Skip to main content
SpringerLink
Log in

On p-adic L-function associated with cusp forms on \(\mathrm{GL}_2\)

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We construct p-adic L-functions of nearly p-ordinary cusp forms on \(\mathrm{GL}_2\) over number fields. As a consequence, we prove a result on the non-vanishing of the central value of L-function of \(\mathrm{GL_2}\) with cyclotomic twists.

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

Similar content being viewed by others

References

  1. Amice, Y., Vélu, J.: Distributions \(p\)-adiques associeés aux series de Hecke. Astérisque 25–26, 119–131 (1975)

    MATH  Google Scholar 

  2. Borel, A., Wallach, N.: Continuous Cohomology, Discrete Subgroups and Representations of Reductive Groups. Ann. Math. Stud. 94, Princeton Univ. Press, Princeton (1980)

  3. Bredon, G.E.: Sheaf Theory, Graduate Texts in Mathematics, vol. 170, 2nd edn. Springer, Berlin (1997)

    Google Scholar 

  4. Casselman, W.: On some results of Atkin and Lehner. Math. Ann. 201, 301–314 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  5. Deligne, P.: Valeurs de fonctions L et périodes d’intégrales. In: Automorphic Forms, Representations, and \(L\)-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Proc. Symp. Pure Math. 33 (1979) Part II, Am. Math. Soc., Providence (1979), pp. 247–289

  6. Dimitrov, M.: Automorphic symbols, \(p\)-adic \(L\)-functions and ordinary cohomology of Hilbert modular varieties. Am. J. Math. 135(4), 1117–1155 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Haran, S.: \(p\)-adic \(L\)-function for modular forms. Compos. Math. 62(1), 31–46 (1987)

    MathSciNet  MATH  Google Scholar 

  8. Harder, G.: Eisenstein cohomology of arithmetic groups. The case \({\rm GL}(2)\). Invent. Math. 89(1), 37–118 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hida, H.: \(p\)-ordinary cohomology groups for \({\rm SL}(2)\) over number fields. Duke Math. J. 69(2), 259–314 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hida, H.: On the critical values of \(L\)-functions of \({\rm GL}(2)\) and \({\rm GL}(2) \times {\rm GL}(2)\). Duke Math. J. 74(2), 431–529 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hida, H.: Non-critical values of adjoint L-functions for \({\rm SL}(2)\). In: Automorphic Forms, Automorphic Representations, and Arithmetic (Forth Worth, 1996), Proceedings of Symposium Pure Mathematics 66 (1999) Part I, Am. Math. Soc., Providence, pp. 123–175 (1999)

  12. Jacquet, H., Langlands, R.: Automorphic forms on \({\rm GL}(2)\). Springer Lecture Notes 114, (1970)

  13. Jacquet, H., Shalika, J. A.: A non-vanishing theorem for zeta functions of \({\rm GL}_n\), Invent. Math. 38 (1976/77), no. 1, 1–16

  14. Katz, N.: \(p\)-adic \(L\)-functions for CM fields. Invent. Math. 49(3), 199–297 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kitagawa, K.: On standard \(p\)-adic \(L\)-functions of families of elliptic cusp forms. In: \(p\)-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA 1991), Contemp. Math. 165, Am. Math. Soc. Providence, RI, pp. 81–110 (1994)

  16. Kurcǎnov, P.F.: Local measures connected with Jacquet–Langlands cusp forms over fields of CM-type. Mat. Sb. (N.S.) 108 (150) (1979), no. 4, 483–503, 639. In Russian; translated in Math. USSR Sbornik, 36 (1980), no. 4, 449–467

  17. Manin, Y.I.: Non-Archimedean integration and \(p\)-adic Jacquet-Langlands \(L\)-functions, Uspekhi Mat. Nauk 31, 5–54 (1976). In Russian; translated in Russ. Math. Surv, 31, 5–57 (1976)

  18. Mazur, B., Swinnerton-Dyer, P.: On arithmetic of Weil curves. Invent. Math. 25, 1–61 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mazur, B., Tate, J., Teitelbaum, J.: On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84(1), 1–48 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Namikawa, K.: On a congruence prime criterion for cusp forms on \({\rm GL}_2\) over number fields. J. Reine Angew. Math. 707, 149–207 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Ochiai, T.: Several variables \(p\)-adic \(L\)-functions for Hida families of Hilbert modular forms. Doc. Math. 17, 807–849 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Rohrlich, D.E.: On \(L\)-functions of elliptic curves and cyclotomic towers. Invent. Math. 75(3), 409–423 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rohrlich, D.E.: \(L\)-functions and division towers. Math. Ann. 281(4), 611–632 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tate, J.: Number theoretic back ground. In: Automorphic Forms, Representations, and \(L\)-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Proc. Symp. Pure Math. 33 (1979) Part II, Am. Math. Soc., Providence, 1979, pp. 3–26

  25. Urban, E.: Formes automorphes cuspidales pour \({\rm GL}(2)\) sur un corps quadratique imaginare. Valeurs Spéciales de fonctions \(L\) et congruences. Compos. Math. 99(3), 283–324 (1995)

    MATH  Google Scholar 

  26. Višik, M.M.: Nonarchimedean measures connected with Dirichlet series, Mat. Sb. (N.S.) 99 (141) no. 2, 248–260 (1976), 296. In Russian; translated in Math. USSR Sbornik 28 no. 2, 216–228 (1976)

  27. Washington, L.C.: Introduction to Cyclotomic Fields Graduate Texts in Mathematics, vol. 83, 2nd edn. Springer, Berlin (1997)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Engineering, Tokyo Denki University, 5 Asahi-cho, Senju, Adachi City, Tokyo, 120-8551, Japan

    Kenichi Namikawa

Authors
  1. Kenichi Namikawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kenichi Namikawa.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Namikawa, K. On p-adic L-function associated with cusp forms on \(\mathrm{GL}_2\) . manuscripta math. 153, 563–622 (2017). https://doi.org/10.1007/s00229-016-0904-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-016-0904-5

Mathematics Subject Classification

Search

Navigation