Algebraic modular forms. (English) Zbl 0965.11020

Gross, Benedict H.

Isr. J. Math. 113, 61-93 (1999).

The author develops an algebraic theory of modular forms for connected reductive groups \(G\) over \(\mathbb Q\) with the property that every arithmetic subgroup \(\Gamma\) of \(G({\mathbb Q})\) is finite. This algebraic theory enables the author to define “modular forms mod \(p\)” for such groups. The success of such a theory in this case results from the fact that modular forms in this context are combinatorial objects, i.e., essentially functions on finite sets of points.
What makes the theory non-trivial and arithmetic is the Hecke action that these spaces of modular forms carry. The author makes conjectures about mod \(p\) Galois representations attached to mod \(p\) modular forms that are eigenforms for the Hecke action, even detailing the behaviour of such representations when restricted to the decomposition group at \(p\).
J.-P. Serre [Isr. J. Math. 95, 281-299 (1996; Zbl 0870.11030)] had earlier developed such a theory in the case when \(G\) is a definite quaternion algebra.

Reviewer: Chandrashekhar B.Khare (Mumbai)

MSC:

11F55	Other groups and their modular and automorphic forms (several variables)
11F80	Galois representations
20G30	Linear algebraic groups over global fields and their integers
22E55	Representations of Lie and linear algebraic groups over global fields and adèle rings

Keywords:

algebraic theory of modular forms; connected reductive groups; modular forms \(\text{mod }p\); functions on finite sets of points; Hecke action; \(\text{mod }p\) Galois representations

Citations:

Zbl 0870.11030

References:

[1]	Arthur, J., Lectures on automorphic L-functions, inL-Functions and Arithmetic, London Mathematical Society Lecture Notes, 153, 1-21 (1991)
[2]	Bourbaki, N., Groupes et algèbras de Lie (1982), Paris: Hermann, Paris · Zbl 0505.22006
[3]	Borel, A., Reduction theory for arithmetic groups, inAlgebraic Groups and Discontinuous Subgroups, Proceedings of Symposia in Pure Mathematics, 9, 20-25 (1966) · Zbl 0213.47201
[4]	Borel, A., Automorphic L-functions, inAutomorphic Forms, Representations, and L-Functions, Proceedings of Symposia in Pure Mathematics, 33, 27-61 (1979) · Zbl 0412.10017
[5]	Borel, A.; Harish-Chandra, Arithmetic subgroups of algebraic groups, Annals of Mathematics, 75, 485-535 (1962) · Zbl 0107.14804 · doi:10.2307/1970210
[6]	Borel, A.; Tits, J., Groupes réductifs, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 27, 55-150 (1965) · Zbl 0145.17402 · doi:10.1007/BF02684375
[7]	Bosch, S.; Lütkebohmert, W.; Raynaud, M., Néron models (1990), Berlin: Springer-Verlag, Berlin · Zbl 0705.14001
[8]	Cartier, P., Representations of p-adic groups, inAutomorphic Forms, Representations, and L-Functions, Proceedings of Symposia in Pure Mathematics, 33, 111-155 (1979) · Zbl 0421.22010
[9]	de Siebenthal, J., Sur certains sous-groupes de rang un des groupes de Lie clos, Comptes Rendus de l’Académie des Sciences, Paris, 230, 910-912 (1950) · Zbl 0036.15602
[10]	Gross, B., Heights and L-series, Number Theory CMS Conference Series, 7, 115-188 (1985)
[11]	Gross, B., Groups overZ, Inventiones Mathematicae, 124, 263-279 (1996) · Zbl 0846.20049 · doi:10.1007/s002220050053
[12]	Gross, B., On the motive of a reductive group, Inventiones Mathematicae, 130, 287-313 (1997) · Zbl 0904.11014 · doi:10.1007/s002220050186
[13]	Gross, B., On the motive of G and the principal homomorphism SL_2 →Ĝ, Asian Journal of Mathematics, 1, 208-213 (1997) · Zbl 0942.20031
[14]	B. Gross,On the Satake isomorphism, inGalois Representations in Arithmetic Algebraic Geometry, London Mathematical Society Lecture Notes, Vol. 254, 1998, pp. 223-237. · Zbl 0996.11038
[15]	Gross, B.; Savin, G., Motives with Galois group of type G_2:an exceptional theta correspondence, Compositio Mathematica, 114, 153-217 (1998) · Zbl 0931.11015 · doi:10.1023/A:1000456731715
[16]	Kottwitz, R., Stable trace formula: cuspidal tempered terms, Duke Mathematical Journal, 51, 611-650 (1984) · Zbl 0576.22020 · doi:10.1215/S0012-7094-84-05129-9
[17]	Mazur, B., Modular curves and the Eisenstein ideal, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 47, 33-186 (1977) · Zbl 0394.14008 · doi:10.1007/BF02684339
[18]	D. Mumford,Abelian varieties, Publications of the Tata Institute5, 1974.
[19]	Platonov, V.; Rapinchuk, A., Algebraic Groups and Number Theory (1944), New York: Academic Press, New York · Zbl 0806.11002
[20]	Serre, J.-P., Abelian ℓ-adic representations and elliptic curves (1968), New York: Benjamin, New York · Zbl 0186.25701
[21]	Serre, J.-P., Lettre de J-P. Serre à J. Tate, 7 août 1987, Israel Journal of Mathematics, 95, 282-291 (1996) · Zbl 0870.11030 · doi:10.1007/BF02761043
[22]	Serre, J.-P., Propriétés conjecturates des groupes de Galois motiviques et des représentations ℓ-adiques, inMotives, Proceedings of Symposia in Pure Mathematics, 55, 377-400 (1991) · Zbl 0812.14002
[23]	Siegel, C., Berechnung von Zetafunktionen als ganzzählige Stellen, Nachricthen der Akademie der Wissenschaften in Göttingen. II. Mathematisch-Physikalische Klasse, 10, 87-102 (1969) · Zbl 0186.08804
[24]	G. Shimura,Arithmetic Theory of Automorphic Functions, Princeton University Press, 1971. · Zbl 0221.10029
[25]	Steinberg, R., Representations of algebraic groups, Nagoya Mathematical Journal, 22, 33-56 (1963) · Zbl 0271.20019
[26]	Tate, J., A review of non-Archimedean elliptic functions, inElliptic Curves, Modular Forms, and Fermat’s Last Theorem, International Press Series in Number Theory, 1, 162-184 (1995) · Zbl 1071.11508
[27]	Tits, J., Reductive groups over local fields, inAutomorphic Forms, Representations, and L-Functions, Proceedings of Symposia in Pure Mathematics, 33, 29-69 (1979) · Zbl 0415.20035
[28]	Tits, J., Représentations linéares irréductibles d’un groupe reductif sur un corps quelconque, Journal für die reine und angewandte Mathematik, 247, 196-220 (1971) · Zbl 0227.20015 · doi:10.1515/crll.1971.247.196