On icosahedral Artin representations

By Kevin Buzzard, Mark Dickinson, N. Shepherd-barron, Richard Taylor. 2001

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If ρ : Gal(Qac/Q) → GL2(C) is a continuous odd irreducible representation with nonsolvable image, then under certain local hypotheses we prove that ρ is the representation associated to a weight 1 modular form and hence that the L-function of ρ has an analytic continuation to the entire complex plane. Introduction E. Artin [A] conjectured that theL-seriesL(r, s) of any continuous representation r : Gal(Qac/Q) −→ GLn(C) is entire except possibly for a pole at s = 1 whenr contains the trivial representation. The case when n = 1 is simply a restatement of the Kronecker-Weber theorem and standard results on the analytic continuation of Dirichlet L-series. Artin proved his conjecture when r is induced from a 1-dimensional representation of an open subgroup of Gal (Qac/Q). Moreover, R. Brauer [ Br] was able to show in general that L(r, s) is meromorphic on the whole complex plane. Since then, the only real progress has been for n = 2, although very recently D. Ramakrishnan [Ra] has dealt with somen = 4 cases. Whenn = 2, such representations can be classified according to the image of the projectivised representation proj : Gal(Qac/Q) −→ PGL2(C). This image is either cyclic, dihedral, the alternating group A4 (the tetrahedral case), the symmetric group S4 (the octahedral case), or the alternating group A5 (the icosahedral case). When the image of projr is cyclic, thenr is reducible and Artin’s conjecture follows from the n = 1 case. When the image of proj is dihedral, thenr is induced from a character of an open subgroup of index 2, and so Artin himself proved the conjecture in this case, although the result is implicit in earlier work of E. Hecke [ H ]. R. Langlands [Langl] proved Artin’s conjecture for tetrahedral and some octahedral representations. J. Tunnell [Tu] extended this to all octahedral representations. These results are based on Langlands’s theory of cyclic base change for automorphic representations of GL 2, DUKE MATHEMATICAL JOURNAL Vol. 109, No. 2, c ©2001 Received 29 December 1999. Revision received 13 October 2000. 2000Mathematics Subject Classification . Primary 11F11, 11F80; Secondary 11F33, 11G18, 14G22. Taylor’s work partially supported by National Science Foundation grant number DMS-9702885 and by the Miller Institute at the University of California at Berkeley.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/62cafea296f91b74665b4db6ce3c3667a91b50da

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Citations in the corpus

Modularity of 2-adic Barsotti-Tate representations (2009, result): “We remark that Artin’s conjecture for odd, two dimensional representations was previously known in many cases, thanks to the work of Langlands [19], Tunnel [27], and Buzzard–Dickinson–Shepherd-Barron–Taylor [ 7 ].”
Serre’s modularity conjecture (II) (2009, background): “2 of [51], we deduce that there is a number field F such that”
Nearly ordinary Galois deformations over arbitrary number fields (2007, background): “ed ﬁeld of ρ|K. Then the representation ρ|K does not deform to a one-parameter family of nearly ordinary Galois representations arising from classical automorphic forms over K. 50 Note that the proof [6] of the Artin conjecture for certain odd representations of Gal(Q/Q) requires the following idea:”
p-adic families of Siegel modular cuspforms (2012, background): “ght two Mazur-Tate-Teitelbaum conjecture by R. Greenberg and G. Stevens in [GS] and in the proof of certain cases of the Artin conjecture by K. Buzzard, M. Dickinson, N. Shepherd-Barron, R. Taylor in [BDS-BT]. An important turn in its history was marked by R. Coleman’s construction of ﬁnite slope p-”