All Outputs (22)

On Jacobi–Weierstrass mock modular forms (2025)
Journal Article
Alfes, C., Funke, J., Mertens, M. H., & Rosu, E. (2025). On Jacobi–Weierstrass mock modular forms. Advances in Mathematics, 465, Article 110147. https://doi.org/10.1016/j.aim.2025.110147

We construct harmonic weak Maass forms that map to cusp forms of weight k ≥ 2 with rational coefficients under the ξ-operator. This generaliz... Read More about On Jacobi–Weierstrass mock modular forms.

The Shimura-Shintani correspondence via singular theta lifts and currents (2023)
Journal Article
Crawford, J., & Funke, J. (2023). The Shimura-Shintani correspondence via singular theta lifts and currents. International Journal of Number Theory, 19(10), https://doi.org/10.1142/s1793042123501178

Indefinite theta series: the case of an N-gon (2023)
Journal Article
Funke, J., & Kudla, S. (2023). Indefinite theta series: the case of an N-gon. Pure and Applied Mathematics Quarterly, 19 (2023)(1), 191-231. https://doi.org/10.4310/pamq.2023.v19.n1.a8

The construction of Green currents and singular theta lifts for unitary groups (2021)
Journal Article
Funke, J., & Hofmann, E. (2021). The construction of Green currents and singular theta lifts for unitary groups. Transactions of the American Mathematical Society, 374(4), 2909-2947. https://doi.org/10.1090/tran/8289

With applications in the Kudla program in mind we employ singular theta lifts for the reductive dual pair $ \mathrm {U}(p,q)\times \mathrm {U}(1,1)$ to construct two different kinds of Green forms for codimension $ q$-cycles in Shimura varieties asso... Read More about The construction of Green currents and singular theta lifts for unitary groups.

On some incomplete theta integrals (2019)
Journal Article
Funke, J., & Kudla, S. (2019). On some incomplete theta integrals. Compositio Mathematica, 155(9), 1711-1746. https://doi.org/10.1112/s0010437x19007504

In this paper we construct indefinite theta series for lattices of arbitrary signature as ‘incomplete’ theta integrals, that is, by integrating the theta forms constructed by the second author with J. Millson over certain singular -chains in the asso... Read More about On some incomplete theta integrals.

Modularity of generating series of winding numbers (2018)
Journal Article
Bruinier, J., Funke, J., Imamoḡlu, Ö., & Li, Y. (2018). Modularity of generating series of winding numbers. Research in the Mathematical Sciences, 5(2), Article 23. https://doi.org/10.1007/s40687-018-0140-6

The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of h... Read More about Modularity of generating series of winding numbers.

Mock modular forms and geometric theta functions for indefinite quadratic forms (2017)
Journal Article
Funke, J., & Kudla, S. S. (2017). Mock modular forms and geometric theta functions for indefinite quadratic forms. Journal of Physics A: Mathematical and Theoretical, 50(40), Article 404001. https://doi.org/10.1088/1751-8121/aa848b

Theta functions for indefinite quadratic forms are an important tool to construct modular forms and Mock modular forms. In this note, we recall the representation-theoretic background in the construction of theta series with emphasis on the theory de... Read More about Mock modular forms and geometric theta functions for indefinite quadratic forms.

Degenerate Whittaker functions for Sp_n(R) (2016)
Journal Article
Bruinier, J., Funke, J., & Kudla, S. (2018). Degenerate Whittaker functions for Sp_n(R). International Mathematics Research Notices, 2018(1), 1-56. https://doi.org/10.1093/imrn/rnw218

In this paper, we construct Whittaker functions with exponential growth for the degenerate principal series of the symplectic group of genus n induced from the Siegel parabolic subgroup. This is achieved by explicitly constructing a certain Goodman–W... Read More about Degenerate Whittaker functions for Sp_n(R).

The geometric theta correspondence for Hilbert modular surfaces (2014)
Journal Article
Funke, J., & Millson, J. (2014). The geometric theta correspondence for Hilbert modular surfaces. Duke Mathematical Journal, 163(1), 65-116. https://doi.org/10.1215/00127094-2405279

We give a new proof and an extension of the celebrated theorem of Hirzebruch and Zagier [17] that the generating function for the intersection numbers of the Hirzebruch-Zagier cycles in (certain) Hilbert modular surfaces is a classical modular form o... Read More about The geometric theta correspondence for Hilbert modular surfaces.

