Skip to main content
SpringerLink
Log in

Diagonal restrictions of p-adic Eisenstein families

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We compute the diagonal restriction of the first derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series attached to a general (odd) character of the narrow class group of a real quadratic field, and express the Fourier coefficients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of Darmon and Vonk (Duke Math J, to appear, 2020) at appropriate real quadratic points of Drinfeld’s p-adic upper half-plane. This can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier (J Reine Angew Math 355:191–220, 1985, §7) which arose in their “analytic proof” of the factorisation of differences of singular moduli, and whose inspiration can be traced to Siegel’s proof of the rationality of the values at negative integers of the Dedekind zeta function of a totally real field. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic fields based on RM values of rigid meromorphic cocycles, and leads to an expression for the p-adic logarithms of Gross–Stark units and Stark–Heegner points in terms of the first derivatives of certain twisted Rankin triple product p-adic L-functions.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Notes

  1. Its restriction to \(\hom (\mathbb {Z}_p^\times , 1+p\mathbb {Z}_p)\) is customarily denoted \(L_p(F, \psi \omega _p, s)\) in the literature.

  2. Note that this differs from the notion of reducedness defined by Gauß in his Disquisitiones Arithmeticae [15]. Reduced forms in his sense are always reduced in our sense, but the converse is not true.

  3. Note that this form may fail to be primitive.

References

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

    Article  MathSciNet  Google Scholar 

  2. Bertolini, M., Seveso, M., Venerucci, R.: Reciprocity laws for diagonal classes and rational points on elliptic curves (submitted)

  3. Buzzard, K., Calegari, F.: The 2-adic eigencurve is proper. Doc. Math. Extra Volume, pp. 211–232 (2006)

  4. Darmon, H.: Integration on \({\cal{H}}_p\times {\cal{H}}\) and arithmetic applications. Ann. Math. (2) 154(3), 589–639 (2001)

    Article  MathSciNet  Google Scholar 

  5. Darmon, H., Dasgupta, S.: Elliptic units for real quadratic fields. Ann. Math. (2) 163(1), 301–346 (2006)

    Article  MathSciNet  Google Scholar 

  6. Darmon, H., Lauder, A., Rotger, V.: Stark points and p-adic iterated integrals attached to modular forms of weight one. Forum of Mathematics, Pi (2015), Vol. 3, e8, 95 pages

  7. Darmon, H., Pollack, R.: Efficient calculation of Stark–Heegner points via overconvergent modular symbols. Isr. J. Math. 153, 319–354 (2006)

    Article  MathSciNet  Google Scholar 

  8. Darmon, H., Pozzi, A., Vonk, J.: The values of the Dedekind–Rademacher cocycle at real multiplication points (in progress)

  9. Darmon, H., Rotger, V.: Stark–Heegner points and generalised Kato classes (submitted)

  10. Darmon, H., Vonk, J.: Singular moduli for real quadratic fields: a rigid analytic approach. Duke Math. J. (2020) (to appear)

  11. Darmon, H., Vonk, J.: A real quadratic Borcherds lift (in progress)

  12. Dasgupta, S., Darmon, H., Pollack, R.: Hilbert modular forms and the Gross–Stark conjecture. Ann. Math. (2) 174(1), 439–484 (2011)

    Article  MathSciNet  Google Scholar 

  13. Dasgupta, S., Kakde, M.: Explicit Formulae for Brumer–Stark Units and Hilbert’s 12th Problem (in progress)

  14. Dasgupta, S., Kakde, M.: On constant terms of Eisenstein series (preprint)

  15. Gauss, C.F.: Disquisitiones Arithmeticae (1801)

  16. Goren, E., Kassaei, P.: Canonical subgroups over Hilbert modular varieties. J. Reine Angew. Math. 670, 1–63 (2012)

    Article  MathSciNet  Google Scholar 

  17. Gross, B., Zagier, D.: On singular moduli. J. Reine Angew. Math. 355, 191–220 (1985)

    MathSciNet  MATH  Google Scholar 

  18. Guitart, X., Masdeu, M.: Overconvergent cohomology and quaternionic Darmon points. J. Lond. Math. Soc. (2) 90(2), 495–524 (2014)

    Article  MathSciNet  Google Scholar 

  19. Kudla, S., Rapoport, M., Yang, T.: On the derivative of an Eisenstein series of weight one. Int. Math. Res. Not. 7, 347–385 (1999)

    Article  MathSciNet  Google Scholar 

  20. Lauder, A.: Computations with classical and p-adic modular forms. LMS J. Comput. Math. 14, 214–231 (2011)

    Article  MathSciNet  Google Scholar 

  21. Lauder, A., Vonk, J.: Computing p-adic L-functions for totally real fields (in preparation)

  22. Li, Y.: Restriction of coherent Hilbert Eisenstein series. Math. Ann. 368(1–2), 317–338 (2017)

    Article  MathSciNet  Google Scholar 

  23. Mazur, B.: On the arithmetic of special values of L functions. Invent. Math. 55(3), 207–240 (1979)

    Article  MathSciNet  Google Scholar 

  24. Mok, C.-P.: On a theorem of Bertolini–Darmon about rationality of Stark–Heegner points over genus fields of real quadratic fields (preprint)

  25. Schneider, P.: Rigid-analytic L-transforms. Number theory, Noordwijkerhout 1983, pp. 216–230, Lecture Notes in Mathematics, vol. 1068. Springer, Berlin (1984)

  26. Serre, J.-P.: Formes modulaires et fonctions zêta p-adiques. Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), pp. 191–268. Lecture Notes in Mathematics, vol. 350. Springer, Berlin (1973)

  27. Shih, S.-C.: On congruence modules related to Hilbert Eisenstein series. Math. Z. (2020) (to appear)

  28. Shimura, G.: Introduction to Arithmetic Theory of Automorphic Functions. Iwanami Shoten/Princeton University Press, Tokyo/Princeton (1971)

    MATH  Google Scholar 

  29. Teitelbaum, J.T.: Values of p-adic L-functions and a p-adic Poisson kernel. Invent. Math. 101(2), 395–410 (1990)

    Article  MathSciNet  Google Scholar 

  30. Zagier, D.: A Kronecker limit formula for real quadratic fields. Math Ann. 213, 153–184 (1975)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Marc Masdeu for numerically verifying the global nature of the point constructed in the final example of this paper. They also thank Francis Brown, Yingkun Li, David Loeffler, Don Zagier, and Sarah Zerbes for their support and for their interest in this work. The research of the Henri Darmon was supported by an NSERC discovery grant. Alice Pozzi was supported by the ERC-COG Grant 523950 ‘Euler Systems’. Jan Vonk was supported by ERC-COG Grant 724638 ‘GALOP’, the Carolyn and Franco Gianturco Fellowship at Linacre College (Oxford), the Max-Planck-Institut für Mathematik (Bonn), and NSF Grant No. DMS-1638352, during various stages of this project.

Author information

Authors and Affiliations

  1. McGill University, Montreal, Canada

    Henri Darmon

  2. Imperial College, London, UK

    Alice Pozzi

  3. Leiden University, Leiden, Netherlands

    Jan Vonk

Authors
  1. Henri Darmon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alice Pozzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jan Vonk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Henri Darmon.

Additional information

Communicated by Wei Zhang.

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

Darmon, H., Pozzi, A. & Vonk, J. Diagonal restrictions of p-adic Eisenstein families. Math. Ann. 379, 503–548 (2021). https://doi.org/10.1007/s00208-020-02086-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-020-02086-2

Mathematics Subject Classification

Search

Navigation