A p-adic Maass–Shimura operator on Mumford curves

Springer Science and Business Media LLC - Tập 47 Số 1 - Trang 139-175 - 2023
Matteo Longo1
1Dipartimento di Matematica Tullio Levi-Civita, Università degli Studi di Padova, Via Trieste 63, 35121, Padua, Italy

Tóm tắt

AbstractWe study a p-adic Maass–Shimura operator in the context of Mumford curves defined by [15]. We prove that this operator arises from a splitting of the Hodge filtration, thus answering a question in [15]. We also study the relation of this operator with generalized Heegner cycles, in the spirit of [1, 4, 19, 28].

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Tài liệu tham khảo

Fabrizio Andreatta and Adrian Iovita, Katz type $$p$$-adic $$l$$-function for primes $$p$$ non split in the cm field, preprint (2019).

Pierre Berthelot, Lawrence Breen, and William Messing, Théorie de Dieudonné cristalline. II, Lecture Notes in Mathematics, vol. 930, Springer-Verlag, Berlin, 1982.

J.-F. Boutot and H. Carayol, Uniformisation $$p$$-adique des courbes de Shimura: les théorèmes de čerednik et de Drinfel’ d, Astérisque (1991), no. 196-197, 7, 45–158 (1992), Courbes modulaires et courbes de Shimura (Orsay, 1987/1988).

Massimo Bertolini, Henri Darmon, and Kartik Prasanna, Generalized Heegner cycles and $$p$$-adic Rankin $$L$$-series, Duke Math. J. 162 (2013), no. 6, 1033–1148, With an appendix by Brian Conrad.

Siegfried Bosch and Ulrich Görtz, Coherent modules and their descent on relative rigid spaces, J. Reine Angew. Math. 495 (1998), 119–134.

Pierre Berthelot and Arthur Ogus, Notes on crystalline cohomology, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978.

P. Berthelot and A. Ogus, $$F$$-isocrystals and de Rham cohomology. I, Invent. Math. 72 (1983), no. 2, 159–199.

Siegfried Bosch, Lectures on formal and rigid geometry, , vol. 2105, Springer, Cham, 2014.

Kevin Buzzard, Integral models of certain Shimura curves, Duke Math. J. 87 (1997), no. 3, 591–612.

Antoine Chambert-Loir, Cohomologie cristalline: un survol, Exposition. Math. 16 (1998), no. 4, 333–382.

Henri Darmon, Rational points on modular elliptic curves, CBMS Regional Conference Series in Mathematics, vol. 101, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004.

A. J. de Jong, Barsotti-Tate groups and crystals, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), no. Extra Vol. II, 1998, pp. 259–265.

V. G. Drinfel’d, Coverings of $$p$$-adic symmetric domains, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 29–40.

Gerd Faltings, Crystalline cohomology of semistable curve—the $${\bf Q}_p$$-theory, J. Algebraic Geom. 6 (1997), no. 1, 1–18.

Cameron Franc, Nearly rigid analytic modular forms and their values at CM points, ProQuest LLC, Ann Arbor, MI, 2011, Thesis (Ph.D.)–McGill University (Canada).

Alexandre Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné, Les Presses de l’Université de Montréal, Montreal, Que., 1974, Séminaire de Mathématiques Supérieures, No. 45 (Été, 1970).

Michael Harris, Special values of zeta functions attached to Siegel modular forms, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 1, 77–120.

Ki-ichiro Hashimoto, Explicit form of quaternion modular embeddings, Osaka J. Math. 32 (1995), no. 3, 533–546.

Ernest Hunter Brooks, Shimura curves and special values of $$p$$-adic $$L$$-functions, Int. Math. Res. Not. IMRN (2015), no. 12, 4177–4241.

Haruzo Hida, Elementary theory of $$L$$-functions and Eisenstein series, London Mathematical Society Student Texts, vol. 26, Cambridge University Press, Cambridge, 1993.

Luc Illusie, Cohomologie cristalline (d’après P. Berthelot), 53–60. Lecture Notes in Math., Vol. 514.

Adrian Iovita and Michael Spiess, Logarithmic differential forms on $$p$$-adic symmetric spaces, Duke Math. J. 110 (2001), no. 2, 253–278.

Adrian Iovita and Michael Spieß, Derivatives of $$p$$-adic $$L$$-functions, Heegner cycles and monodromy modules attached to modular forms, Invent. Math. 154 (2003), no. 2, 333–384.

Bruce W. Jordan and Ron A. Livné, Local Diophantine properties of Shimura curves, Math. Ann. 270 (1985), no. 2, 235–248.

Nicholas M. Katz, On the differential equations satisfied by period matrices, Inst. Hautes Études Sci. Publ. Math. (1968), no. 35, 223–258.

Nicholas M. Katz, $$p$$-adic $$L$$-functions for CM fields, Invent. Math. 49 (1978), no. 3, 199–297.

Nicholas M. Katz and Tadao Oda, On the differentiation of de Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ. 8 (1968), 199–213.

Daniel Kriz, A new $$p$$-adic Maass-Shimura operator and supersingular Rankin-Selberg $$p$$-adic $$l$$-functions, preprint available arxiv:1805.03605.pdf (2018).

Matteo Longo and Maria Rosaria Pati, Exceptional zero formulae for anticyclotomic $$p$$-adic $$L$$-functions of elliptic curves in the ramified case, J. Number Theory 190 (2018), 187–211.

Matteo Longo and Maria Rosaria Pati, Generalized Heegner cycles on Mumford curves, Math. Z. 297 (2021), no. 1-2, 483–515.

Matteo Longo and Stefano Vigni, On the vanishing of Selmer groups for elliptic curves over ring class fields, J. Number Theory 130 (2010), no. 1, 128–163.

Matteo Longo and Stefano Vigni, A refined Beilinson-Bloch conjecture for motives of modular forms, Trans. Amer. Math. Soc. 369 (2017), no. 10, 7301–7342.

Matteo Longo and Stefano Vigni, Kolyvagin systems and Iwasawa theory of generalized Heegner cycles, Kyoto J. Math. 59 (2019), no. 3, 717–746.

Marc Masdeu, CM cycles on Shimura curves, and $$p$$-adic $$L$$-functions, Compos. Math. 148 (2012), no. 4, 1003–1032.

William Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, , Vol. 264, Springer-Verlag, Berlin-New York, 1972.

B. Mazur and William Messing, Universal extensions and one dimensional crystalline cohomology, Lecture Notes in Mathematics, Vol. 370, Springer-Verlag, Berlin-New York, 1974.

Andrea Mori, Power series expansions of modular forms and their interpolation properties, Int. J. Number Theory 7 (2011), no. 2, 529–577.

Tadao Oda, The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École Norm. Sup. (4) 2 (1969), 63–135.

Arthur Ogus, $$F$$-isocrystals and de Rham cohomology. II. Convergent isocrystals, Duke Math. J. 51 (1984), no. 4, 765–850.

Peter Schneider, The cohomology of local systems on $$p$$-adically uniformized varieties, Math. Ann. 293 (1992), 623–650.

Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6 (1955–1956), 1–42.

Goro Shimura, On some arithmetic properties of modular forms of one and several variables, Ann. of Math. (2) 102 (1975), no. 3, 491–515.

Jeremy Teitelbaum, On Drinfel’d’s universal formal group over the $$p$$-adic upper half plane, Math. Ann. 284 (1989), no. 4, 647–674.

Thomas Zink, Cartiertheorie kommutativer formaler Gruppen, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 68, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1984, With English, French and Russian summaries.