Selecta Mathematica

, Volume 22, Issue 3, pp 1073–1115 | Cite as

Eisenstein congruences for split reductive groups

  • Jonas Bergström
  • Neil Dummigan


We present a general conjecture on congruences between Hecke eigenvalues of parabolically induced and cuspidal automorphic representations of split reductive groups, modulo divisors of critical values of certain L-functions. We examine the consequences in several special cases and use the Bloch–Kato conjecture to further motivate a belief in the congruences.


Congruences of modular forms Harder’s conjecture Bloch–Kato conjecture 

Mathematics Subject Classification

11F33 11F46 11F67 11F75 


  1. 1.
    Arakawa, T.: Vector valued Siegel’s modular forms of degree two and the associated Andrianov \(L\)-functions. Manuscr. Math. 44, 155–185 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ash, A., Pollack, D.: Everywhere unramified automorphic cohomology for \(\text{ SL }(3,{\mathbb{Z}})\). Int. J. Number Theory 4, 663–675 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bergström, J., Dummigan, N., Mégarbané, T.: Eisenstein congruences for \(\text{ SO }(4,3), \text{ SO }(4,4)\)-values, preprint (2015).
  4. 4.
    Bergström, J., Faber, C., van der Geer, G.: Siegel modular forms of degree three and the cohomology of local systems. Sel. Math. (N.S.) 20, 83–124 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bergström, J., Faber, C., van der Geer, G.: Siegel modular forms of genus 2 and level 2: cohomological computations and conjectures. Int. Math. Res. Not. (2008). Art. ID rnn 100Google Scholar
  6. 6.
    Bloch, S., Kato, K.: \(L\)-functions and Tamagawa Numbers of Motives, The Grothendieck Festschrift Volume I. Progress in Mathematics, vol. 86. Birkhäuser, Boston (1990)zbMATHGoogle Scholar
  7. 7.
    Böcherer, S., Satoh, T., Yamazaki, T.: On the pullback of a differential operator and its application to vector valued Eisenstein series. Comment. Math. Univ. St. Paul 42, 1–22 (1992)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Borel, A.: Automorphic \(L\)-functions. AMS Proc. Symp. Pure Math. 33(2), 27–61 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Borel, A., Jacquet, H.: Automorphic forms and automorphic representations. AMS Proc. Symp. Pure Math. 33(1), 189–202 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brown, J.: Saito-Kurokawa lifts and applications to the Bloch–Kato conjecture. Compos. Math. 143, 290–322 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Buzzard, K., Gee, T.: The conjectural connections between automorphic representations and Galois representations. arXiv:1009.0785
  12. 12.
    Cartier, P.: Representations of \({\mathfrak{p}}\)-adic groups. AMS Proc. Symp. Pure Math. 33(1), 111–155 (1979)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Clozel, L.: Motifs et formes automorphes: applications du principe de functorialité. In: Clozel, L., Milne, J.S. (eds.) Automorphic Forms, Shimura Varieties and L-functions, vol. I, pp. 77–159. Academic Press, London (1990)Google Scholar
  14. 14.
    Cremona, J.E., Mazur, B.: Visualizing elements in the Shafarevich–Tate group. Exp. Math. 9, 13–28 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Danielsen, T.H.: The work of Harish-Chandra.
  16. 16.
    Deligne, P.: Valeurs de Fonctions \(L\) et Périodes d’Intégrales. AMS Proc. Symp. Pure Math. 33(2), 313–346 (1979)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Deligne, P.: Variétés de Shimura. AMS Proc. Symp. Pure Math. 33(2), 247–290 (1979)CrossRefzbMATHGoogle Scholar
  18. 18.
    Diamond, F., Flach, M., Guo, L.: The Tamagawa number conjecture of adjoint motives of modular forms. Ann. Sci. École Norm. Sup. (4) 37, 663–727 (2004)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Diamond, F., Taylor, R.: Non-optimal levels of mod \(l\) modular representations. Invent. Math. 115, 435–462 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dummigan, N.: Symmetric square \(L\)-functions and Shafarevich–Tate groups, II. Int. J. Number Theory 5, 1321–1345 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dummigan, N.: A simple trace formula for algebraic modular forms. Exp. Math. 22(2), 123–131 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dummigan, N.: Symmetric square \(L\)-functions and Shafarevich–Tate groups. Exp. Math. 10, 383–400 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Dummigan, N.: Eisenstein congruences for unitary groups, preprint (2014).
