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A p-ADIC HERMITIAN MAASS LIFT

  • TOBIAS BERGER (a1) and KRZYSZTOF KLOSIN (a2)
Abstract

For K, an imaginary quadratic field with discriminant −DK, and associated quadratic Galois character χK, Kojima, Gritsenko and Krieg studied a Hermitian Maass lift of elliptic modular cusp forms of level DK and nebentypus χK via Hermitian Jacobi forms to Hermitian modular forms of level one for the unitary group U(2, 2) split over K. We generalize this (under certain conditions on K and p) to the case of p-oldforms of level pDK and character χK. To do this, we define an appropriate Hermitian Maass space for general level and prove that it is isomorphic to the space of special Hermitian Jacobi forms. We then show how to adapt this construction to lift a Hida family of modular forms to a p-adic analytic family of automorphic forms in the Maass space of level p.

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1. Andreatta, F., Iovita, A. and Pilloni, V., p-adic families of Siegel modular cuspforms, Ann. Math. 181 (2) (2015), 623697.
2. Atkin, A. O. L. and Li, W. C. W., Twists of newforms and pseudo-eigenvalues of W-operators, Invent. Math. 48 (3) (1978), 221243.
3. Atobe, H., Pullbacks of Hermitian Maass lifts, J. Number Theory 153 (2015), 158229.
4. Banerjee, D., Ghate, E. and Narasimha Kumar, V. G., Λ-adic forms and the Iwasawa main conjecture, in Guwahati Workshop on Iwasawa Theory of Totally Real Fields, Ramanujan Mathematical Society Lecture Notes Series, vol. 12 (Ramanujan Mathematical Society, Mysore, 2010), 1547.
5. Böcherer, S. and Schmidt, C.-G., p-adic measures attached to Siegel modular forms, Ann. Inst. Fourier (Grenoble) 50 (5) (2000), 13751443.
6. Chenevier, G., Familles p-adiques de formes automorphes pour GLn, J. R. Angew. Math. 570 (2004), 143217.
7. Emerton, M., Pollack, R. and Weston, T., Variation of Iwasawa invariants in Hida families, Invent. Math. 163 (3) (2006), 523580.
8. Eischen, E. and Wan, X., p-adic L-functions of finite slope forms on unitary groups and Eisenstein series. J. Inst. Math. Jussieu. 15 (2016), 471510.
9. Eichler, M. and Zagier, D., The theory of Jacobi forms, Progress in Mathematics, vol. 55 (Birkhäuser Boston, Inc., Boston, MA, 1985).
10. Gritsenko, V. A., The Maass space for SU(2,2). The Hecke ring, and zeta functions, Trudy Mat. Inst. Steklov. 183 (1990), 68–78, 223225. Translated in Galois theory, rings, algebraic groups and their applications (Russian), Proc. Steklov Inst. Math. 4 (1991), 75–86.
11. Grosche, J., Über verallgemeinerte Hermitesche Modulgruppen, J. R. Angew. Math. 302 (1978), 137166.
12. Guerzhoy, P., On p-adic families of Siegel cusp forms in the Maaß Spezialschar, J. R. Angew. Math. 523 (2000), 103112.
13. Haverkamp, K., Hermitesche Jacobiformen, Schriftenreihe des Mathematischen Instituts der Universität Münster. 3. Serie, vol. 15 (Univ. Münster, Münster, 1995), 105.
14. Hida, H., Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms, Invent. Math. 85 (3) (1986), 545613.
15. Hida, H., Elementary theory of L-functions and Eisenstein series, London Mathematical Society Student Texts, vol. 26, (Cambridge University Press, Cambridge, 1993).
16. Harris, M., Li, J.-S. and Skinner, C. M., The Rallis inner product formula and p-adic L-functions, in Proceedings of the Conference on Automorphic Representations, L-functions and Applications: Progress and Prospects, Ohio State University Mathematics Research Institute Publ., vol. 11 (Walter de Gruyter, Berlin, 2005), 225255.
17. Ibukiyama, T., Saito-Kurokawa liftings of level N and practical construction of Jacobi forms, Kyoto J. Math. 52 (1) (2012), 141178.
18. Ikeda, T., On the lifting of Hermitian modular forms, Compos. Math. 144 (5) (2008), 11071154.
19. Kawamura, H.-A., On certain constructions of p-adic Siegel modular forms of even genus, preprint (2010) arXiv:1011.6476.
20. Klingen, H., Bemerkung über Kongruenzuntergruppen der Modulgruppe n-ten Grades, Arch. Math. 10 (1959), 113122.
21. Klosin, K., The Maass space for U(2,2) and the Bloch–Kato conjecture for the symmetric square motive of a modular form, J. Math. Soc. Jpn. 67 (2) (2015), 797860.
22. Kikuta, T. and Mizuno, Y., On p-adic Hermitian Eisenstein series and p-adic Siegel cusp forms, J. Number Theory 132 (9) (2012), 19491961.
23. Kohnen, W., Newforms of half-integral weight, J. R. Angew. Math. 333 (1982), 3272.
24. Kojima, H., An arithmetic of Hermitian modular forms of degree two, Invent. Math. 69 (2) (1982), 217227.
25. Krieg, A., The Maaß spaces on the Hermitian half-space of degree 2, Math. Ann. 289 (4) (1991), 663681.
26. Li, Z., On Λ-adic Saito-Kurokawa lifting and its application, PhD thesis (Columbia University, 2009).
27. Miyake, T., Modular forms (Springer-Verlag, Berlin, 1989), Translated from the Japanese by Yoshitaka Maeda.
28. Manickam, M., Ramakrishnan, B., and Vasudevan, T. C., On Saito-Kurokawa descent for congruence subgroups, Manuscr. Math. 81 (1–2) (1993), 161182.
29. Mazur, B. and Wiles, A., Class fields of abelian extensions of Q, Invent. Math. 76 (2) (1984), 179330.
30. Shintani, T., On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83126.
31. Stevens, G., Λ-adic modular forms of half-integral weight and a Λ-adic Shintani lifting, in Arithmetic geometry (Tempe, AZ, 1993) (Childress, N. and Jones, J. W., Editors), Contemporary Mathematics, vol. 174 (American Mathematics Society, Providence, RI, 1994), 129151.
32. Skinner, C. and Urban, E., The Iwasawa main conjectures for GL 2, Invent. Math. 195 (1) (2014), 1277.
33. Taylor, R., On congruences between modular forms, PhD Thesis, (Princeton University, Princeton, 1988).
34. Urban, E., Eigenvarieties for reductive groups, Ann. Math. (2) 174 (3) (2011), 16851784.
35. Wiles, A., On ordinary Λ-adic representations associated to modular forms, Invent. Math. 94 (3) (1988), 529573.
36. Wiles, A., The Iwasawa conjecture for totally real fields, Ann. Math. 131 (3) (1990), 493540.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
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