On two geometric theta lifts

By Jan H. Bruinier, Jens Funke. 2002

469 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/65c95dc0d0d6821d851b8e7fc29d52c0427218b6

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Citations in the corpus

Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms (2011, methodology): “This nonholomorphic form has weight 0, and is a weak Maass form (for more on weak Maass forms, see [8]).”
Automorphic Properties of Generating Functions for Generalized Rank Moments and Durfee Symbols (2008, methodology): “First we present some background, beginning with definitions of harmonic weak Maass forms (of half-integral weight), mock theta functions, and mock modular forms [20, 34, 36].”
Zagier-type dualities and lifting maps for harmonic Maass–Jacobi forms (2010, background): “[11] J. Bruinier, J. Funke, On two geometric theta lifts, Duke Math.”
The f(q) mock theta function conjecture and partition ranks (2006, background): “2 of [11]) show that if g is a weak Maass form of weight k for the group Γ0(4N) with Nebentypus χ, then ξk(g) is a weakly holomorphic modular form of weight 2 − k on Γ0(4N) with”
Hecke‐type double sums, Appell–Lerch sums, and mock theta functions, I (2012, background): “Zwegers thus established the modularity theory for mock theta functions, and his work allows mock theta functions to be cast as the holomorphic parts of weak Maass forms [11, 30, 9, 10]).”
An extension of the Hardy-Ramanujan circle method and applications to partitions without sequences (2011, background): “4); although the coefficients of χ(q) grow asymptotically with some exponential factor eC √ n (the same is true for any mock theta function due to their nontrivial principal parts [13], [25]), the exponential growth of p2(n) is the same as that of r(n).”
Faltings heights of CM cycles and derivatives of L-functions (2008, methodology): “We compute IT (f ) combining the ideas of [37] and [8].”
Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues (2008, background): “Weakly holomorphic modular forms naturally sit in spaces of harmonic weak Maass forms (see Section 2 for definitions), more general automorphic forms which have been a source of recent interest due to their connection to Ramanujan’s mock theta functions, Borcherds products, derivatives of modular L-”
MOCK THETA FUNCTIONS AND QUANTUM MODULAR FORMS (2013, background): “They are holomorphic parts of certain weight-1/2 harmonic weakMaass forms of Bruinier and Funke [18].”
Heegner divisors, $L$-functions and harmonic weak Maass forms (2007, methodology): “theta lifts of weakly holomorphic modular forms in a diﬀerent way. However, his automorphic products are of higher level, and only twists by even real Dirichlet characters are considered. In [Br] and [BF], the (untwisted) regularized theta lift was studied on harmonic weak Maass forms and was used t”
On the converse theorem for Borcherds products (2012, background): “ruct a linear map Λ : S1+n/2,L −→ H 1, (X Γ) from the space of cusp forms of weight 1+n/2 with representation ρL for the group Mp2(Z) tosquare integrable harmonic (1,1)formson XΓ, see Section 4.1. In [BF1] itwas shown that this map is adjoint to the corresponding Kudla-Millson lift [KM]. The convers”
ON THE EXPLICIT CONSTRUCTION OF HIGHER DEFORMATIONS OF PARTITION STATISTICS (2007, background): “In a series of papers the author and Ono [12, 13] connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions which are related to modular forms and which were first considered in [11].”
Borcherds products and arithmetic intersection theory on Hilbert modular surfaces (2003, methodology): “cusp form of weight 2 for ΓK, essentially the DoiNaganuma lift of the m-th Poincar´e series in S+ 2 (D,χD). Green functions as Gm are investigated from the point of view of the Weil representation in [BrFu]. Borcherds products and arithmetic intersection theory 29 3 Star products on Hilbert modular ”