  24. 24.
    Dummigan, N., Fretwell, D.: Ramanujan-style congruences of local origin. J. Number Theory 143, 248–261 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Dummigan, N., Ibukiyama, T., Katsurada, H.: Some Siegel modular standard \(L\)-values, and Shafarevich–Tate groups. J. Number Theory 131, 1296–1330 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Faber, C., van der Geer, G.: Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes, I, II. C. R. Math. Acad. Sci. Paris 338, 381–384 and 467–470 (2004)Google Scholar
  27. 27.
    Fontaine, J.-M.: Valeurs spéciales des fonctions \(L\) des motifs, Séminaire Bourbaki, vol. 1991/92. Astérisque 206, Exp. No. 751, 4, 205–249 (1992)Google Scholar
  28. 28.
    Gritsenko, V.: Arithmetical lifting and its applications. In: David, S. (ed.) Number theory (Paris, 1992–1993), London Mathematical Society Lecture Note Series, vol. 215, pp. 103–126. Cambridge University Press, Cambridge (1995)Google Scholar
  29. 29.
    Gross, B.H.: On the Satake transform. In: Scholl, A.J., Taylor, R.L. (eds.) Galois Representations in Arithmetic Algebraic Geometry, London Mathematical Society Lecture Note Series, vol. 254, pp. 223–237. Cambridge University Press, Cambridge (1998)Google Scholar
  30. 30.
    Gross, B.H., Savin, G.: Motives with Galois group of type \(G_2\): an exceptional theta-correspondence. Compos. Math. 114, 153–217 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Harder, G.: A congruence between a Siegel and an elliptic modular form. In: Ranestad, K. (ed.) The 1-2-3 of Modular Forms, pp. 247–262. Springer, Berlin (2008)CrossRefGoogle Scholar
  32. 32.
    Harder, G.: Secondary operations in the cohomology of Harish-Chandra modules.
  33. 33.
    Harder, G.: Eisensteinkohomologie und die Konstruktion gemischter Motive. Lecture Notes in Mathematics, vol. 1562. Springer, Berlin (1993)Google Scholar
  34. 34.
    Harder, G.: Arithmetic aspects of rank one Eisenstein cohomology. In: Srinivas, V. (ed.) Cycles, Motives and Shimura Varieties. Tata Institute of Fundamental Research Studies in Mathematics, pp. 131–190. Tata Institute of Fundamental Research, Mumbai (2010)Google Scholar
  35. 35.
    Harder, G.: A short note which owes its existence to some discussions with J. Bergström, C. Faber, G. van der Geer, A. Mellit and J. Schwermer.
  36. 36.
    Harder, G.: Cohomology in the language of Adeles, Chapter III of book in preparation.
  37. 37.
    Ibukiyama, T., Katsurada, H., Poor, C., Yuen, D.: Congruences to Ikeda–Miyawaki lifts and triple \(L\)-values of elliptic modular forms. J. Number Theory 134, 142–180 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Ikeda, T.: Pullback of lifting of elliptic cusp forms and Miyawakis conjecture. Duke Math. J. 131, 469–497 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Katsurada, H.: Congruence of Siegel modular forms and special values of their standard zeta functions. Math. Z. 259, 97–111 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Katsurada, H., Mizumoto, S.: Congruences for Hecke eigenvalues of Siegel modular forms. Abh. Math. Semin. Univ. Hambg. 82, 129–152 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Kim, H.H.: Automorphic \(L\)-functions. In: Cogdell, J.W., Kim, H.H., Murty, M.R. (eds.) Lectures on Automorphic L-functions. Fields Institute Monographs, vol. 20, pp. 97–201. American Mathematical Society, Providence (2004)Google Scholar
  42. 42.
    Klingen, H.: Zum Darstellungssatz fur Siegelsche Modulformen. Math. Z. 102, 30–43 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Knapp, A.W.: Lie Groups Beyond an Introduction, 2nd edn. Birkhäuser, Boston (2002)zbMATHGoogle Scholar
