Jan Hendrik Bruinier, Benjamin Howard, Stephen S. Kudla, Michael Rapoport, Tonghai Yang,
We form generating series, valued in the Chow group and the arithmetic Chow group, of special divisors on the compactified integral model of a Shimura variety associated to a unitary group of signature pn ´1, 1q, and prove their modularity.The main ingredient in the proof is the calculation of vertical components appearing in the divisor of a Borcherds product on the integral model.
Tópico(s): Molecular spectroscopy and chirality
2020 - Société Mathématique de France | Astérisque
Jan Hendrik Bruinier, Martin Raum,
We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogs of Fourier–Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties.
Tópico(s): Algebraic structures and combinatorial models
2015 - Cambridge University Press | Forum of Mathematics Pi
Jan Hendrik Bruinier, Ken Ono, Andrew V. Sutherland,
We give algorithms for computing the singular moduli of suitable nonholomorphic modular functions F(z). By combining the theory of isogeny volcanoes with a beautiful observation of Masser concerning the nonholomorphic Eisenstein series E2⁎(z), we obtain CRT-based algorithms that compute the class polynomials HD(F;x), whose roots are the discriminant D singular moduli for F(z). By applying these results to a specific weak Maass form Fp(z), we obtain a CRT-based algorithm for computing partition class ...
Tópico(s): Algebraic Geometry and Number Theory
2015 - Elsevier BV | Journal of Number Theory
Jan Hendrik Bruinier, Stephan Ehlen, Eberhard Freitag,
We prove that there are only finitely many isometry classes of even lattices $L$ of signature $(2,n)$ for which the space of cusp forms of weight $1+n/2$ for the Weil representation of the discriminant group of $L$ is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of $L$ can be realized as the divisor of a Borcherds product. We obtain similar classification results in greater generality for finite quadratic modules.
Tópico(s): Algebraic structures and combinatorial models
2015 - American Mathematical Society | Mathematics of Computation
Jan Hendrik Bruinier, Benjamin Howard, Tonghai Yang,
Tópico(s): Advanced Mathematical Identities
2014 - Springer Science+Business Media | Inventiones mathematicae
Jan Hendrik Bruinier, Ken Ono,
We prove that the coefficients of certain weight −1/2 harmonic Maass forms are "traces" of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight −2 harmonic weak Maass forms to spaces of weight −1/2 vector-valued harmonic weak Maass forms on Mp2(Z), a result which is of independent interest. We then prove a general theorem which guarantees (with bounded denominator) when such Maass singular moduli are algebraic. As an example of these results, we ...
Tópico(s): Advanced Algebra and Geometry
2013 - Elsevier BV | Advances in Mathematics
We prove a new converse theorem for Borcherds' multiplicative theta lift which improves the previously known results. To this end we develop a newform theory for vector valued modular forms for the Weil representation, which might be of independent interest. We also derive lower bounds for the ranks of the Picard groups and the spaces of holomorphic top degree differential forms of modular varieties associated to orthogonal groups.
Tópico(s): Advanced Mathematical Identities
2013 - Elsevier BV | Journal of Algebra
Jan Hendrik Bruinier, Jens Funke, Özlem Imamoğlu,
Abstract In this paper, we use regularized theta liftings to construct weak Maass forms of weight 1/2 as lifts of weak Maass forms of weight 0. As a special case we give a new proof of some of recent results of Duke, Toth and the third author on cycle integrals of the modular j -invariant and extend these to any congruence subgroup. Moreover, our methods allow us to settle the open question of a geometric interpretation for periods of j along infinite geodesics in the upper half plane. In particular, ...
Tópico(s): Analytic Number Theory Research
2013 - De Gruyter | Journal für die reine und angewandte Mathematik (Crelles Journal)
Jan Hendrik Bruinier, Stephen S. Kudla, Tai Yang,
We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big complex multiplication (CM) points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d,2). Our approach depends on the fact that the Green functions in question are constructed as regularized theta lifts of harmonic weak Maass forms, and it involves the Siegel–Weil formula and the central derivatives of incoherent Eisenstein series for ...
Tópico(s): Analytic Number Theory Research
2011 - Oxford University Press | International Mathematics Research Notices
Jan Hendrik Bruinier, Ken Ono,
Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions.Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as "generating functions" for central values and derivatives of quadratic twists of weight 2 modular L-functions.To obtain these results, we construct differentials of the third kind with twisted Heegner divisor by suitably generalizing the Borcherds ...
Tópico(s): Analytic Number Theory Research
2010 - Princeton University | Annals of Mathematics
Jan Hendrik Bruinier, Ken Ono, Robert C. Rhoades,
Tópico(s): Advanced Algebra and Geometry
2009 - Springer Nature | Mathematische Annalen
Elliptic Modular Forms and Their Applications.- Hilbert Modular Forms and Their Applications.- Siegel Modular Forms and Their Applications.- Congruence Between a Siegel and an Elliptic Modular Form.
