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Generalized Heegner cycles on Mumford curves

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Abstract

We study generalised Heegner cycles, originally introduced by Bertolini–Darmon–Prasanna for modular curves in Bertolini et al. (Duke Math J 162(6):1033–1148, 2013), in the context of Mumford curves. The main result of this paper relates generalized Heegner cycles with the two variable anticyclotomic p-adic L-function attached to a Coleman family \(f_\infty \) and an imaginary quadratic field K, constructed in Bertolini and Darmon (Invent Math 168(2):371–431, 2007) and Seveso (J Reine Angew Math 686:111–148, 2014). While in Bertolini and Darmon (Invent Math 168(2):371–431, 2007) and Seveso (J Reine Angew Math 686:111–148, 2014) only the restriction to the central critical line of this 2 variable p-adic L-function is considered, our generalised Heegner cycles allow us to study the restriction of this function to non-central critical lines. The main result expresses the derivative along the weight variable of this anticyclotomic p-adic L-function restricted to non necessarily central critical lines as a combination of the image of generalized Heegner cycles under a p-adic Abel–Jacobi map. In studying generalised Heegner cycles in the context of Mumford curves, we also obtain an extension of a result of Masdeu (Compos Math 148(4):1003–1032, 2012) for the (one variable) anticyclotomic p-adic L-function of a modular form f and K at non-central critical integers.

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References

  1. Bertolini, M., Darmon, H., Prasanna, K.: Generalized Heegner cycles and \(p\)-adic Rankin \(L\)-series. Duke Math. J. 162(6), 1033–1148 (2013) (With an appendix by Brian Conrad)

  2. Bertolini, M., Darmon, H.: Heegner points on Mumford–Tate curves. Invent. Math. 126(3), 413–456 (1996)

    Article  MathSciNet  Google Scholar 

  3. Bertolini, M., Darmon, H.: Heegner points, \(p\)-adic \(L\)-functions, and the Cerednik–Drinfeld uniformization. Invent. Math. 131(3), 453–491 (1998)

    Article  MathSciNet  Google Scholar 

  4. Bertolini, M., Darmon, H.: Hida families and rational points on elliptic curves. Invent. Math. 168(2), 371–431 (2007)

    Article  MathSciNet  Google Scholar 

  5. Bertolini, M., Darmon, H.: The rationality of Stark–Heegner points over genus fields of real quadratic fields. Ann. Math. (2) 170(1), 343–370 (2009)

    Article  MathSciNet  Google Scholar 

  6. Bertolini, M., Darmon, H., Iovita, A., Spiess, M.: Teitelbaum’s exceptional zero conjecture in the anticyclotomic setting. Am. J. Math. 124(2), 411–449 (2002)

    Article  MathSciNet  Google Scholar 

  7. Besser, A.: CM cycles over Shimura curves. J. Algebraic Geom. 4(4), 659–691 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Bloch, S., Kato, K.: \(L\)-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, vol. I, Progr. Math., vol. 86, pp. 333–400. Birkhäuser, Boston (1990)

  9. Boutot, J.-F., Carayol, H.: Uniformisation \(p\)-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfeld, Astérisque (1991) [no. 196-197, 7, 45–158 (1992), Courbes modulaires et courbes de Shimura (Orsay, 1987/1988)]

  10. Brinon, O., Conrad, B.: Cmi summer school notes on p-adic hodge theory. http://math.stanford.edu/~conrad/papers/notes.pdf

  11. Chenevier, G.: Une correspondance de Jacquet–Langlands \(p\)-adique. Duke Math. J. 126(1), 161–194 (2005)

    Article  MathSciNet  Google Scholar 

  12. Coleman, R.F.: Dilogarithms, regulators and \(p\)-adic \(L\)-functions. Invent. Math. 69(2), 171–208 (1982)

    Article  MathSciNet  Google Scholar 

  13. Darmon, H.: 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)

  14. Dasgupta, S., Teitelbaum, J.: The \(p\)-adic upper half plane, \(p\)-adic geometry, Univ. Lecture Ser., vol. 45, pp. 65–121. American Mathematical Society, Providence, RI (2008)

  15. de Shalit, E.: Eichler cohomology and periods of modular forms on \(p\)-adic Schottky groups. J. Reine Angew. Math. 400, 3–31 (1989)

    MathSciNet  MATH  Google Scholar 

  16. Deninger, C., Murre, J.: Motivic decomposition of abelian schemes and the Fourier transform. J. Reine Angew. Math. 422, 201–219 (1991)

    MathSciNet  MATH  Google Scholar 

  17. Fontaine, J.M., Ouyang, Y.: Theory of p-adic galois representations (2008). https://www.math.u-psud.fr/~fontaine/galoisrep.pdf

