Abstract
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.
References
Arakawa, T.: Vector valued Siegel’s modular forms of degree two and the associated Andrianov \(L\)-functions. Manuscr. Math. 44, 155–185 (1983)
Ash, A., Pollack, D.: Everywhere unramified automorphic cohomology for \(\text{ SL }(3,{\mathbb{Z}})\). Int. J. Number Theory 4, 663–675 (2008)
Bergström, J., Dummigan, N., Mégarbané, T.: Eisenstein congruences for \(\text{ SO }(4,3), \text{ SO }(4,4)\)-values, preprint (2015). http://neil-dummigan.staff.shef.ac.uk/papers.html
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)
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 100
Bloch, S., Kato, K.: \(L\)-functions and Tamagawa Numbers of Motives, The Grothendieck Festschrift Volume I. Progress in Mathematics, vol. 86. Birkhäuser, Boston (1990)
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)
Borel, A.: Automorphic \(L\)-functions. AMS Proc. Symp. Pure Math. 33(2), 27–61 (1979)
Borel, A., Jacquet, H.: Automorphic forms and automorphic representations. AMS Proc. Symp. Pure Math. 33(1), 189–202 (1979)
Brown, J.: Saito-Kurokawa lifts and applications to the Bloch–Kato conjecture. Compos. Math. 143, 290–322 (2007)
Buzzard, K., Gee, T.: The conjectural connections between automorphic representations and Galois representations. arXiv:1009.0785
Cartier, P.: Representations of \({\mathfrak{p}}\)-adic groups. AMS Proc. Symp. Pure Math. 33(1), 111–155 (1979)
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)
Cremona, J.E., Mazur, B.: Visualizing elements in the Shafarevich–Tate group. Exp. Math. 9, 13–28 (2000)
Danielsen, T.H.: The work of Harish-Chandra. http://www.math.ku.dk/~thd/Harish-Chandra
Deligne, P.: Valeurs de Fonctions \(L\) et Périodes d’Intégrales. AMS Proc. Symp. Pure Math. 33(2), 313–346 (1979)
Deligne, P.: Variétés de Shimura. AMS Proc. Symp. Pure Math. 33(2), 247–290 (1979)
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)
Diamond, F., Taylor, R.: Non-optimal levels of mod \(l\) modular representations. Invent. Math. 115, 435–462 (1994)
Dummigan, N.: Symmetric square \(L\)-functions and Shafarevich–Tate groups, II. Int. J. Number Theory 5, 1321–1345 (2009)
Dummigan, N.: A simple trace formula for algebraic modular forms. Exp. Math. 22(2), 123–131 (2013)
Dummigan, N.: Symmetric square \(L\)-functions and Shafarevich–Tate groups. Exp. Math. 10, 383–400 (2001)
Dummigan, N.: Eisenstein congruences for unitary groups, preprint (2014). http://neil-dummigan.staff.shef.ac.uk/papers.html
Dummigan, N., Fretwell, D.: Ramanujan-style congruences of local origin. J. Number Theory 143, 248–261 (2014)
Dummigan, N., Ibukiyama, T., Katsurada, H.: Some Siegel modular standard \(L\)-values, and Shafarevich–Tate groups. J. Number Theory 131, 1296–1330 (2011)
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)
Fontaine, J.-M.: Valeurs spéciales des fonctions \(L\) des motifs, Séminaire Bourbaki, vol. 1991/92. Astérisque 206, Exp. No. 751, 4, 205–249 (1992)
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)
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)
Gross, B.H., Savin, G.: Motives with Galois group of type \(G_2\): an exceptional theta-correspondence. Compos. Math. 114, 153–217 (1998)
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)
Harder, G.: Secondary operations in the cohomology of Harish-Chandra modules. http://www.math.uni-bonn.de/people/harder/Manuscripts/Eisenstein/SecOPs
Harder, G.: Eisensteinkohomologie und die Konstruktion gemischter Motive. Lecture Notes in Mathematics, vol. 1562. Springer, Berlin (1993)
