correspondence, we generalize this result to traces of CM values of (weakly holomorphic) modular functions on modular curves of arbitrary genus. We also study the theta lift for the weight 0 Eisenstein series for SL2(ℤ) and realize a certain generating series of arithmetic intersection numbers as the derivative of Zagiers Eisenstein series of weight 3/2. This recovers a result of Kudla, Rapoport and Yang.

The arithmetic of the values of modular functions and the divisors of modular forms

We investigate the arithmetic and combinatorial significance of the values of the polynomials j n ( x ) defined by the q -expansion \[\sum_{n=0}^{\infty}j_n(x)q^n:=\frac{E_4(z)^2E_6(z)}{\Delta(z)}\cdot\frac{1}{j(z)-x}.\] They allow us to provide an explicit description of the action of the Ramanujan Theta-operator on modular forms. There are a substantial number of consequences for this result. We obtain recursive formulas for coefficients of modular forms, formulas for the infinite product exponents of modular forms, and new p -adic class number formulas.

On Borcherds Products Associated with Lattices of Prime Discriminant

We show that certain spaces of vector valued modular forms are isomorphic to spaces of scalar valued modular forms whose Fourier coefficients are supported on suitable progressions.As an application we give a very explicit description of Borcherds products on Hilbert modular surfaces.

Coefficients of half-integral weight modular forms

Abstract In this paper, we study the distribution of the coefficients a ( n ) of half-integral weight modular forms modulo odd integers M . As a consequence, we obtain improvements of indivisibility results for the central critical values of quadratic twists of L -functions associated with integral weight newforms established in Ono and Skinner (Fourier coefficients of half-integral weight modular forms modulo l, Ann. of Math. 147 (1998) 453–470). Moreover, we find a simple criterion for proving cases of Newmans conjecture for the partition function.

Integrals of automorphic Green's functions associatedto Heegner divisors

In the present paper we find explicit formulas for the degrees of Heegner divisors on arithmetic quotients of the orthogonal group

Heights of Kudla–Rapoport divisors and derivatives of \(L\)-functions

\Orth(2,p)

On the Rank of Picard Groups of Modular Varieties Attached to Orthogonal Groups

and for the integrals of certain automorphic Greens functions associated with Heegner divisors. The latter quantities are important in the study of the arithmetic degrees of Heegner divisors in the context of Arakelov geometry. In particular, we obtain a different proof and a generalization of results of Kudla relating these quantities to the Fourier coefficients of certain non-holomorphic Eisenstein series of weight

Hilbert Modular Forms and Their Applications

1+p/2