Regularized theta liftings and periods of modular functions (2013)
Journal Article
Bruinier, J., Funke, J., & Imamoglu, O. (2015). Regularized theta liftings and periods of modular functions. Journal für die reine und angewandte Mathematik, 2015(703), 43-93. https://doi.org/10.1515/crelle-2013-0035

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 integra... Read More about Regularized theta liftings and periods of modular functions.

Boundary behaviour of special cohomology classes arising from the Weil representation (2012)
Journal Article
Funke, J., & Millson, J. (2013). Boundary behaviour of special cohomology classes arising from the Weil representation. Journal of the Institute of Mathematics of Jussieu, 12(3), 571-634. https://doi.org/10.1017/s1474748012000795

In our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Sie... Read More about Boundary behaviour of special cohomology classes arising from the Weil representation.

Arithmetic Geometry and Automorphic forms, Volume in honor of the 60th birthday of Stephen S. Kudla. (2011)
Book
Funke, J., Cogdell, J., Rapoprt, M., & Yang, T. (Eds.). (2011). birthday of Stephen S. Kudla. International Press and the Higher Education Press of China

Spectacle cycles with coefficients and modular forms of half-integral weight (2011)
Book Chapter
Funke, J., & Millson, J. (2011). Spectacle cycles with coefficients and modular forms of half-integral weight. In J. Cogdell, J. Funke, M. Rapoport, & T. Yang (Eds.), Arithmetic geometry and automorphic forms (91-154). International Press

In this paper we present a geometric way to extend the Shintani lift from even weight cusp forms for congruence subgroups to arbitrary modular forms, in particular Eisenstein series. This is part of our eorts to extend in the noncompact situation the... Read More about Spectacle cycles with coefficients and modular forms of half-integral weight.

On the injectivity of the Kudla-Millson lift and surjectivity of the Borcherds lift (2010)
Book Chapter
Bruinier, J., & Funke, J. (2010). On the injectivity of the Kudla-Millson lift and surjectivity of the Borcherds lift. In J. Lepowsky, J. McKay, & M. Tuite (Eds.), Moonshine : the first quarter century and beyond : proceedings of a workshop on the moonshine conjectures and vertex algebras (12-39). Cambridge University Press. https://doi.org/10.1017/cbo9780511730054.004

Special cohomology classes for the Weil representation (2008)
Presentation / Conference Contribution
Funke, J. (2008, October). Special cohomology classes for the Weil representation. Presented at Automorphic Representations, Automorphic Forms, L-functions, and Related Topics, Kyoto

CM points and weight 3/2 modular forms (2007)
Presentation / Conference Contribution
Funke, J. (2007, December). CM points and weight 3/2 modular forms. Presented at Analytic Number Theory. A Tribute to Gauss and Dirichlet, G

Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms (2006)
Journal Article
Funke, J., & Millson, J. (2006). Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms. American Journal of Mathematics, 128(4), 899-948. https://doi.org/10.1353/ajm.2006.0032

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 t... Read More about Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms.

Traces of CM values of modular functions (2006)
Journal Article
Bruinier, J., & Funke, J. (2006). Traces of CM values of modular functions. Journal für die reine und angewandte Mathematik, 594, 1-33. https://doi.org/10.1515/crelle.2006.034

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 correspo... Read More about Traces of CM values of modular functions.

On two geometric theta lifts (2004)
Journal Article
Bruinier, J., & Funke, J. (2004). On two geometric theta lifts. Duke Mathematical Journal, 125(1), 45-90. https://doi.org/10.1215/s0012-7094-04-12513-8

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 Ku... Read More about On two geometric theta lifts.

Heegner divisors and non-holomorphic modular forms (2002)
Journal Article
Funke, J. (2002). Heegner divisors and non-holomorphic modular forms. Compositio Mathematica, 133(3), 289-321. https://doi.org/10.1023/a%3A1020002121978

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 com... Read More about Heegner divisors and non-holomorphic modular forms.