  44. 44.
    Kurokawa, N.: Congruences between Siegel modular forms of degree \(2\). Proc. Jpn. Acad. 55A, 417–422 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Li, J.-S., Schwermer, J.: On the cuspidal cohomology of arithmetic groups. Am. J. Math. 131, 1431–1464 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math. IHES 47, 33–186 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Mazur, B., Wiles, A.: Class fields of abelian extensions of \({\mathbb{Q}}\). Invent. Math. 76, 179–330 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Miyawaki, I.: Numerical examples of Siegel cusp forms of degree 3 and their zeta functions. Mem. Fac. Sci. Kyushu Univ. 46, 307–339 (1992)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Mizumoto, S.: Congruences for eigenvalues of Hecke operators on Siegel modular forms of degree two. Math. Ann. 275, 149–161 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Moriyama, T.: Representations of \(\text{ GSp }(4,{{\mathbb{R}}})\) with emphasis on discrete series. In: Furusawa, M. (ed.) Automorphic Forms on GSp(4), Proceedings of the 9th Autumn Workshop on Number Theory, pp. 199–209, 6–10 Nov 2006, Hakuba, JapanGoogle Scholar
  51. 51.
    Petersen, D.: Cohomology of local systems on the moduli of principally polarized abelian surfaces. Pac. J. Math. 275, 39–61 (2015)Google Scholar
  52. 52.
    Ribet, K.: A modular construction of unramified \(p\)-extensions of \({\mathbb{Q}}(\mu _p)\). Invent. Math. 34, 151–162 (1976)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Ribet, K.: On modular representations of \(\text{ Gal }(\overline{\mathbb{Q}}/{\mathbb{Q}})\) arising from modular forms. Invent. Math. 100, 431–476 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Satoh, T.: On certain vector valued Siegel modular forms of degree two. Math. Ann. 274, 335–352 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Skinner, C., Urban, E.: The Iwasawa main conjectures for \(\text{ GL }_2\). Invent. Math. 195, 1–277 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Skoruppa, N.-P., Zagier, D.: Jacobi forms and a certain space of modular forms. Invent. Math. 94, 113–146 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Springer, T.A.: Reductive groups. AMS Proc. Symp. Pure Math. 33(1), 3–28 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Urban, E.: Selmer groups and the Eisenstein–Klingen ideal. Duke Math. J. 106, 485–525 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Urban, E.: Groupes de Selmer et Fonctions L p-adiques pour les representations modulaires adjointes, preprint (2006).
  60. 60.
    van der Geer, G.: Siegel modular forms and their applications. In: Ranestad, K. (ed.) The 1-2-3 of Modular Forms. Springer, Berlin, pp. 181–245 (2008)Google Scholar
  61. 61.
    Wallach, N.R.: Real Reductive Groups I. Academic Press, Edinburgh (1988)zbMATHGoogle Scholar
  62. 62.
    Weissauer, R.: The trace of Hecke operators on the space of classical holomorphic Siegel modular forms of genus two, preprint (2009). arXiv:0909.1744
  63. 63.
    Weissauer, R.: Existence of Whittaker models related to four dimensional symplectic Galois representations. In: Edixhoven, B., van der Geer, G., Moonen, B. (eds.) Modular Forms on Schiermonnikoog, pp. 285–310. Cambridge University Press, Cambridge (2008)Google Scholar
  64. 64.
    Weissauer, R.: Four dimensional Galois representations. In: Tilouine, J., Carayol, H., Harris, M., Vigneras, M.-F. (eds.) Formes automorphes. II. Le cas du groupe GSp(4). Astérisque 302, 67–150 (2005)Google Scholar
  65. 65.
    Yoo, H.: Non-optimal levels of a reducible mod \(\ell \) modular representation, preprint (2014). arXiv:1409.8342

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© Springer International Publishing 2015

Authors and Affiliations

  1. 1.Matematiska institutionenStockholms universitetStockholmSweden
  2. 2.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK

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