Annales mathématiques du Québec

, Volume 40, Issue 2, pp 397–434 | Cite as

p-adic families of modular forms and p-adic Abel-Jacobi maps

  • Matthew Greenberg
  • Marco Adamo Seveso


We show that p-adic families of modular forms give rise to certain p-adic Abel-Jacobi maps at their p-new specializations. We introduce the concept of differentiation of distributions, using it to give a new description of the Coleman-Teitelbaum cocycle that arises in the context of the \(\mathcal {L}\)-invariant.


Nous associons certaines applications p-adiques d’Abel-Jacobi aux familles analytiques de formes modulaires à ses poids nouveaux en p. Nous introduisons le concept de la dérivée d’une distribution. Utilisant ce concept, nous donnons une nouvelle perspective sur le cocycle de Coleman-Teitelbaum dans le contexte de l’invariant \(\mathcal {L}\).


  1. 1.
    Andreatta, F., Iovita, A., Stevens, G.: Overconvergent modular sheaves and modular forms for \(GL_{2/F}\), Israel J. Math. 201, 299–359Google Scholar
  2. 2.
    Andreatta, F., Iovita, A., Stevens, G.: Overconvergent Eichler-Shimura isomorphisms. To appear in J. Inst. Math. Jussieu, available on CJO2014. doi: 10.1017/S1474748013000364
  3. 3.
    Ash, A., Stevens, G.: \(p\)-adic deformation of arithmetic cohomology, draft dated 29.09.2008Google Scholar
  4. 4.
    Bertolini, M., Darmon, H.: Hida families and rational points on elliptic curves. Invent. Math. 168(2), 371–431 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bertolini, M., Darmon, H., Iovita, A.: Families of automorphic forms on definite quaternion algebras and Teitelbaum’s conjecture. Astérisque 331, 29–64 (2010)MathSciNetMATHGoogle Scholar
  6. 6.
    Bosch, S., Lutkebohmert, W., Raynaud, M.: Néron models, Springer-Verlag (1990)Google Scholar
  7. 7.
    Bourbaki, N.: Groupes et Algébres de Lie, Hermann, Paris (1972)Google Scholar
  8. 8.
    Brown, K.S.: Cohomology of groups, Springer (1980)Google Scholar
  9. 9.
    Buzzard, K.: On \(p\)-adic families of automorphic forms. In: Modular Curves and Abelian Varieties, Progr. Math., vol. 224, Birkhäuser Verlag, Basel, pp 23–44 (2004)Google Scholar
  10. 10.
    Coleman, R.F.: \(p\)-adic Banach spaces and families of modular forms. Invent. Math. 127(3), 417–479Google Scholar
  11. 11.
    Coleman, R.F., Mazur, B.: The eigencurve. In: Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, pp 1–113Google Scholar
  12. 12.
    Darmon, H.: Integration on \({{\cal H}_{p}\times {\cal H}}\) and arithmetic applications. Ann. of Math. 154(2), 589–639 (2001)Google Scholar
  13. 13.
    Darmon, H.: Rational points on modular elliptic curves, CBMS Regional Conference Series in Mathematics, 101. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (2004)Google Scholar
  14. 14.
    Dasgupta, S.: Gross-Stark units, Stark-Heegner points, and class fields of real quadratic fields, PhD thesisGoogle Scholar
  15. 15.
    Dasgupta, S.: Stark-Heegner points on modular Jacobians. Ann. Scient. Ec. Norm. Sup. 38, 427–469 (2005)MathSciNetMATHGoogle Scholar
  16. 16.
    Dasgupta, S., Greenberg, M.: \({\cal L}\)-invariants and Shimura curves, to appear in Algebra and Number TheoryGoogle Scholar
  17. 17.
    Dasgupta, S., Teitelbaum, J.: The \(p\)-adic upper half plane. In: Savitt D, Thakur, D (ed) \(p\)-adic Geometry, Lectures from the 2007 Arizona Winter School. University Lecture Series 45, Amer. Math. Soc., Providence, RI (2008)Google Scholar
  18. 18.
    Fresnel, J., van der Put, M.: Rigid analytic geometry and its applications, Birkhäuser Boston (2004)Google Scholar
  19. 19.
    Greenberg, M.: Stark-Heegner points and the cohomology of quaternionic Shimura varieties. Duke Math. J. 147(3), 541–575 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Greenberg, M., Seveso, M.A.: \(p\)-adic families of cohomological modular forms for indefinite quaternion algebras and the Jacquet-Langlands correspondence. Canad. J. Math (2015). doi: 10.4153/CJM-2015-062-x Google Scholar
  21. 21.
