Geodesic
KUDLA’S MODULARITY CONJECTURE AND FORMAL FOURIER–JACOBI SERIES
Forum of Mathematics, Pi, Tome 3 (2015)

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
@article{10_1017_fmp_2015_6,
     author = {JAN HENDRIK BRUINIER and MARTIN WESTERHOLT-RAUM},
     title = {KUDLA{\textquoteright}S {MODULARITY} {CONJECTURE} {AND} {FORMAL} {FOURIER{\textendash}JACOBI} {SERIES}},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {3},
     year = {2015},
     doi = {10.1017/fmp.2015.6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2015.6/}
}
TY  - JOUR
AU  - JAN HENDRIK BRUINIER
AU  - MARTIN WESTERHOLT-RAUM
TI  - KUDLA’S MODULARITY CONJECTURE AND FORMAL FOURIER–JACOBI SERIES
JO  - Forum of Mathematics, Pi
PY  - 2015
VL  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2015.6/
DO  - 10.1017/fmp.2015.6
LA  - en
ID  - 10_1017_fmp_2015_6
ER  - 
%0 Journal Article
%A JAN HENDRIK BRUINIER
%A MARTIN WESTERHOLT-RAUM
%T KUDLA’S MODULARITY CONJECTURE AND FORMAL FOURIER–JACOBI SERIES
%J Forum of Mathematics, Pi
%D 2015
%V 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2015.6/
%R 10.1017/fmp.2015.6
%G en
%F 10_1017_fmp_2015_6
JAN HENDRIK BRUINIER; MARTIN WESTERHOLT-RAUM. KUDLA’S MODULARITY CONJECTURE AND FORMAL FOURIER–JACOBI SERIES. Forum of Mathematics, Pi, Tome 3 (2015). doi: 10.1017/fmp.2015.6

[1] Andrianov, A. N., ‘Modular descent and the Saito–Kurokawa conjecture’, Invent. Math. 53(3) (1979), 267–280.Google Scholar

[2] Aoki, H., ‘Estimating Siegel modular forms of genus 2 using Jacobi forms’, J. Math. Kyoto Univ. 40(3) (2000), 581–588.Google Scholar

[3] Ash, A., Mumford, D., Rapoport, M. and Tai, Y.-S., ‘Smooth compactification of locally symmetric varieties’, in: Lie Groups: History, Frontiers and Applications, IV (Mathematical Science Press, Brookline, MA, 1975).Google Scholar

[4] Blichfeldt, H. F., ‘The minimum value of quadratic forms, and the closest packing of spheres’, Math. Ann. 101(1) (1929), 605–608.Google Scholar

[5] Borcherds, R. E., ‘The Gross–Kohnen–Zagier theorem in higher dimensions’, Duke Math. J. 97(2) (1999), 219–233.Google Scholar

[6] Borel, A. and Wallach, N. R., Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Annals of Mathematics Studies, 94 (Princeton University Press and University of Tokyo Press, Princeton, NJ and Tokyo, 1980).Google Scholar

[7] Braun, H., ‘Hermitian modular functions’, Ann. of Math. (2) 50(2) (1949), 827–855.Google Scholar

[8] Bruinier, J. H., ‘Vector valued formal Fourier–Jacobi series’, Proc. Amer. Math. Soc. 143(2) (2015), 505–512.Google Scholar

[9] Bruinier, J. H., Van Der Geer, G., Harder, G. and Zagier, D. B., The 1-2-3 of Modular Forms, Lectures from the Summer School on Modular Forms and their Applications held in Nordfjordeid, June 2004 (Universitext, Springer, Berlin, 2008).Google Scholar | DOI

[10] Eichler, M., ‘Über die Anzahl der linear unabhängigen Siegelschen Modulformen von gegebenem Gewicht’, Math. Ann. 213 (1975), 281–291. erratum; ibid. (1975), 195.Google Scholar

[11] Eichler, M. and Zagier, D. B., The Theory of Jacobi Forms, Progress in Mathematics, 55 (Birkhäuser Boston Inc., Boston, MA, 1985).Google Scholar

[12] Farkas, G., Grushevsky, S., Salvati Manni, R. and Verra, A., ‘Singularities of theta divisors and the geometry of A’, J. Eur. Math. Soc. (JEMS) 16(9) (2014), 1817–1848.Google Scholar

[13] Freitag, E. and Kiehl, R., Étale Cohomology and the Weil Conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 13 (Springer, Berlin, 1988), Translated from the German by B. S. Waterhouse and W. C. Waterhouse, with an historical introduction by J. A. Dieudonné.Google Scholar

[14] Fritzsche, K. and Grauert, H., From Holomorphic Functions to Complex Manifolds, Graduate Texts in Mathematics, 213 (Springer, New York, 2002).Google Scholar

[15] Grauert, H. and Remmert, R., ‘Komplexe Räume’, Math. Ann. 136 (1958), 245–318.Google Scholar

