Vector valued formal Fourier-Jacobi series
HTML articles powered by AMS MathViewer
- by Jan Hendrik Bruinier PDF
- Proc. Amer. Math. Soc. 143 (2015), 505-512 Request permission
Abstract:
H. Aoki showed that any symmetric formal Fourier-Jacobi series for the symplectic group $\mathrm {Sp}_2(\mathbb {Z})$ is the Fourier-Jacobi expansion of a holomorphic Siegel modular form. We prove an analogous result for vector valued symmetric formal Fourier-Jacobi series, by combining Aoki’s theorem with facts about vector valued modular forms. Recently, this result was also proved independently by M. Raum using a different approach. As an application, by means of work of W. Zhang, modularity results for special cycles of codimension $2$ on Shimura varieties associated to orthogonal groups can be derived.References
- Hiroki Aoki, Estimating Siegel modular forms of genus 2 using Jacobi forms, J. Math. Kyoto Univ. 40 (2000), no. 3, 581–588. MR 1794522, DOI 10.1215/kjm/1250517682
- Richard E. Borcherds, The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J. 97 (1999), no. 2, 219–233. MR 1682249, DOI 10.1215/S0012-7094-99-09710-7
- Atish Dabholkar and Suresh Nampuri, Quantum black holes, Strings and fundamental physics, Lecture Notes in Phys., vol. 851, Springer, Heidelberg, 2012, pp. 165–232. MR 2920326, DOI 10.1007/978-3-642-25947-0_{5}
- Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735, DOI 10.1007/978-1-4684-9162-3
- E. Freitag, Siegelsche Modulfunktionen, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 254, Springer-Verlag, Berlin, 1983 (German). MR 871067, DOI 10.1007/978-3-642-68649-8
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
- B. Gross, W. Kohnen, and D. Zagier, Heegner points and derivatives of $L$-series. II, Math. Ann. 278 (1987), no. 1-4, 497–562. MR 909238, DOI 10.1007/BF01458081
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- T. Ibukiyama, C. Proor, and D. Yuen, Jacobi forms that characterize paramodular forms, preprint (2012), arXiv:1209.3438 [math.NT].
- Stephen S. Kudla, Algebraic cycles on Shimura varieties of orthogonal type, Duke Math. J. 86 (1997), no. 1, 39–78. MR 1427845, DOI 10.1215/S0012-7094-97-08602-6
- Stephen S. Kudla, Integrals of Borcherds forms, Compositio Math. 137 (2003), no. 3, 293–349. MR 1988501, DOI 10.1023/A:1024127100993
- Stephen S. Kudla, Special cycles and derivatives of Eisenstein series, Heegner points and Rankin $L$-series, Math. Sci. Res. Inst. Publ., vol. 49, Cambridge Univ. Press, Cambridge, 2004, pp. 243–270. MR 2083214, DOI 10.1017/CBO9780511756375.009
- Stephen S. Kudla and John J. Millson, Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables, Inst. Hautes Études Sci. Publ. Math. 71 (1990), 121–172. MR 1079646
- M. Raum, Formal Fourier Jacobi expansions and special cycles of codimension $2$, preprint (2013), arXiv:1302.0880 [math.NT].
- B. Steinle, Fixpunktmannigfaltigkeiten symplektischer Matrizen, Acta Arith. 20 (1972), 63–106.
- Wei Zhang, Modularity of generating functions of special cycles on Shimura varieties, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–Columbia University. MR 2717745
Additional Information
- Jan Hendrik Bruinier
- Affiliation: Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, D–64289 Darmstadt, Germany
- MR Author ID: 641446
- Email: bruinier@mathematik.tu-darmstadt.de
- Received by editor(s): March 15, 2013
- Received by editor(s) in revised form: May 16, 2013
- Published electronically: September 19, 2014
- Additional Notes: The author was partially supported by DFG grants BR-2163/2-2 and FOR 1920.
- Communicated by: Kathrin Bringmann
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 505-512
- MSC (2010): Primary 11F46, 11F50
- DOI: https://doi.org/10.1090/S0002-9939-2014-12272-6
- MathSciNet review: 3283640