Abstract
Adapting the work of Kudla and Millson we obtain a lifting of cuspidal cohomology classes for the symmetric space associated to GO(V) for an indefinite rational quadratic space V of even dimension to holomorphic Siegel modular forms on GSp n (A). For n = 2 we prove the Thom Lemma for hyperbolic 3-space, which together with results of Kudla and Millson imply an interpretation of the Fourier coefficients of the theta lift as period integrals of the cohomology class over certain cycles, and relates those over infinite geodesics to L-values of cusp forms for GL2 over imaginary quadratic fields. This allows us to prove, for almost all primes p, the p-integrality of the lift for a particular choice of Schwartz function. We further calculate the Hecke eigenvalues (including for some “bad” places) for this choice in the case of V of signature (3,1).
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Agarwal, M., Klosin, K.: Yoshida lifts and the Bloch–Kato conjecture for the convolution L-function. J. Number Theory (2013) (to appear)
Böcherer S., Dummigan N., Schulze-Pillot R.: Yoshida lifts and Selmer groups. J. Math. Soc. Jpn. 64(4), 1353–1405 (2012)
Böcherer S., Schulze-Pillot R.: Siegel modular forms and theta series attached to quaternion algebras. Nagoya Math. J. 121, 35–96 (1991)
Bygott, J.: Modular forms and modular symbols over imaginary quadratic fields. Ph.D. thesis, University of Exeter (1999). Available for download at http://www.warwick.ac.uk/staff/J.E.Cremona/
Cremona J.E., Whitley E.: Periods of cusp forms and elliptic curves over imaginary quadratic fields. Math. Comput. 62(205), 407–429 (1994)
Eichler, M.: Quadratische Formen und orthogonale Gruppen. Springer, Berlin (1974) (Zweite Auflage, Die Grundlehren der mathematischen Wissenschaften, Band 63)
Faltings, G., Chai, C.L.: Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22. Springer, Berlin (1990). With an appendix by David Mumford
Funke J., Millson J.: Cycles in hyperbolic manifolds of non-compact type and Fourier coefficients of Siegel modular forms. Manuscr. Math. 107(4), 409–444 (2002)
Funke J., Millson J.: Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms. Am. J. Math. 128(4), 899–948 (2006)
Gan W.T., Ichino A.: On endoscopy and the refined Gross–Prasad conjecture for (SO5, SO4). J. Inst. Math. Jussieu 10(2), 235–324 (2011)
Ghate E.: Critical values of the twisted tensor L-function in the imaginary quadratic case. Duke Math. J. 96(3), 595–638 (1999)
Harder G.: Eisenstein cohomology of arithmetic groups. The case GL2. Invent. Math. 89(1), 37–118 (1987)
Harris M.: Eisenstein series on Shimura varieties. Ann. Math. 119(1), 59–94 (1984)
Harris M., Kudla S.S.: Arithmetic automorphic forms for the nonholomorphic discrete series of GSp(2). Duke Math. J. 66(1), 59–121 (1992)
Harris M., Soudry D., Taylor R.: l-adic representations associated to modular forms over imaginary quadratic fields. I. Lifting to GSp4(Q). Invent. Math. 112(2), 377–411 (1993)
Kitaoka, Y.: Arithmetic of quadratic forms, Cambridge Tracts in Mathematics, vol. 106. Cambridge University Press, Cambridge (1993)
Kojima H.: Fourier coefficients of modular forms of half integral weight, periods of modular forms and the special values of zeta functions. Hiroshima Math. J. 27(2), 361–371 (1997)
Kudla S.S.: Algebraic cycles on Shimura varieties of orthogonal type. Duke Math. J. 86(1), 39–78 (1997)
Kudla S.S., Millson J.J.: Geodesic cyclics and the Weil representation. I. Quotients of hyperbolic space and Siegel modular forms. Compositio Math. 45(2), 207–271 (1982)
Kudla S.S., Millson J.J.: The theta correspondence and harmonic forms. I. Math. Ann. 274(3), 353–378 (1986)
Kudla S.S., Millson J.J.: The theta correspondence and harmonic forms. II. Math. Ann. 277(2), 267–314 (1987)
Kudla S.S., Millson J.J.: Tubes, cohomology with growth conditions and an application to the theta correspondence. Canad. J. Math. 40(1), 1–37 (1988)
Kudla S.S., Millson J.J.: 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, 121–172 (1990)
Kurčanov P.F.: The cohomology of discrete groups and Dirichlet series that are related to Jacquet-Langlands cusp forms. Izv. Akad. Nauk SSSR Ser. Mat. 42(3), 588–601 (1978)
Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and theorems for the special functions of mathematical physics. Springer, New York (1966)
Prasanna K.: Integrality of a ratio of Petersson norms and level-lowering congruences. Ann. Math. 163(3), 901–967 (2006)
Prasanna, K.: Arithmetic aspects of the theta correspondence and periods of modular forms. In: Eisenstein series and applications, Progr. Math., vol. 258, pp. 251–269. Birkhäuser, Boston (2008)
Prasanna K.: Arithmetic properties of the Shimura–Shintani–Waldspurger correspondence. Invent. Math. 176(3), 521–600 (2009)
Rallis S.: Langlands’ functoriality and the Weil representation. Am. J. Math. 104(3), 469–515 (1982)
Roberts B.: The theta correspondence for similitudes. Israel J. Math. 94, 285–317 (1996)
Roberts B.: Global L-packets for GSp(2) and theta lifts. Doc. Math. 6, 247–314 (2001) (electronic)
Sugano T.: On holomorphic cusp forms on quaternion unitary groups of degree 2. J. Fac. Sci. Univ. Tokyo Sect 1A Math. 31(3), 521–568 (1985)
Taylor R.: Galois representations associated to Siegel modular forms of low weight. Duke Math. J. 63(2), 281–332 (1991)
Urban E.: Formes automorphes cuspidales pour GL2 sur un corps quadratique imaginaire. Valeurs spéciales de fonctions L et congruences. Compos. Math. 99(3), 283–324 (1995)
Urban E.: Module de congruences pour GL(2) d’un corps imaginaire quadratique et théorie d’Iwasawa d’un corps CM biquadratique. Duke Math. J. 92(1), 179–220 (1998)
Vignéras, M.F.: Arithmétique des algèbres de quaternions. In: Lecture Notes in Mathematics, vol. 800. Springer, Berlin (1980)
Yoshida H.: Siegel’s modular forms and the arithmetic of quadratic forms. Invent. Math. 60(3), 193–248 (1980)
Yoshida H.: On Siegel modular forms obtained from theta series. J. Reine Angew. Math. 352, 184–219 (1984)
Yoshida, H.: The action of Hecke operators on theta series. In: Algebraic and topological theories (Kinosaki, 1984), pp. 197–238. Kinokuniya, Tokyo (1986)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Berger, T. Arithmetic properties of similitude theta lifts from orthogonal to symplectic groups. manuscripta math. 143, 389–417 (2014). https://doi.org/10.1007/s00229-013-0628-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-013-0628-8