Hi to everybody! I am studying right now the work of Serre and Deligne about the modularity of 2-dimensional complex Galois representations. I know that if $\rho \colon G_{\mathbb Q} \to GL_2(\mathbb C)$ is an irreducible Galois representation s.t. $\pi(\rho(G_{\mathbb Q}))$ is dihedral or isomorphic to $S_4$ or $A_4$ (where $\pi \colon GL_2(\mathbb C) \to PGL_2(\mathbb C)$ is the projection onto the quotient), then the Artin conjecture holds for $L(s,\rho)$, namely such function has an holomorphic continuation to the entire complex plane. So my question is: has the conjecture been proven (or disproven) in the case $\pi(\rho(G_{\mathbb Q})) \cong A_5$? And moreover, I read that it is expected a more general result to be true, something called "strong Artin conjecture" which deals with cuspidal representations. Where can I find something introductory to this topic?
2 Answers
The remaining $A_5$ case of the strong Artin conjecture was recently settled by Khare and Wintenberger, see Corollary 10.2 in their paper Serre's modularity conjecture (I), Invent. Math. 178 (2009), 485-504. I should add that the proof is very complicated and builds on the deep work of many others (I started to make a list here, but then stopped as it would be quite long and probably incomplete).
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7$\begingroup$ You need $\rho$ to be "odd", i.e. $\mathrm{det}\rho(c)=-1$ on $c$ complex conjugation, for Khare-Wintenberger to apply. The even $A_5$ case is wide open. $\endgroup$ Commented Mar 25, 2011 at 20:07
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$\begingroup$ You are absolutely right, thanks. Actually, I made a similar comment myself, on mathoverflow.net/questions/58901/… $\endgroup$ Commented Mar 25, 2011 at 23:20
Rogawski's article "Functoriality and the Artin Conjecture", available from his website, is a good introduction. It was published in 1997, but I think that Khare-Wintenberger is the only advance since then.
A bit more is known -- the Strong Artin Conjecture holds for $n$-dimensional representations of nilpotent Galois groups by Arthur-Clozel ("Simple algebras, base change, and the advanced theory of the trace formula").
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4$\begingroup$ I think it is worth mentioning Taylor's largely successful program, which has since been supplanted by Khare-Wintenberger. Taylor's idea was to view an odd icosahedral representation $\rho$ as a $2$-adic Galois representation, whose mod-$2$ image matches the projective image of $\rho$ using the coincidence $A_5 \simeq SL_2(F_4)$. Then you can try to prove that mod-$2$ representations are modular, and then that $2$-adic lifts thereof which are unramified at $2$ are modular and correspond to weight one modular forms (all with some technical hypothesis, of course).... $\endgroup$ Commented Mar 26, 2011 at 2:19
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4$\begingroup$ ... This was carried out in a series of papers: Shepherd-Barron and Taylor proved the mod-$2$ modularity statement by a variant of Wiles's $3-5$ switch, first switching to the prime $5$ and then to the prime $3$; Dickinson proved the (very difficult) $2$-adic modularity lifting theorem by careful application of Wiles's techniques; and Buzzard and Taylor proved that unramified-at-$l$ $l$-adic lifts of modular mod-$l$ things are modular of weight one by gluing together overconvergent eigenforms defined on overlapping subsets of a rigid-analytic model of some modular curve. A tour de force! $\endgroup$ Commented Mar 26, 2011 at 2:26