## Prof. Dr. Jan Hendrik Bruinier

### Contact

bruinier@mathematik.tu-...

work +49 6151 16-22462

Work
S2|15 411

Schlossgartenstraße 7

64289
Darmstadt

### Research Interests

- Automorphic Forms
- Number Theory
- Complex and Algebraic Geometry

#### Research papers and preprints

- Arithmetic volumes of unitary Shimura varieties (with B. Howard), preprint (2021). pdf (opens in new tab).
- Special cycles on toroidal compactifications of orthogonal Shimura varieties (with S. Zemel), Math. Ann., to appear. pdf (opens in new tab).
- CM values of higher automorphic Green functions for orthogonal groups (with S. Ehlen and T. Yang), Invent. Math. 225 (2021), 693-785. pdf (opens in new tab).
- Theta lifts for Lorentzian lattices and coefficients of mock theta functions (with M. Schwagenscheidt), Mathematische Zeitschrift 297 (2021), 1633-1657. pdf (opens in new tab).
- A converse theorem for Borcherds products on X_0(N) (with M. Schwagenscheidt), Nagoya Mathematical Journal 240 (2020), 237-256. pdf (opens in new tab).
- Arithmetic degrees of special cycles and derivatives of Siegel Eisenstein series (with T. Yang), Journal of the European Mathematical Society (JEMS) 23 (2021), 1613–1674. pdf (opens in new tab).
- Modularity of generating series of winding numbers (with J. Funke, O. Imamoglu, Yingkun Li), Research in the Mathematical Sciences 5:23 (2018). pdf (opens in new tab).
- Modularity of generating series of divisors on unitary Shimura varieties II: arithmetic applications (with B. Howard, S. Kudla, M. Rapoprt, and T. Yang), Astérisque 421 (2020), 127-186. pdf (opens in new tab).
- Cones of Heegner divisors (with M. Möller), Journal of Algebraic Geometry 28 (2019), 497-517. pdf (opens in new tab).
- Modularity of generating series of divisors on unitary Shimura varieties (with B. Howard, S. Kudla, M. Rapoport, and T. Yang), Astérisque 421 (2020), 7-125. pdf (opens in new tab).
- Algebraic formulas for the coefficients of mock theta functions and Weyl vectors of Borcherds products (with M. Schwagenscheidt), Journal of Algebra 478 (2017), 38-57. pdf (opens in new tab).
- Borcherds products with prescribed divisor, Bull. London Math. Soc. 49 (2017), 979-987. pdf (opens in new tab).
- Degenerate Whittaker functions for Sp_n(R) (with J. Funke and S. Kudla), Int. Math. Res. Not. IMRN (2018), no. 1, 1–56. pdf (opens in new tab).
- Heegner divisors in generalized Jacobians and traces of singular moduli (with Y. Li), Algebra and Number Theory 10 (2016), 1277-1300. pdf (opens in new tab).
- Kudla's modularity conjecture and formal Fourier-Jacobi series (with M. Raum), Forum of Math. Pi, vol. 3 (2015), 30 pp. pdf.
- Lattices with many Borcherds products (with S. Ehlen and E. Freitag), Mathematics of Computation, vol. 85, no. 300 (2016), 1953-1981. pdf (opens in new tab).
- Vector valued formal Fourier-Jacobi series, Proc. Amer. Math. Soc. 143 (2015), 505-512. pdf (opens in new tab).
- Heights of Kudla-Rapoport divisors and derivatives of L-functions (with B. Howard and T. Yang), Invent. Math. 201 (2015), 1-95. pdf (opens in new tab).
- Class polynomials for nonholomorphic modular functions (with K. Ono and A. Sutherland), Journal of Number Theory 161 (2016), 204-229. pdf (opens in new tab).
- On the converse theorem for Borcherds products, Journal of Algebra 397 (2014), 315-342. pdf (opens in new tab).
- Regularized theta liftings and periods of modular functions (with J. Funke and O. Imamoglu), J. Reine Angew. Math. 703 (2015), 43-93. pdf (opens in new tab).
- Harmonic Maass forms and periods, Math. Ann. 357 (2013), 1363-1387. pdf (opens in new tab).
- Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms (with K. Ono), Advances in Mathematics 246 (2013), 198-219. pdf (opens in new tab).
- Computation of harmonic weak Maass forms (with F. Strömberg), Experimental Mathematics 21:2 (2012), 117-131. pdf (opens in new tab).
- Special values of Green functions at big CM points (with S. Kudla and T. Yang), Int. Math. Res. Not. 2012:9 (2012), 1917-1967. pdf (opens in new tab).
