zbMATH — the first resource for mathematics

Functoriality and the inverse Galois problem. II: Groups of type \(B_n\) and \(G_2\). (English) Zbl 1194.11063
The authors continue their work from [Compos. Math. 144, No. 3, 541–564 (2008; Zbl 1194.11062)] and show similar results for groups of type \(B_n\) and \(G_2.\) In a final section they point out several minor errata for loc. cit. and give a slight reformulation of their main result.

11F80 Galois representations
12F12 Inverse Galois theory
11F12 Automorphic forms, one variable
11R32 Galois theory
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
Full Text: DOI Numdam EuDML arXiv
[1] Carter (R.).— Finite Groups of Lie Type, Wiley Classics Library, New York, 1993. · Zbl 0567.20023
[2] Clozel (L.).— Représentations galoisiennes associées aux représentations automorphes autoduales de \({\rm GL}(n)\). Inst. Hautes Études Sci. Publ. Math. No. 73, p. 97-145 (1991). · Zbl 0739.11020
[3] Cogdell (J.), Kim (H.), Piatetski-Shapiro (I.) and Shahidi (F.).— Functoriality for the classical groups. Publ. Math. Inst. Hautes Études Sci. No. 99, p. 163-233 (2004). · Zbl 1090.22010
[4] DeBacker (S.) and Reeder (M.).— Depth zero supercuspidal \(L\)-packets and their stability. Annals of Math. 169, No. 3, p. 795-901 (2009). · Zbl 1193.11111
[5] Gan (W. T.).— Exceptional Howe correspondences over finite fields. Compositio Math. 118, p. 323-344 (1999). · Zbl 0939.20010
[6] Gan (W. T.) and Savin (G.).— Real and global lifts from \({\text{PGL}}_3\) to \({\text{G}}_2\). Inter. Math. Res. Not. 50, p. 2699-2724 (2003). · Zbl 1037.22033
[7] Gan (W. T.) and Savin (G.).— Endoscopic lifts from \({\text{PGL}}_3\) to \({\text{G}}_2\). Compositio Math. 140, p. 793-808 (2004). · Zbl 1071.22007
[8] Ginzburg (D.), Rallis (S.) and Soudry (D.).— A tower of theta correspondences for \(G_2\). Duke Math. J. 88, p. 537-624 (1997). · Zbl 0881.11051
[9] Gross (B. H.) and Reeder (M.).— Arithmetic invariants of discrete Langlands parameters. In preparation. · Zbl 1207.11111
[10] Gross (B. H.) and Savin (G.).— Motives with Galois group of type \(G_2\): an exceptional theta correspondence. Compositio Math. 114, p. 153-217 (1998). · Zbl 0931.11015
[11] Harris (M.), Taylor (R.).— The geometry and cohomology of some simple Shimura varieties. Annals of Mathematics Studies, 151. Princeton University Press, Princeton, NJ, 2001. viii+276 pp. · Zbl 1036.11027
[12] Harris (M.).— Potential automorphy of odd-dimensional symmetric powers of elliptic curves, and applications. to appear in Algebra, Arithmetic, and Geometry: Manin Festschrift (Birkhuser, in press). · Zbl 1234.11068
[13] Huang (J. S.), Pandžić (P.) and Savin (G.).— New dual pair correspondences. Duke Math. J. 82, p. 447-471 (1996). · Zbl 0865.22009
[14] Humphreys (J. E.).— Linear Algebraic Groups. Graduate Texts in Mathematics, 21. Springer-Verlag, New York, 1975. · Zbl 0325.20039
[15] Humphreys (J. E.).— Conjugacy classes in semi-simple algebraic groups. Mathematical Surveys and Monographs, 43. American Mathematical Society, Providence, RI, 1995. · Zbl 0834.20048
[16] Jiang (D.), Soudry (D.).— The local converse theorem for \({\text{SO}}(2n+1)\) and applications. Ann. of Math. (2) 157 (2003), no. 3, 743-806. · Zbl 1049.11055
[17] Jiang (D.), Soudry (D.).— Lecture at the workshop on Automorphic Forms, Geometry and Arithmetic. Oberwolfach, February 2008. Announcement available at
[18] Khare (C.), Larsen (M.) and Savin (G.).— Functoriality and the inverse Galois problem. Compositio Math. 144 (2008), 541-564. · Zbl 1194.11062
[19] Kostrikin (A. I.) and Tiep (P. H.).— Orthogonal Decompositions and Integral Lattices, De Gruyter Expositions in Mathematics 15, Walter de Gruyter, Berlin - New York, 1994. · Zbl 0855.11033
[20] Khare (C.) and Wintenberger (J-P.).— Serre’s modularity conjecture (I), Invent Math. 178, p. 485-504 (2009). · Zbl 1304.11041
[21] Larsen (M.).— Maximality of Galois actions for compatible systems. Duke Math. J. 80, no. 3, p. 601-630 (1995). · Zbl 0912.11026
[22] Larsen (M.) and Pink (R.).— Finite subgroups of algebraic groups. preprint available at
[23] Magaard (K.) and Savin (G.).— Exceptional theta correspondences. Compositio Math. 107, p. 89-123 (1997). · Zbl 0878.22011
[24] Moy (A.).— The irreducible orthogonal and symplectic Galois representations of a \(p\)-adic field (the tame case). Journal of Number Theory 10, p. 341-344 (1984). · Zbl 0546.12009
[25] Muić (G.).— The unitary dual of \(p\)-adic \(G_2\). Duke Math. J. 90, p. 465-493 (1997). · Zbl 0896.22006
[26] Savin (G.).— \(K\)-types of minimal representations \((p\)-adic case). Glasnik Mat. Vol. 31(51), p. 93-99. · Zbl 0856.22020
[27] Savin (G.).— Lifting of generic depth zero representations of classical groups. J. of Algebra 319, p. 3244-3258 (2008). · Zbl 1147.22008
[28] Sug Woo Shin.— Galois representations arising from some compact Shimura varieties. Preprint, IAS, (2008). · Zbl 1269.11053
[29] Tadić (M.).— Representations of \(p\)-adic symplectic groups. Compositio Math. 90, p. 123-181 (1994). · Zbl 0797.22008
[30] Taylor (R.).— Galois representations. Ann. Fac. Sci. Toulouse Math. 13, p. 73-119 (2004). · Zbl 1074.11030
[31] Wiese (G.).— On projective linear groups over finite fields as Galois groups over the rational numbers. Modular Forms on Schiermonnikoog edited by Bas Edixhoven, Gerard van der Geer and Ben Moonen. Cambridge University Press, p. 343-350 (2008). · Zbl 1217.12004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.