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)
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. http://www.math.uni-bonn.de/people/harder/Manuscripts/Eisenstein/g=3
Harder, G.: Cohomology in the language of Adeles, Chapter III of book in preparation. http://www.math.uni-bonn.de/people/harder/Manuscripts/buch/chap3-2014
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)
Ikeda, T.: Pullback of lifting of elliptic cusp forms and Miyawakis conjecture. Duke Math. J. 131, 469–497 (2006)
Katsurada, H.: Congruence of Siegel modular forms and special values of their standard zeta functions. Math. Z. 259, 97–111 (2008)
Katsurada, H., Mizumoto, S.: Congruences for Hecke eigenvalues of Siegel modular forms. Abh. Math. Semin. Univ. Hambg. 82, 129–152 (2012)
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)
Klingen, H.: Zum Darstellungssatz fur Siegelsche Modulformen. Math. Z. 102, 30–43 (1967)
Knapp, A.W.: Lie Groups Beyond an Introduction, 2nd edn. Birkhäuser, Boston (2002)
Kurokawa, N.: Congruences between Siegel modular forms of degree \(2\). Proc. Jpn. Acad. 55A, 417–422 (1979)
Li, J.-S., Schwermer, J.: On the cuspidal cohomology of arithmetic groups. Am. J. Math. 131, 1431–1464 (2009)
Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math. IHES 47, 33–186 (1977)
Mazur, B., Wiles, A.: Class fields of abelian extensions of \({\mathbb{Q}}\). Invent. Math. 76, 179–330 (1984)
Miyawaki, I.: Numerical examples of Siegel cusp forms of degree 3 and their zeta functions. Mem. Fac. Sci. Kyushu Univ. 46, 307–339 (1992)
Mizumoto, S.: Congruences for eigenvalues of Hecke operators on Siegel modular forms of degree two. Math. Ann. 275, 149–161 (1986)
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, Japan
Petersen, D.: Cohomology of local systems on the moduli of principally polarized abelian surfaces. Pac. J. Math. 275, 39–61 (2015)
Ribet, K.: A modular construction of unramified \(p\)-extensions of \({\mathbb{Q}}(\mu _p)\). Invent. Math. 34, 151–162 (1976)
Ribet, K.: On modular representations of \(\text{ Gal }(\overline{\mathbb{Q}}/{\mathbb{Q}})\) arising from modular forms. Invent. Math. 100, 431–476 (1990)
Satoh, T.: On certain vector valued Siegel modular forms of degree two. Math. Ann. 274, 335–352 (1986)
Skinner, C., Urban, E.: The Iwasawa main conjectures for \(\text{ GL }_2\). Invent. Math. 195, 1–277 (2014)
Skoruppa, N.-P., Zagier, D.: Jacobi forms and a certain space of modular forms. Invent. Math. 94, 113–146 (1988)
Springer, T.A.: Reductive groups. AMS Proc. Symp. Pure Math. 33(1), 3–28 (1979)
Urban, E.: Selmer groups and the Eisenstein–Klingen ideal. Duke Math. J. 106, 485–525 (2001)
Urban, E.: Groupes de Selmer et Fonctions L p-adiques pour les representations modulaires adjointes, preprint (2006). http://www.math.columbia.edu/~urban/EURP.html
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)
Wallach, N.R.: Real Reductive Groups I. Academic Press, Edinburgh (1988)
Weissauer, R.: The trace of Hecke operators on the space of classical holomorphic Siegel modular forms of genus two, preprint (2009). arXiv:0909.1744
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)
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)
Yoo, H.: Non-optimal levels of a reducible mod \(\ell \) modular representation, preprint (2014). arXiv:1409.8342
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bergström, J., Dummigan, N. Eisenstein congruences for split reductive groups. Sel. Math. New Ser. 22, 1073–1115 (2016). https://doi.org/10.1007/s00029-015-0211-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-015-0211-0