The construction of Green currents and singular theta lifts for unitary groups

Funke, Jens; Hofmann, Eric

doi:10.1090/tran/8289

The construction of Green currents and singular theta lifts for unitary groups
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by Jens Funke and Eric Hofmann PDF

Trans. Amer. Math. Soc. 374 (2021), 2909-2947 Request permission

Abstract:

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 associated to unitary groups. We establish an adjointness result between our singular theta lift and the Kudla-Millson lift. Further, we compare the two Greens forms and obtain modularity for the generating function of the difference of the two Green forms. Finally, we show that the Green forms obtained by the singular theta lift satisfy an eigenvalue equation for the Laplace operator and conclude that our Green forms coincide with the ones constructed by Oda and Tsuzuki by different means.

References

Jeffrey Adams, The theta correspondence over $\Bbb R$, Harmonic analysis, group representations, automorphic forms and invariant theory, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 12, World Sci. Publ., Hackensack, NJ, 2007, pp. 1–39. MR 2401808, DOI 10.1142/9789812770790_{0}001
Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. MR 1625724, DOI 10.1007/s002220050232
Kathrin Bringmann, Stephan Ehlen, and Markus Schwagenscheidt, On the modular completion of certain generating functions, International Mathematics Research Notices, to appear.
Jan H. Bruinier, Borcherds products on O(2, $l$) and Chern classes of Heegner divisors, Lecture Notes in Mathematics, vol. 1780, Springer-Verlag, Berlin, 2002. MR 1903920, DOI 10.1007/b83278
Jan Hendrik Bruinier, Regularized theta lifts for orthogonal groups over totally real fields, J. Reine Angew. Math. 672 (2012), 177–222. MR 2995436, DOI 10.1515/crelle.2011.163
Jan Hendrik Bruinier and Jens Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45–90. MR 2097357, DOI 10.1215/S0012-7094-04-12513-8
Jan-Hendrik Bruinier, Ben Howard, Stephen S. Kudla, Michael Rapoport, and Tonghai Yang, Modularity of generating series of divisors on unitary Shimura varieties, Astériques, to appear.
Jan-Hendrik Bruinier, Ben Howard, Stephen S. Kudla, Michael Rapoport, and Tonghai Yang. Modularity of generating series of divisors on unitary Shimura varieties II: arithmetic applications. Astériques, to appear.
Jan Hendrik Bruinier, Benjamin Howard, and Tonghai Yang, Heights of Kudla-Rapoport divisors and derivatives of $L$-functions, Invent. Math. 201 (2015), no. 1, 1–95. MR 3359049, DOI 10.1007/s00222-014-0545-9
Jan Hendrik Bruinier and Ulf Kühn, Integrals of automorphic Green’s functions associated to Heegner divisors, Int. Math. Res. Not. 31 (2003), 1687–1729. MR 1981482, DOI 10.1155/S1073792803204165
Jan Hendrik Bruinier and Tonghai Yang, Arithmetic degrees of special cycles and derivatives of Siegel Eisenstein series, Journal of the European Mathematical Society (JEMS), to appear.
Stephan Ehlen and Siddarth Sankaran, On two arithmetic theta lifts, Compos. Math. 154 (2018), no. 10, 2090–2149. MR 3867297, DOI 10.1112/s0010437x18007327
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. I, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0061695
Mogens Flensted-Jensen, Discrete series for semisimple symmetric spaces, Ann. of Math. (2) 111 (1980), no. 2, 253–311. MR 569073, DOI 10.2307/1971201
Jens Funke and Stephen S. Kudla, Mock modular forms and geometric theta functions for indefinite quadratic forms, J. Phys. A 50 (2017), no. 40, 404001, 19. MR 3708083, DOI 10.1088/1751-8121/aa848b
Jens Funke and John Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, Amer. J. Math. 128 (2006), no. 4, 899–948. MR 2251589
Luis E. Garcia and Siddarth Sankaran, Green forms and the arithmetic Siegel-Weil formula, Invent. Math. 215 (2019), no. 3, 863–975. MR 3935034, DOI 10.1007/s00222-018-0839-4
Jeffrey A. Harvey and Gregory Moore, Algebras, BPS states, and strings, Nuclear Phys. B 463 (1996), no. 2-3, 315–368. MR 1393643, DOI 10.1016/0550-3213(95)00605-2
Eric Hofmann, Borcherds products on unitary groups, Math. Ann. 358 (2014), no. 3-4, 799–832. MR 3175141, DOI 10.1007/s00208-013-0966-6
Tobias Hufler, Automorphe Formen auf orthogonalen und unitären Gruppen, Ph.D. thesis, TU Darmstadt, 2017.
Stephen S. Kudla, Central derivatives of Eisenstein series and height pairings, Ann. of Math. (2) 146 (1997), no. 3, 545–646. MR 1491448, DOI 10.2307/2952456
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, The theta correspondence and harmonic forms. I, Math. Ann. 274 (1986), no. 3, 353–378. MR 842618, DOI 10.1007/BF01457221
Stephen S. Kudla and John J. Millson, The theta correspondence and harmonic forms. II, Math. Ann. 277 (1987), no. 2, 267–314. MR 886423, DOI 10.1007/BF01457364
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
Yifeng Liu, Arithmetic theta lifting and $L$-derivatives for unitary groups, I, Algebra Number Theory 5 (2011), no. 7, 849–921. MR 2928563, DOI 10.2140/ant.2011.5.849
Takayuki Oda and Masao Tsuzuki, The secondary spherical functions and automorphic Green currents for certain symmetric pairs, Pure Appl. Math. Q. 5 (2009), no. 3, Special Issue: In honor of Friedrich Hirzebruch., 977–1028. MR 2532712, DOI 10.4310/PAMQ.2009.v5.n3.a4
Takuro Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83–126. MR 389772