Tópico(s): Analytic Number Theory Research
2009 - Association of College and Research Libraries | Choice Reviews Online
Jan Hendrik Bruinier, Tonghai Yang,
We study the Faltings height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a Rankin L-function. We prove the conjecture in certain cases where the Shimura variety has dimension 0, 1, or 2. In particular, we obtain a new proof of the Gross-Zagier formula.
Tópico(s): Analytic Number Theory Research
2009 - Springer Science+Business Media | Inventiones mathematicae
Jan Hendrik Bruinier, Oliver Stein,
We define Hecke operators for vector valued modular forms transforming with the Weil representation associated to a discriminant form. We describe the properties of the corresponding algebra of Hecke operators and study the action on modular forms.
Tópico(s): Advanced Combinatorial Mathematics
2008 - Springer Science+Business Media | Mathematische Zeitschrift
Jan Hendrik Bruinier, Ken Ono, Robert C. Rhoades,
For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ2-k (resp. D k-1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are “dual” under ξ2-k to newforms with vanishing Hecke eigenvalues (such as CM ...
Tópico(s): Advanced Algebra and Geometry
2008 - Springer Nature | Mathematische Annalen
Jan Hendrik Bruinier, José Ignacio Burgos Gil, Ulf Kühn,
We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight 2. Moreover, we determine the arithmetic self-intersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and we study Faltings heights of arithmetic Hirzebruch-Zagier divisors
Tópico(s): Algebraic structures and combinatorial models
2007 - Duke University Press | Duke Mathematical Journal
Jan Hendrik Bruinier, Jens Funke,
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 ...
Tópico(s): Analytic Number Theory Research
2006 - De Gruyter | Journal für die reine und angewandte Mathematik (Crelles Journal)
Jan Hendrik Bruinier, Paul A. Jenkins, Ken Ono,
Tópico(s): Advanced Algebra and Geometry
2005 - Springer Nature | Mathematische Annalen
Jan Hendrik Bruinier, Tonghai Yang,
Tópico(s): Advanced Topics in Algebra
2005 - Springer Science+Business Media | Inventiones mathematicae
Jan Hendrik Bruinier, Jens Funke,
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 ...
Tópico(s): Geometric and Algebraic Topology
2004 - Duke University Press | Duke Mathematical Journal
Jan Hendrik Bruinier, Winfried Kohnen, Ken Ono,
We investigate the arithmetic and combinatorial significance of the values of the polynomials jn(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 ...
Tópico(s): Analytic Number Theory Research
2004 - Cambridge University Press | Compositio Mathematica
Jan Hendrik Bruinier, Ken Ono,
Tópico(s): Advanced Topics in Algebra
2003 - Springer Nature | Mathematische Annalen
Jan Hendrik Bruinier, Ken Ono,
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 Newman' ...
Tópico(s): Advanced Mathematical Identities
2003 - Elsevier BV | Journal of Number Theory
Jan Hendrik Bruinier, Michael Bundschuh,
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.
Tópico(s): Finite Group Theory Research
2003 - Springer Nature | Developments in mathematics
Tópico(s): Mathematical Analysis and Transform Methods
2002 - Springer Nature | Lecture notes in mathematics
Jan Hendrik Bruinier, Eberhard Freitag,
The local Picard group at a generic point of the one-dimensional Baily-Borel boundary of a Hermitean symmetric quotient of type O(2,n) is computed. The main ingredient is a local version of Borcherds’ automorphic products. The local obstructions for a Heegner divisor to be principal are given by certain theta series with harmonic coefficients. Sometimes they generate Borcherds’ space of global obstructions. In these particular cases we obtain a simple proof of a result due to the first author: Suppose ...
Tópico(s): Algebraic Geometry and Number Theory
2001 - Association of the Annals of the Fourier Institute | Annales de l’institut Fourier
Jan Hendrik Bruinier, Michael Kuß,
Tópico(s): Algebraic Geometry and Number Theory
2001 - Springer Science+Business Media | manuscripta mathematica
Tópico(s): Analytic Number Theory Research
1999 - Duke University Press | Duke Mathematical Journal
Tópico(s): Finite Group Theory Research
1999 - Springer Science+Business Media | Inventiones mathematicae
Jan Hendrik Bruinier, Markus Schwagenscheidt,
We present some applications of the Kudla–Millson and the Millson theta lift. The two lifts map weakly holomorphic modular functions to vector valued harmonic Maass forms of weight 3/2 and 1/2, respectively. We give finite algebraic formulas for the coefficients of Ramanujan's mock theta functions f(q) and ω(q) in terms of traces of CM-values of a weakly holomorphic modular function. Further, we construct vector valued harmonic Maass forms whose shadows are unary theta functions, and whose holomorphic ...
Tópico(s): Analytic Number Theory Research
2017 - Elsevier BV | Journal of Algebra