- CM values of automorphic Green functions on orthogonal groups over totally real fields (with T. Yang). In: Arithmetic Geometry and Automorphic Forms, Eds.: J. Cogdell et. al., Higher Education Press and International Press (2011), 1-54. pdf (opens in new tab).
- Regularized theta lifts for orthogonal groups over totally real fields, J. Reine Angew. Math. 672 (2012), 177-222. pdf (opens in new tab).
- Identities and congruences for Ramanujan's omega(q) (with K. Ono), Ramanujan J. 23 (2010), 151-157. pdf (opens in new tab).
- Faltings heights of CM cycles and derivatives of L-functions (with T. Yang), Invent. Math. 177 (2009), 631-681. pdf (opens in new tab).
- Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues (with K. Ono and R. Rhoades), Math. Ann. 342 (2008), 673-693. pdf (opens in new tab).
- Heegner divisors, L-functions and harmonic weak Maass forms (with K. Ono), Annals of Math. 172 (2010), 2135-2181. pdf (opens in new tab).
- Sign changes of coefficients of half integral weight modular forms (with W. Kohnen). In: Modular forms on Schiermonnikoog, Eds.: B. Edixhoven, G. van der Geer and B. Moonen, Cambridge University Press (2008), 57-66. pdf (opens in new tab).
- The Weil representation and Hecke operators for vector valued modular forms (with O. Stein), Mathematische Zeitschrift 264 (2010), 249-270. pdf (opens in new tab).
- On the injectivity of the Kudla-Millson lift and surjectivity of the Borcherds lift (with J. Funke). In: Moonshine -- the first quarter century and beyond, Eds.: J. Lepowski, J. McKay, M. Tuite, Cambridge University Press (2010), 12-39. pdf (opens in new tab).
- Twisted Borcherds products on Hilbert modular surfaces and their CM values (with T. Yang), Amer. J. Math. 129 (2007), 807-841. pdf (opens in new tab).
- Hilbert class polynomials and traces of singular moduli (with P. Jenkins and K. Ono), Math. Ann. 334 (2006), 373-393. pdf (opens in new tab).
- Traces of CM-values of modular functions (with Jens Funke), J. Reine Angew. Math. 594 (2006), 1-33. pdf (opens in new tab).
- CM values of Hilbert modular functions (with T. Yang), Invent. Math. 163 (2006), 229-288. pdf (opens in new tab).
- Borcherds products and arithmetic intersection theory on Hilbert modular surfaces (with J. Burgos Gil and U. Kühn), Duke Math. Journal 139 (2007), 1-88. pdf (opens in new tab).
- On Two Geometric Theta Lifts (with J. Funke), Duke Math. Journal 125 (2004), 45-90. pdf (opens in new tab).
- Two applications of the curve lemma for orthogonal groups, Math. Nachr. 274-275 (2004), 19-31. pdf (opens in new tab).
- Integrals of automorphic Green's functions associated to Heegner divisors (with U. Kühn), Int. Math. Res. Not. 2003:31 (2003), 1687-1729. pdf (opens in new tab).
- Coefficients of half-integral weight modular forms (with K. Ono), J. Number Theory 99 (2003), 164-179. pdf (opens in new tab).
- The arithmetic of the values of modular functions and the divisors of modular forms (with W. Kohnen and K. Ono), Compos. Math. 140 (2004), 552-566. pdf (opens in new tab).
- On Borcherds products associated with lattices of prime discriminant (with M. Bundschuh), Ramanujan J. 7 (2003), 49-61. pdf (opens in new tab).
- The arithmetic of Borcherds' exponents (with K. Ono), Math. Ann. 327 (2003), 293-303. pdf (opens in new tab).
- On the rank of Picard groups of modular varieties attached to orthogonal groups, Compos. Math. 133 (2002), 49-63. pdf (opens in new tab).
- Local Borcherds products (with E. Freitag), Ann. Inst. Fourier 51 (2001), 1-27. pdf (opens in new tab).
- Eisenstein series attached to lattices and modular forms on orthogonal groups (with M. Kuss), Manuscr. Math. 106 (2001), 443-459. pdf (opens in new tab).
- Borcherds products and Chern classes of Hirzebruch-Zagier divisors, Invent. Math. 138 (1999), 51-83. pdf (opens in new tab).
- Non-vanishing modulo l of Fourier coefficients of half-integral weight modular forms, Duke Math. Journal 98 (1999), 595-611. pdf (opens in new tab).
- Non-vanishing of scalar products of Fourier-Jacobi coefficients of Siegel cusp forms (with S. Böcherer and W. Kohnen), Math. Ann. 313 (1999), 1-13. pdf (opens in new tab).
- On a theorem of Vigneras, Abh. Math. Sem. Univ. Hamburg 68 (1998), 163-168. pdf (opens in new tab).