  18. Hunter Brooks, E.: Shimura curves and special values of \(p\)-adic \(L\)-functions. Int. Math. Res. Not. IMRN (12), 4177–4241 (2015)

  19. Iovita, A., Spieß, M.: Derivatives of \(p\)-adic \(L\)-functions, Heegner cycles and monodromy modules attached to modular forms. Invent. Math. 154(2), 333–384 (2003)

    Article  MathSciNet  Google Scholar 

  20. Jannsen, U.: Mixed Motives and Algebraic \(K\)-Theory. Lecture Notes in Mathematics, vol. 1400. Springer, Berlin (1990)

  21. Jeremy, T.: Teitelbaum, values of \(p\)-adic \(L\)-functions and a \(p\)-adic Poisson kernel. Invent. Math. 101(2), 395–410 (1990)

    MathSciNet  MATH  Google Scholar 

  22. Jordan, B.W., Livné, R.A.: Local Diophantine properties of Shimura curves. Math. Ann. 270(2), 235–248 (1985)

    Article  MathSciNet  Google Scholar 

  23. Longo, M., Mao, Z.: Kohnen’s formula and a conjecture of Darmon and Tornaría. Trans. Am. Math. Soc. 370(1), 73–98 (2018)

    Article  Google Scholar 

  24. Longo, M., Nicole, M.-H.: The Saito–Kurokawa lifting and Darmon points. Math. Ann. 356(2), 469–486 (2013)

    Article  MathSciNet  Google Scholar 

  25. Longo, M., Nicole, M.-H.: The \(p\)-adic variation of the Gross-Kohnen-Zagier theorem. Forum Math. 31(4), 1069–1084 (2019)

  26. Longo, M., Pati, M.R.: Exceptional zero formulae for anticyclotomic \(p\)-adic \(L\)-functions of elliptic curves in the ramified case. J. Number Theory 190, 187–211 (2018)

    Article  MathSciNet  Google Scholar 

  27. Longo, M., Vigni, S.: Quaternion algebras, Heegner points and the arithmetic of Hida families. Manuscr. Math. 135(3–4), 273–328 (2011)

    Article  MathSciNet  Google Scholar 

  28. Longo, M., Vigni, S.: A note on control theorems for quaternionic Hida families of modular forms. Int. J. Number Theory 8(6), 1425–1462 (2012)

    Article  MathSciNet  Google Scholar 

  29. Longo, M., Vigni, S.: The rationality of quaternionic Darmon points over genus fields of real quadratic fields. Int. Math. Res. Not. IMRN (13), 3632–3691 (2014)

  30. Longo, M., Vigni, S.: Quaternionic Darmon points on abelian varieties. Riv. Math. Univ. Parma (N.S.) 7(1), 39–70 (2016)

  31. Masdeu, M.: CM cycles on Shimura curves, and \(p\)-adic \(L\)-functions. Compos. Math. 148(4), 1003–1032 (2012)

    Article  MathSciNet  Google Scholar 

  32. Milne, J.S.: Étale cohomology, Princeton Mathematical Series, vol. 33. Princeton University Press, Princeton, N.J. (1980)

    Google Scholar 

  33. Nekovář, J.: \(p\)-adic Abel–Jacobi maps and \(p\)-adic heights, The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), CRM Proc. Lecture Notes, vol. 24, pp. 367–379. American Mathematical Society, Providence, RI (2000)

  34. Scholl, A.J.: Classical motives, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, pp. 163–187. American Mathematical Society, Providence, RI (1994)

  35. Seveso, M.A.: Heegner cycles and derivatives of \(p\)-adic \(L\)-functions. J. Reine Angew. Math. 686, 111–148 (2014)

    MathSciNet  MATH  Google Scholar 

  36. Venerucci, R.: Exceptional zero formulae and a conjecture of Perrin–Riou. Invent. Math. 203(3), 923–972 (2016)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We are grateful to A. Iovita, S. Vigni, D. Loeffler, M. Masdeu and M. Seveso for many interesting discussions about the topics of this paper. M.L. is supported by PRIN 2017, INdAM, DOR University of Padova, and M.R.P. is supported by BIRD 2017 University of Padova. We would like to thank the referee for her/his comments which lead us to improve the clarity of the paper.

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  1. Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121, Padova, Italy

    Matteo Longo & Maria Rosaria Pati

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Longo, M., Pati, M.R. Generalized Heegner cycles on Mumford curves. Math. Z. 297, 483–515 (2021). https://doi.org/10.1007/s00209-020-02522-8

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