Groupes et algébres de lie

Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)

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 Borcherds's 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.

/pdf/on-two-geometric-theta-lifts-ev5wb3yl7w.pdf

On two geometric theta lifts

Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has been researching harmonic Maass forms. These non-holomorphic modular forms play central roles in many subjects: arithmetic geometry, combinatorics, modular forms, and mathematical physics. Here we outline the general facets of the theory, and we give several applications to number theory: partitions and $q$-series, modular forms, singular moduli, Borcherds products, extensions of theorems of Kohnen-Zagier and Waldspurger on modular $L$-functions, and the work of Bruinier and Yang on Gross-Zagier formulae. What is surprising is that this story has an unlikely beginning: the pursuit of the solution to a great mathematical mystery.

/pdf/unearthing-the-visions-of-a-master-harmonic-maass-forms-and-55usxzjfvs.pdf

Unearthing the visions of a master: harmonic Maass forms and number theory

In 1944, Freeman Dyson initiated the study of ranks of integer partitions. Here we solve the classical problem of obtaining formulas for N
 e
 (n) (resp. N
 o
 (n)), the number of partitions of n with even (resp. odd) rank. Thanks to Rademacher’s celebrated formula for the partition function, this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f(q), a problem with its own long history dating to Ramanujan’s last letter to Hardy. Little was known about this problem until Dragonette in 1952 obtained asymptotic results. In 1966, G.E. Andrews refined Dragonette’s results, and conjectured an exact formula for the coefficients of f(q). By constructing a weak Maass-Poincare series whose “holomorphic part” is q
 -1
 f(q
 24), we prove the Andrews-Dragonette conjecture, and as a consequence obtain the desired formulas for N
 e
 (n) and N
 o
 (n).

/pdf/the-f-q-mock-theta-function-conjecture-and-partition-ranks-52wjvauc48.pdf

The f(q) mock theta function conjecture and partition ranks

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() and realize a certain generating series of arithmetic intersection numbers as the derivative of Zagier's Eisenstein series of weight 3/2. This recovers a result of Kudla, Rapoport and Yang.

/pdf/traces-of-cm-values-of-modular-functions-2kvs3w5tma.pdf

Traces of CM values of modular functions

We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p, 2) and associate to it a nonholomorphic elliptic modular form by integrating a certain theta function over the modular curve. We compute the Fourier expansion and identify the generating series of the (suitably defined) intersection numbers of the Heegner divisors in M with the modular curve as the holomorphic part of the modular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.

/pdf/heegner-divisors-and-nonholomorphic-modular-forms-3te0rjaqph.pdf

Heegner Divisors and Nonholomorphic Modular Forms

The purpose of this paper is to generalize the relation between intersection numbers of cycles in locally symmetric spaces of orthogonal type and Fourier coefficients of Siegel modular forms to the case where the cycles have local coefficients. Now the correspondence will involve vector-valued Siegel modular forms.

/pdf/cycles-with-local-coefficients-for-orthogonal-groups-and-bjg5yrutpw.pdf

Cycles with local coefficients for orthogonal groups and vector- valued siegel modular forms

The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. In this paper, we establish for O(p,2) an adjointness result between Borcherds' singular theta lift and the Kudla-Millson theta lift. We extend this result to arbitrary signature by introducing a new Borcherds 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.

Jens Funke

Papers

On two geometric theta lifts

Traces of CM values of modular functions

Heegner Divisors and Nonholomorphic Modular Forms

Cycles with local coefficients for orthogonal groups and vector- valued siegel modular forms

On Two Geometric Theta Lifts