#### Some expository papers

- Recent work on the partition function (with A. Folsom, Z. Kent, and K. Ono), The Legacy of Srinivasa Ramanujan, RMS-Lecture Notes Series No. 20 (2013), pp. 139–151. pdf (opens in new tab).
- The Gross-Zagier formula and the Borcherds lift, Proceedings of the National Institute of Mathematical Sciences 3, No. 10 (2008), 1-8. pdf (opens in new tab).
- Traces of CM-values of modular functions and related topics.

In: Proceedings of the Conference on “Automorphic Forms and Automorphic L-Functions”, RIMS Kokyuroku 1468, Kyoto (2006). pdf (opens in new tab). - Infinite products in number theory and geometry, Jahresber. Dtsch. Math. Ver. 106 (2004), Heft 4, 151-184. pdf (opens in new tab).
- Arithmetic Hirzebruch-Zagier divisors and modular forms.

In: Mathematisches Institut, Seminars, Ed.: Y. Tschinkel, Universität Göttingen (2004), 201-209. pdf (opens in new tab). - Local Picard groups and theta series.

In: Proceedings of the Japanese-German Seminar “Explicit Structures of Modular Forms and Zeta Functions”, Eds.: T. Ibukiyama and W. Kohnen, Ryushi-do (2002), 69-73. pdf (opens in new tab). - Primzahlen, Teilersummen und die Riemannsche Vermutung, Math. Semesterber. 48 (2001), 79-92. pdf (opens in new tab).
- Congruence properties of values of L-functions and applications (with K. James, W. Kohnen, K. Ono, C. Skinner, and V. Vatsal).

In: Topics in Number Theory (Eds.: S. D. Ahlgren et al.), Kluwer Academic Publishers (1999), 115-125.

#### Books

- L-Functions and Automorphic Forms. Proceedings of the conference LAF, Heidelberg, February 22-26, 2016 (jointly edited with W. Kohnen). Contributions in Mathematical and Computational Sciences, vol. 10, Springer-Verlag (2017).
- Hilbert modular forms and their applications. Notes of the lectures at the summer school on “Modular Forms and their Applications” at the Sophus Lie Conference Center, Nordfjordeid, Norway. Appeared in: The 1-2-3 of Modular Forms.
- Borcherds products on O(2,l) and Chern classes of Heegner divisors, Springer Lecture Notes in Mathematics 1780, Springer-Verlag (2002), 152 pages. This is a revised version of my Habilitationsschrift (Universität Heidelberg, May 2000). pdf (opens in new tab).

#### Thesis

- Borcherdsprodukte und Chernsche Klassen von Hirzebruch-Zagier-Zykeln, Dissertation, Universität Heidelberg (1998). pdf (opens in new tab).
- Modulformen halbganzen Gewichtes und Beziehungen zu Dirichletreihen, Diploma thesis, Universität Heidelberg (1997).

### Things of interest

- Teaching winter semester 2021/22: Lineare Algebra 1, Seminar Riemannsche Flächen.
- TU Darmstadt Algebra Group.
- I am deputy spokesperson of the CRC/TRR 326 GAUS.
- I am coordinator of the LOEWE-Schwerpunkt Uniformisierte Strukturen in Arithmetik und Geometrie.
- Member of the editorial boards of: Forum Mathematicum, Research in Number Theory.

### Useful Links

- Richard Borcherds, Eberhard Freitag, Jens Funke, Ben Howard, Steve Kudla, Ulf Kühn, Ken Ono, Nils Scheithauer, Tonghai Yang
- The 1-2-3 of Modular Forms
- Database of L-functions and modular forms LMFDB, Magma, PARI/GP
- Zentralblatt MATH Database, MathSciNet, arXiv.org e-print archive, Number Theory Web
- Deutsche Mathematiker-Vereinigung, AMS
- Mathematisches Institut Köln, Mathematisches Institut Heidelberg, MPI Bonn