Skip to main content
Log in

\(\Lambda \)-adic families of Jacobi forms

  • Research
  • Published:
Research in Number Theory Aims and scope Submit manuscript


We show that Hida’s families of p-adic elliptic modular forms generalize to p-adic families of Jacobi forms. We also construct p-adic versions of theta lifts from elliptic modular forms to Jacobi forms. Our results extend to Jacobi forms previous works by Hida and Stevens on the related case of half-integral weight modular forms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Bertolini, M., Darmon, H.: The rationality of Stark–Heegner points over genus fields of real quadratic fieldseegner points over genus fields of real quadratic fields. Ann. Math. 170, 343–369 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Boylan, H.: Jacobi Forms, Finite Quadratic Modules and Weil Representations over Number Fields. Lecture Notes in Mathematics, vol. 2130. Springer, Cham (2015)

  3. Candelori, L.: The algebraic functional equation of Riemann’s theta function. Ann. Inst. Fourier. Preprint 2016. arXiv:1512.04415

  4. Candelori, L.: The Transformation Laws of Algebraic Theta Functions. Preprint. arXiv:1609.04486

  5. Candelori, L.: Metaplectic stacks and vector-valued modular forms of half-integral weight. PhD Thesis (2014)

  6. Darmon, H.: Integration on \({\cal{H}}_p\times {\cal{H}}\) and arithmetic applications. Ann. Math. 154, 589–639 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques, pp. 143–316. Lecture Notes in Mathematics, vol. 349. Springer, Berlin

  8. Darmon, H., Tornarìa, G.: Stark–Heegner points and the shimura correspondence. Compos. Math. 144(5), 1155–1175 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Progress in Mathematics, vol. 55. Birkhäuser, Boston (1985)

  10. Gross, B., Kohnen, W., Zagier, D.: Heegner points and derivatives of \(L\)-series. II. Math. Ann. 278(1–4), 497–562 (1987)

    MathSciNet  MATH  Google Scholar 

  11. Greenberg, R., Stevens, G.: \(p\)-adic \(L\)-functions and \(p\)-adic periods of modular forms. Invent. Math. 111(2), 407–447 (1993)

    MathSciNet  MATH  Google Scholar 

  12. Guerzhoy, P.I.: An approach to the \(p\)-adic theory of Jacobi forms. Int. Math. Res. Not. 1, 31–39 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Guerzhoy, P.I.: Jacobi–Eisenstein series and \(p\)-adic interpolation of symmetric squares of cusp forms. Ann. Inst. Fourier (Grenoble) 45(3), 605–624 (1995)

    MathSciNet  MATH  Google Scholar 

  14. Guerzhoy, P.: On \(p\)-adic families of Siegel cusp forms in the MaaßSpezialschar. J. Reine Angew. Math. 523, 103–112 (2000)

    MathSciNet  MATH  Google Scholar 

  15. Guerzhoy, P.I.: Jacobi forms and a \(p\)-adic \(L\)-function in two variables. Fundam. Prikl. Mat. 6(4), 1007–1021 (2000)

    MathSciNet  Google Scholar 

  16. Hida, H.: Galois representations into \({\rm GL}_2({ Z}_p[[X]])\) attached to ordinary cusp forms. Invent. Math. 85(3), 545–613 (1986)

    MathSciNet  MATH  Google Scholar 

  17. Hida, H.: Iwasawa modules attached to congruences of cusp forms. Ann. Sci. École Norm. Sup. (4) 19(2), 231–273 (1986)

    MathSciNet  MATH  Google Scholar 

  18. Hida, H.: On \(p\)-adic Hecke algebras for \({\rm GL}_2\) over totally real fields. Ann. Math. (2) 128(2), 295–384 (1988)

    MathSciNet  MATH  Google Scholar 

  19. Hida, H.: Elementary Theory of \(L\)-Functions and Eisenstein Series. London Mathematical Society Student Texts, vol. 26. Cambridge University Press, Cambridge (1993)

  20. Hida, H.: On \(\Lambda \)-adic forms of half integral weight for \({\text{SL}}(2)\) /Q, Number theory (Paris, 1992–1993). London Math. Soc. Lecture Note Series, vol. 215, pp. 139–166. Cambridge University Press, Cambridge (1995)

  21. Howard, B.: Variation of Heegner points in Hida families. Invent. Math. 167(1), 91–128 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Ibukiyama, T.: Saito–Kurokawa liftings of level \(N\) and practical construction of Jacobi forms. Kyoto J. Math. 52(1), 141–178 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Kramer, J.: A geometrical approach to the theory of Jacobi forms. Compos. Math. 79(1), 1–19 (1991)

    MathSciNet  MATH  Google Scholar 

  24. Kramer, J.: An arithmetic theory of Jacobi forms in higher dimensions. J. Reine Angew. Math. 458, 157–182 (1995)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  26. Longo, M., Nicole, M.-H.: The \(\Lambda \)-adic Shimura–Shintani–Waldspurger correspondence. Doc. Math. 18, 1–21 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Longo, M., Nicole, M.-H.: On the \(p\)-adic variation of the Gross–Kohnen—Zagier Theorem. Forum Math. 31(4), 1069–1084 (2019)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  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)

    MathSciNet  MATH  Google Scholar 

  30. Longo, M., Martin, K., Yan, H.: Rationality of darmon points over genus fields of non-maximal orders. Ann. Math. Québec 44(1), 173–195 (2020)

    MathSciNet  MATH  Google Scholar 

  31. Moret-Bailly, L.: Sur l’équation fonctionnelle de la fonction thêta de Riemann. Compos. Math. 75(2), 203–217 (1990)

    MathSciNet  MATH  Google Scholar 

  32. Manickam, M., Ramakrishnan, B., Vasudevan, T.C.: On Saito–Kurokawa descent for congruence subgroups. Manuscr. Math. 81(1–2), 161–182 (1993)

    MathSciNet  MATH  Google Scholar 

  33. Manickam, M., Ramakrishnan, B.: On Shimura, Shintani and Eichler-Zagier correspondences. Trans. Am. Math. Soc. 352(6), 2601–2617 (2000)

    MathSciNet  MATH  Google Scholar 

  34. Nekovář, J.: Selmer complexes. Astérisque 310, 1–559 (2006)

    MathSciNet  MATH  Google Scholar 

  35. Ramsey, N.: Geometric and \(p\)-adic modular forms of half-integral weight. Ann. Inst. Fourier (Grenoble) 56(3), 599–624 (2006)

    MathSciNet  MATH  Google Scholar 

  36. Stevens, G.: \(\Lambda \)-adic modular forms of half-integral weight and a \(\Lambda \)-adic Shintani lifting. In: Arithmetic Geometry (Tempe, AZ, 1993), Contemporary Mathematics, vol. 174, pp. 129–151. American Mathematical Society, Providence (1994)

Download references


Part of this work was done during visits of M.-H.N. at the Mathematics Department of the University of Padova whose congenial hospitality he is grateful for, and also during a visit of M.L. in Montréal supported by the grant of the CRM-Simons professorship held by M.-H.N. in 2017-2018 at the Centre de recherches mathématiques (C.R.M., Montréal).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Matteo Longo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Longo, M., Nicole, MH. \(\Lambda \)-adic families of Jacobi forms. Res. number theory 6, 21 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: