Galois representations with conjectural connections to arithmetic cohomology. (English) Zbl 1023.11025

Ash, Avner; Doud, Darrin; Pollack, David

Duke Math. J. 112, No. 3, 521-579 (2002).

In [Duke Math. J. 54, 179-230 (1987; Zbl 0641.10026)], J.-P. Serre published a conjecture relating continuous \(2\)-dimensional absolutely irreducible Galois representations (in characteristic \(p\)) to the reductions modulo \(p\) of modular forms. In a joint paper [Duke Math. J. 105, 1-24 (2000; Zbl 1015.11018)], the first author and W. Sinnott stated another conjecture that relates niveau \(1\) Galois representations of arbitrary dimension \(n\) and cohomology groups of congruence subgroups of \(\text{GL}(n,\mathbb Z)\). In the case \(n = 2\), this conjecture of Ash and Sinnott is closely related to Serre’s conjecture. An expanded version of the Ash-Sinnott conjecture is given in the present paper that includes Galois representations of higher niveau. Computational evidence for the last conjecture in the case \(n = 3\) is presented (in the form of Galois representations for which the coefficients of the characteristic polynomials of the images of the Frobenius match the Hecke eigenvalues for the corresponding cohomology of congruence subgroups).
Using the results of A. Ash and Pham Huu Tiep [J. Algebra 222, 376-399 (1999; Zbl 0949.11032)], the authors also show that the conjecture holds for the symmetric squares of certain \(2\)-dimensional Galois representations. Some computational tests for the conjecture in the case \(n = 4\) have recently been done in A. Ash, P. E. Gunnells and M. McConnell [J. Number Theory 94, 181-212 (2002; Zbl 1006.11026)].

Reviewer: Pham Huu Tiep (Gainesville)

MSC:

11F75	Cohomology of arithmetic groups
11F80	Galois representations

Keywords:

cohomology of arithmetic groups; Galois representations

Citations:

Zbl 0641.10026; Zbl 1015.11018; Zbl 0949.11032; Zbl 1006.11026

Software:

PARI/GP

References:

[1]	G. Allison, A. Ash, and E. Conrad, Galois representations, Hecke operators, and the mod-\(p\) cohomology of \({\GL}(3,\mathbb{Z})\) with twisted coefficients, Experiment. Math. 7 (1998), 361–390. · Zbl 0923.11083 · doi:10.1080/10586458.1998.10504381
[2]	A. Ash, Galois representations attached to mod \(p\) cohomology of {\(\GL(n,\mathbb{Z})\) , Duke Math. J. 65 (1992), 235–255.} · Zbl 0774.11024 · doi:10.1215/S0012-7094-92-06510-0
[3]	A. Ash and M. McConnell, Experimental indications of three-dimensional Galois representations from the cohomology of {\({\mathrm SL}(3,\mathbb{Z})\) , Experiment. Math. 1 (1992), 209–223.} · Zbl 0780.11029 · doi:10.1080/10586458.1992.10504259
[4]	A. Ash and W. Sinnott, An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod \(p\) cohomology of {\(\GL(n,\mathbb{Z})\) , Duke Math. J. 105 (2000), 1–24.} · Zbl 1015.11018 · doi:10.1215/S0012-7094-00-10511-X
[5]	A. Ash and G. Stevens, Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues , J. Reine Angew. Math. 365 (1986), 192–220. · Zbl 0596.10026
[6]	–. –. –. –., Modular forms in characteristic \(\ell\) and special values of their \(L\) -functions, Duke Math. J. 53 (1986), 849–868. · Zbl 0618.10026 · doi:10.1215/S0012-7094-86-05346-9
[7]	A. Ash and P. H. Tiep, Modular representations of \(\gl(3,\mathbb{F}_p)\) , symmetric squares, and mod-\(p\) cohomology of {\(\gl(3,\mathbb{Z})\), J. Algebra 222 (1999), 376–399.} · Zbl 0949.11032 · doi:10.1006/jabr.1999.7988
[8]	S. Beckmann, On finding elements in inertia groups by reduction modulo \(p\) , J. Algebra 164 (1994), 415–429. · Zbl 0810.11067 · doi:10.1006/jabr.1994.1067
[9]	L. Clozel, “Motifs et formes automorphes: Applications du principe de fonctorialité” in Automorphic Forms, Shimura Varieties, and \(L\)-Functions (Ann Arbor, Mich., 1988) , Perspect. Math. 10 , Academic Press, Boston, 1990, 77–159. · Zbl 0705.11029
[10]	S. R. Doty and G. Walker, The composition factors of \({\mathbb F}_ p[x_ 1,x_ 2,x_ 3]\) as a \(\GL(3,p)\)-module, J. Algebra 147 (1992), 411–441. · Zbl 0789.20039 · doi:10.1016/0021-8693(92)90213-6
[11]	D. Doud, \(S_4\) and \(\widetilde S_4\) extensions of \(\mathbb{Q}\) ramified at only one prime, J. Number Theory 75 (1999), 185–197. · Zbl 0928.11049 · doi:10.1006/jnth.1998.2345
[12]	——–, “Three-dimensional Galois representations with conjectural connections to arithmetic cohomology” to appear in Number Theory for the Millennium (Urbana-Champaign, Ill., 2000) , ed. B. Berndt et al., Peters, Boston, 2002.
[13]	B. Edixhoven, “‘Serre’s conjecture” in Modular Forms and Fermat’s Last Theorem (Boston, 1995) , Springer, New York, 1997, 209–242. \CMP1 638 480 · Zbl 0918.11023
[14]	J.-M. Fontaine and G. Laffaile, Construction de représentations \(p\)-adiques , Ann. Sci. École Norm. Sup. (4) 15 (1982), 547–608. · Zbl 0579.14037
[15]	J. E. Humphreys, Linear Algebraic Groups , Grad. Texts in Math. 21 , Springer, New York, 1975.
[16]	G. D. James, The Representation Theory of the Symmetric Groups , Lecture Notes in Math. 682 , Springer, Berlin, 1978. · Zbl 0393.20009
[17]	S. Lang, Algebraic Number Theory , 2d ed., Grad. Texts in Math. 110 , Springer, New York, 1994. · Zbl 0811.11001
[18]	The PARI-Group, {PARI/GP, Version 2.1.0 , available from}
[19]	J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques , Invent. Math. 15 (1972), 259–331. · Zbl 0235.14012 · doi:10.1007/BF01405086
[20]	–. –. –. –., “Modular forms of weight one and Galois representations” in Algebraic Number Fields: \(L\)-Functions and Galois Properties (Durham, England, 1975) , Academic Press, London, 1977, 193–268. · Zbl 0366.10022
[21]	——–, Local Fields , Grad. Texts. in Math. 67 , Springer, New York, 1979.
[22]	–. –. –. –., Sur les représentations modulaires de degré \(2\) de {\(\gal(\overline{\mathbb{Q}}/\mathbb{Q})\) , Duke Math. J. 54 (1987), 179–230.} · Zbl 0641.10026 · doi:10.1215/S0012-7094-87-05413-5
[23]	G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions , Kanô Memorial Lectures 1 , Publ. Math. Soc. Japan 11 , Iwanami Shoten, Tokyo; Princeton Univ. Press, Princeton, 1971. · Zbl 0221.10029
[24]	J. Tunnell, Artin’s conjecture for representations of octahedral type , Bull. Amer. Math. Soc. (N.S.) 5 (1981), 173–175. · Zbl 0475.12016 · doi:10.1090/S0273-0979-1981-14936-3