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$p$-ADIC EISENSTEIN SERIES AND $L$-FUNCTIONS OF CERTAIN CUSP FORMS ON DEFINITE UNITARY GROUPS

Part of: Arithmetic algebraic geometry Algebraic number theory: local and $p$-adic fields Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  21 November 2014

Ellen Eischen  and
Xin Wan
Ellen Eischen
Affiliation:
Department of Mathematics, The University of North Carolina at Chapel Hill, CB #3250, Chapel Hill, NC 27599-3250, USA (eeischen@email.unc.edu)
Xin Wan
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10025, USA (xw2295@math.columbia.edu)
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Abstract

We construct $p$-adic families of Klingen–Eisenstein series and $L$-functions for cusp forms (not necessarily ordinary) unramified at an odd prime $p$ on definite unitary groups of signature $(r,0)$ (for any positive integer $r$) for a quadratic imaginary field ${\mathcal{K}}$ split at $p$. When $r=2$, we show that the constant term of the Klingen–Eisenstein family is divisible by a certain $p$-adic $L$-function.

Keywords

p-adic automorphic formsautomorphic forms on unitary groupsEisenstein seriesp-adic L-functionsKlingen–Eisenstein series

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms 11F03: Modular and automorphic functions
Secondary: 11F30: Fourier coefficients of automorphic forms 11S40: Zeta functions and $L$-functions 11G18: Arithmetic aspects of modular and Shimura varieties 11F55: Other groups and their modular and automorphic forms (several variables) 11F85: $p$-adic theory, local fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 3 , July 2016 , pp. 471 - 510
Copyright
© Cambridge University Press 2014 