[16] Grushevsky, S., ‘Geometry of A and its compactifications’, in: Algebraic Geometry—Seattle 2005. Part 1, Proceedings of Symposia in Applied Mathematics, 80 (American Mathematical Society, Providence, RI, 2009), 193–234.Google Scholar

[17] Ibukiyama, T., Poor, C. and Yuen, D. S., ‘Jacobi forms that characterize paramodular forms’, Abh. Math. Semin. Univ. Hambg. 83(1) (2013), 111–128.Google Scholar

[18] Kohnen, W., Krieg, A. and Sengupta, J., ‘Characteristic twists of a Dirichlet series for Siegel cusp forms’, Manuscripta Math. 87(4) (1995), 489–499.Google Scholar

[19] Kohnen, W. and Skoruppa, N.-P., ‘A certain Dirichlet series attached to Siegel modular forms of degree two’, Invent. Math. 95(3) (1989), 541–558.Google Scholar

[20] Krieg, A., Modular Forms on Half-Spaces of Quaternions, Lecture Notes in Mathematics, 1143 (Springer, Berlin, 1985).Google Scholar

[21] Kudla, S. S., ‘Algebraic cycles on Shimura varieties of orthogonal type’, Duke Math. J. 86(1) (1997), 39–78.Google Scholar

[22] Kudla, S. S., ‘Special cycles and derivatives of Eisenstein series’, in: Heegner Points and Rankin L-Series, Mathematical Sciences Research Institute Publications, 49 (Cambridge University Press, Cambridge, 2004), 243–270.Google Scholar

[23] Kudla, S. S. and Millson, J., ‘Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables’, Publ. Math. Inst. Hautes Études Sci. 71 (1990), 121–172.Google Scholar

[24] Maass, H., ‘Über eine Spezialschar von Modulformen zweiten Grades’, Invent. Math. 52(1) (1979), 95–104.Google Scholar

[25] Maass, H., ‘Über eine Spezialschar von Modulformen zweiten Grades II’, Invent. Math. 53(3) (1979), 249–253.Google Scholar

[26] Maass, H., ‘Über eine Spezialschar von Modulformen zweiten Grades III’, Invent. Math. 53(3) (1979), 255–265.Google Scholar

[27] Matsumura, H., Commutative Algebra, 2nd edn, Mathematics Lecture Note Series, 56 (Benjamin/Cummings Publishing Co., Reading, MA, 1980).Google Scholar

[28] Namikawa, Y., Toroidal Compactification of Siegel Spaces, Lecture Notes in Mathematics, 812 (Springer, Berlin, 1980).Google Scholar

[29] Raum, M., ‘Formal Fourier Jacobi expansions and special cycles of codimension 2’, Compos. Math. (accepted), Preprint 2013, arXiv:1302.0880.Google Scholar

[30] Runge, B., ‘Theta functions and Siegel–Jacobi forms’, Acta Math. 175(2) (1995), 165–196.Google Scholar

[31] Salvati Manni, R., ‘Modular forms of the fourth degree’, in: Classification of Irregular Varieties (Trento, 1990), Lecture Notes in Mathematics, 1515 (Springer, Berlin, 1992), 106–111.Google Scholar

[32] Shimura, G., ‘The arithmetic of automorphic forms with respect to a unitary group’, Ann. of Math. (2) 107(3) (1978), 569–605.Google Scholar

[33] Siegel, C. L., ‘Die Modulgruppe in einer einfachen involutorischen Algebra’, in: Festschrift zur Feier des zweihundertjährigen Bestehens der Akademie der Wissenschaften in Göttingen, I. Math.-Phys. Kl. (Springer, Berlin, 1951), 157–167.Google Scholar

[34] Taïbi, O., ‘Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula’, Preprint, 2014, arXiv:14064247.Google Scholar

[35] The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2014.Google Scholar

[36] Wang, J., ‘Estimations on dimensions of spaces of Jacobi forms’, Sci. China A 42(2) (1999), 147–153.Google Scholar

[37] Yuan, X., Zhang, S.-W. and Zhang, W., ‘The Gross–Kohnen–Zagier theorem over totally real fields’, Compos. Math. 145(5) (2009), 1147–1162.Google Scholar

[38] Zagier, D. B., ‘Sur la conjecture de Saito–Kurokawa (d’après H Maass)’, in: Seminar on Number Theory, Paris 1979–80, Progress of Mathematics, 12 (Birkhäuser, Boston, 1981), 371–394.Google Scholar

[39] Zhang, W., ‘Modularity of generating functions of special cycles on shimura varieties’, PhD Thesis, Columbia University, 2009.Google Scholar

[40] Ziegler, C. D., ‘Jacobi forms of higher degree’, Abh. Math. Semin. Univ. Hambg. 59 (1989), 191–224.Google Scholar

Cité par Sources :