Abstract
In [10] (C R Acad Sci Paris Ser I Math 323(2) 117–120, 1996), [11] (Math Res Lett 10(1):71–83 2003), [12] (Can J Math 57(6):1215–1223 2005), Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare’s result for arbitrary global fields. In a sequel we shall apply this result to strictly compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Böckle, G.: An Eichler–Shimura isomorphism over function fields between Drinfeld modular forms and cohomology classes of crystals. preprint (2004). http://www.iwr.uni-heidelberg.del~Gebhard.Boeckle/preprints.html
Böckle, G.: Hecke characters and Drinfeld modular forms. Preprint (2008)
Corrales-Rodrigáñez C., Schoof R.: The support problem and its elliptic analogue. J. Number Theory 64(2), 276–290 (1997)
Deligne P., Serre J.-P.: Formes modulaires de poids 1. Ann. Sci. École Norm. Sup. (4) 7, 507–530 (1974)
Goss D.: The algebraist’s upper half-plane. Bull. Am. Math. Soc. (N.S.) 2(3), 391–415 (1980)
Goss, D.: L-series of Grössencharakters of type A 0 for function fields. In p-adic Methods in Number Theory and Algebraic Geometry, vol. 133. Contemp. Math., pp. 119–139. AMS (1992)
Gross, B.: Algebraic Hecke characters for function fields. Seminar on Number Theory Paris 1980–1981. Progr. Math. 22, 87–90
Hartshorne R.: Algebraic Geometry Graduate. Texts in Mathematics No. 52. Springer, New York (1977)
Henniart, G.: Représentations l-adiques abéliennes. Seminar on Number Theory, Paris 1980/1981, Progr. Math., 22, 107–126 (1982)
Khare C.: Compatible systems of mod p Galois representations. C. R. Acad. Sci. Paris Sér. I Math. 323(2), 117–120 (1996)
Khar. C.: Compatible systems of mod p Galois representations and Hecke characters. Math. Res. Lett. 10(1), 71–83 (2003)
Khare C.: Reciprocity law for compatible systems of abelian mod p Galois representations. Canad. J. Math. 57(6), 1215–1223 (2005)
MacRae E.R.: On the density of conjugate fixed points of a correspondence. Illinois J. Math. 18, 66–78 (1974)
Schinzel A.: On power residues and exponential congruences. Acta Arith. 27, 397–420 (1975)
Serre, J.-P.: Abelian l-adic representations and elliptic curves. Revised reprint, Research Notes in Mathematics vol. 7. A K Peters, Ltd., Wellesley (1998)
Waldschmidt, M.: Sur certains caractères du groupe des classes d’idèles d’un corps de nombres. Seminar on Number Theory, Paris 1980–1981, Progr. Math., 22, 323–335 (1982)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Böckle, G. Algebraic Hecke characters and strictly compatible systems of abelian mod p Galois representations over global fields. manuscripta math. 140, 303–331 (2013). https://doi.org/10.1007/s00229-012-0543-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-012-0543-4