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References

Barnet-Lamb, T., Geraghty, D., Harris, M. and Taylor, R., A family of Calabi-Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci. 47(1) (2011), 29–98, MR 2827723 (2012m:11069).CrossRefGoogle Scholar
Casselman, W., Introduction to the theory of admissible representations of $p$-adic reductive groups, 1995. Available at http://www.math.ubc.ca/∼cass/research/pdf/p-adic-book.pdf.Google Scholar
Eischen, E. E., p-adic differential operators on automorphic forms on unitary groups, Ann. Inst. Fourier (Grenoble) 62(1) (2012), 177–243, MR 2986270.CrossRefGoogle Scholar
Eischen, E. E., A p-adic Eisenstein measure for unitary groups, J. Reine Angew. Math. (Crelle’s Journal) (accepted for publication), 32 pages, doi:10.1515/crelle-2013-0008.Google Scholar
Eischen, E. E., A p-adic Eisenstein measure for vector-weight automorphic forms, Algebra Number Theory (accepted for publication). Also available at http://arxiv.org/pdf/1302.7229.pdf, 32 pages.Google Scholar
Gelbart, S., Piatetski-Shapiro, I. and Rallis, S., Explicit Constructions of Automorphic L-functions, Lecture Notes in Mathematics, Volume 1254 (Springer-Verlag, Berlin, 1987), MR 892097 (89k:11038).CrossRefGoogle Scholar
Harris, M., Eisenstein series on Shimura varieties, Ann. of Math. (2) 119(1) (1984), 59–94, MR 736560 (85j:11052).CrossRefGoogle Scholar
Harris, M., A simple proof of rationality of Siegel-Weil Eisenstein series, in Eisenstein Series and Applications, Progress in Mathematics, Volume 258, pp. 149–185 (Birkhäuser, Boston, MA, 2008), MR 2402683 (2009g:11061).CrossRefGoogle Scholar
Hida, H., On p-adic L-functions of GL(2) ×GL(2) over totally real fields, Ann. Inst. Fourier (Grenoble) 41(2) (1991), 311–391, MR 1137290 (93b:11052).CrossRefGoogle Scholar
Hida, H., Geometric Modular Forms and Elliptic Curves (World Scientific Publishing Co. Inc., River Edge, NJ, 2000), MR 1794402 (2001j:11022).CrossRefGoogle Scholar
Hida, H., p-adic Automorphic Forms on Shimura Varieties, Springer Monographs in Mathematics (Springer-Verlag, New York, 2004), MR 2055355 (2005e:11054).CrossRefGoogle Scholar
Hida, H., p-adic automorphic forms on reductive groups, Astérisque (298) (2005), 147–254 Automorphic forms. I, MR 2141703 (2006e:11060).Google Scholar
Harris, M., Li, J.-S. and Skinner, C. M., The Rallis inner product formula and p-adic L-functions, in Automorphic Representations, L-functions and Applications: Progress and Prospects, Ohio State University of Mathematics Research Institute Publication, Volume 11, pp. 225–255 (de Gruyter, Berlin, 2005), MR 2192825 (2006k:11096).CrossRefGoogle Scholar
Harris, M., Shepherd-Barron, N. and Taylor, R., A family of Calabi-Yau varieties and potential automorphy, Ann. of Math. (2) 171(2) (2010), 779–813, MR 2630056 (2011g:11106).CrossRefGoogle Scholar
Hsieh, M.-L., Ordinary p-adic Eisenstein series and p-adic L-functions for unitary groups, Ann. Inst. Fourier (Grenoble) 61(3) (2011), 987–1059, MR 2918724.CrossRefGoogle Scholar
Hsieh, M. L., Eisenstein congruence on unitary groups and iwasawa main conjectures for cm fields, J. Amer. Math. Soc. (2013) (accepted for publication), 84 pages.Google Scholar
Jantzen, J. C., Representations of Algebraic Groups, Pure and Applied Mathematics, Volume 131 (Academic Press Inc., Boston, MA, 1987), MR 899071 (89c:20001).Google Scholar
Katz, N. M., p-adic L-functions for CM fields, Invent. Math. 49(3) (1978), 199–297, MR 513095 (80h:10039).CrossRefGoogle Scholar
Kubota, T. and Leopoldt, H.-W., Eine p-adische Theorie der Zetawerte. I. Einführung der p-adischen Dirichletschen L-Funktionen, J. Reine Angew. Math. 214/215 (1964), 328–339, MR 0163900 (29 #1199).Google Scholar
Lan, K.-W., Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties, J. Reine Angew. Math. 664 (2012), 163–228, MR 2980135.Google Scholar
Lan, K.-W., Arithmetic Compactifications of PEL-type Shimura Varieties, London Mathematical Society Monographs, Volume 36 (Princeton University Press, 2013).Google Scholar
Lapid, E. M. and Rallis, S., On the local factors of representations of classical groups, in Automorphic Representations, L-functions and Applications: Progress and Prospects, Ohio State University of Mathematics Research Institute Publication, Volume 11, pp. 309–359 (de Gruyter, Berlin, 2005), MR 2192828 (2006j:11071).CrossRefGoogle Scholar
Panchishkin, A. A., Two variable p-adic L functions attached to eigenfamilies of positive slope, Invent. Math. 154(3) (2003), 551–615, MR 2018785 (2004k:11065).CrossRefGoogle Scholar
Shimura, G., Euler Products and Eisenstein Series, CBMS Regional Conference Series in Mathematics, Volume 93 (Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997), MR 1450866 (98h:11057).CrossRefGoogle ScholarPubMed
Shimura, G., Arithmeticity in the Theory of Automorphic Forms, Mathematical Surveys and Monographs, Volume 82 (American Mathematical Society, Providence, RI, 2000), MR 1780262 (2001k:11086).Google Scholar
Skinner, C. and Urban, E., Vanishing of L-functions and Ranks of Selmer Groups, International Congress of Mathematicians, Volume II, pp. 473–500 (European Mathematical Society, Zürich, 2006), MR 2275606 (2008a:11063).Google Scholar
Skinner, C. and Urban, E., The Iwasawa main conjecture for GL2, Invent. Math. (2013) (accepted for publication).Google Scholar
Urban, E., Nearly overconvergent modular forms, Proceedings of the conference IWASAWA 2012 held at Heidelberg, 2013, in press. Also available at http://www.math.columbia.edu/∼urban/eurp/quasi-surconv.pdf.CrossRefGoogle Scholar
Wan, X., $p$-adic $L$-functions of ordinary forms on unitary groups and Eisenstein series, 2013. Available at http://www.math.columbia.edu/∼xw2295/families%20of%20ordinary%20Eisenstein%20Series.pdf.Google Scholar