    Greenberg, M., Seveso, M.A., Shahabi, S.: Modular \(p\) -adic \(L\)-functions attached to real quadratic fields and arithmetic applications. J. Reine Angew. Math (2014). doi: 10.1515/crelle-2014-0088
  22. 22.
    Greenberg, R., Stevens, G.: \(p\)-adic \(L\)-functions and \(p\) -adic periods of modular forms. Invent. Math. 111, 407–447 (1993)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Hida, H.: Galois representations into \(GL_{\mathbf{2} }\left( \mathbb{Z}_{p}\left[\left[X\right] \right] \right) \) attached to ordinary cusp forms. Invent. Math. 85, 545–613 (1986)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hida, H.: Iwasawa modules attached to congruences of cusp forms. Ann. Scient. Ec. Norm. Sup. 19, 231–273 (1986)MathSciNetMATHGoogle Scholar
  25. 25.
    Longo, M., Vigni, S.: The rationality of quaternionic Darmon points over genus fields of real quadratic fields. IMRN 2014, 3632–3691 (2014)MathSciNetMATHGoogle Scholar
  26. 26.
    Longo, M., Rotger, V., Vigni, S.: On rigid analytic uniformizations of Jacobians of Shimura curves. Amer. J. Math. 135, 1197–1246 (2012)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Mazur, B., Tate, J., Teitelbaum, J.: On \(p\)-adic analogues of the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 84, 1–48 (1986)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Rotger, V., Seveso, M.A.: \(L\)-invariants and Darmon cycles attached to modular forms. J. Eur. Math. Soc. 14(6), 1955–1999 (2012)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Serre, J.P.: Formes modulaires et fonctions zeta \(p\) -adiques. In: Modular functions of one variable III, Lecture Notes in Mathematics 350, Springer, pp. 191–268Google Scholar
  30. 30.
    Serre, J.-P.: Local fields. Springer (1979)Google Scholar
  31. 31.
    Serre, J.-P.: Trees. Springer (1980)Google Scholar
  32. 32.
    Serre, J.P.: Lie algebras and Lie groups. Springer-Verlag (1964)Google Scholar
  33. 33.
    Seveso, M.A.: \(p\)-adic \(L\)-functions and the rationality of Darmon cycles. Canad. J. Math. 64(5), 1122–1181 (2012)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Seveso, M.A.: Heegner cycles and derivatives of p-adic L-functions. J. Reine Angew. Math. 686, 111–148 (2014)MathSciNetMATHGoogle Scholar
  35. 35.
    Seveso, M.A.: The Teitelbaum conjecture in the indefinite setting. Amer. J. Math. 135(6), 1525–1557 (2013)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Papikian, M.: Rigid-analytic geometry and the uniformization of abelian varieties. In: Vakil, R., (ed.) Snowbird lectures in algebraic geometry, Contemp. Math. 388, AMS Providence, pp. 145–160 (2005)Google Scholar
  37. 37.
    de Shalit, E.: Eichler cohomology and periods of modular forms on \(p\)-adic Schottky groups. J. Reine Angew. Math. 400, 3–31 (1989)MathSciNetMATHGoogle Scholar
  38. 38.
    Schneider, P.: Nonarchimedean Functional Analysis. Springer Monographs in Mathematics. Springer-Verlag, Berlin (2002)CrossRefGoogle Scholar
  39. 39.
    Schneider, P.: \(p\)-adic analysis and Lie groups, available at
  40. 40.
    Schneider, P., Teitelbaum, J.: Locally analytic distributions and \(p\)-adic representation theory, with applications to \(\mathbf{GL}_{2}\). J. Amer. Math. Soc. 15, 443–468 (2002)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Schneider, P., Teitelbaum, J.: Continuous and locally analytic representation theory, available at (2004)
  42. 42.
    Stevens, G.: Rigid analytic modular symbols. Preprint dated April 21, 1994Google Scholar
  43. 43.
    Stevens, G.: Coleman’s \({\cal L}\)-invariant and families of modular forms. Asterisque 331, 1–12 (2010)Google Scholar
  44. 44.
    Teitelbaum, J.: Values of \(p\)-adic \(L\) -functions and a \(p\)-adic Poisson kernel. Invent. Math. 101, 395–410 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Fondation Carl-Herz and Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada
  2. 2.Dipartimento di MatematicaUniversità di MilanoMilanItaly

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