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PUBLISHED BY mathematical sciences publishers http://msp.org/ A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2012 by Mathematical Sciences Publishers ALGEBRA AND NUMBER THEORY 6:3(2012) msp The image of complex conjugation in l-adic representations associated to automorphic forms Richard Taylor If F C is a totally real field, if n is an odd integer and if 5 is a regular, algebraic, essentially self-dual, cuspidal automorphic representation of GLn.AFC /, then we calculate the image of any complex conjugation under the l-adic representations rl,ι.5/ associated to 5. Introduction C ∼ Let F denote a totally real number field and fix an isomorphism ι V Ql −! C. It is known that to a regular, algebraic, essentially self-dual, cuspidal automorphic representation 5 of GLn.AFC / one can associate a continuous semisimple Galois representation C C rl,ι.5/ V Gal.F =F / ! GLn.Ql /: (For the definition of “regular, algebraic, essentially self-dual, cuspidal” see the start of Section 1.) This representation is known to be de Rham and its Hodge– Tate numbers are known. (They can be simply calculated from the infinitesimal C character of π1.) For all finite places v of F not dividing l one can calculate the Frobenius semisimplification of the restriction of rl,ι.5/ to a decomposition group above v in terms of πv via the local Langlands correspondence. This uniquely (in fact, over) determines rl,ι.5/. (See[Shin 2011; Clozel et al. 2011; Caraiani 2010; Chenevier and Harris 2011].) The representation rl,ι.5/ is conjectured to be irreducible. This is known if 5 is discrete series at some finite place[Tay- _ ∼ lor and Yoshida 2007]. Moreover rl,ι.5/ D rl,ι.5/ ⊗ µ for some character µ of Gal.FC=FC/ which is either totally odd (takes the value −1 on all complex conjugations) or totally even (takes the value C1 on all complex conjugations). Frank Calegari raised the question as to whether, for an infinite place v of FC C C one can calculate the conjugacy class of rl,ι.5/.cv/, where cv 2 Gal.F =F / is a The author is partially supported by NSF Grant DMS-0600716. MSC2000: 11F80. Keywords: Galois representations. 405 406 Richard Taylor complex conjugation for v. This conjugacy class has order two, so it is semisimple with eigenvalues ±1. The problem is to determine how many C1’s and how many −1’s occur. Because 5 was assumed to be regular, we expect that the number of C1’s and −1’s differ by at most one: jtr rl,ι.5/.cv/j ≤ 1: As we know the determinant of rl,ι.5/ this would completely determine the con- jugacy class of rl,ι.5/.cv/. If µ is totally odd then[Bellaïche and Chenevier 2011] shows that n is even and that rl,ι.5/ preserves an alternating pairing up to multiplier µ. In this case, because GSpn.Ql / has a unique conjugacy class of elements of order two and multiplier −1, we see that tr rl,ι.5/.cv/ D 0 for all vj1. So the problem lies in the case that µ is totally even, i.e., that rl,ι.5/ preserves an orthogonal pairing up to multiplier µ. In this paper we will prove this conjecture in the case n is odd: Proposition 1. Suppose that FC is a totally real field, that n is an odd positive integer and that 5 a regular, algebraic, essentially self-dual, cuspidal automorphic representation 5 of GLn.AFC /. Suppose also that rl,ι.5/ is irreducible. If c 2 Gal.FC=FC/ is a complex conjugation (for some embedding FC ,! C) then jtr rl,ι.5/.c/j ≤ 1: We believe that essentially the same method works if n is even and 5 is discrete series at a finite place, though we haven’t taken the trouble to write the argument down in this case. (One would work with the construction of rl,ι.5/ given in [Harris and Taylor 2001] rather than that given in[Shin 2011].) However we do not see how to treat the general case when n is even. When rl,ι.5/ is reducible one can calculate the trace of r.c/ for some representation of r of Gal.FC=FC/ with the same restriction to Gal.FC=F/, but this does not seem to be very helpful. The construction of rl,ι.5/ is via piecing together twists of representations of Gal.FC=F/ which arise in the cohomology of unitary group Shimura varieties, as F runs over certain imaginary CM fields. For none of these twisted restrictions does complex conjugation make sense. For an infinite place of F one can assign a natural sign to the representations of Gal.FC=F/ that arise in the cohomology of these Shimura varieties, because they are essentially conjugate self-dual. (See [Clozel et al. 2008] or[Bellaïche and Chenevier 2011].) As Calegari has stressed this sign is not related to the image of complex conjugation in our representation of Gal.FC=FC/. This latter image only makes sense for the Galois representations coming from certain automorphic forms on the unitary groups, namely those that arise from an automorphic form on GLn.AFC / by some functoriality. Complex conjugation in l-adic representations 407 In the case that n is odd the unitary groups employed by Shin[2011] have rank n and we are able to use the moduli theoretic interpretation of its Shimura variety to write descent data to the maximal totally real subfield of F. This descent data does not commute with the action of the finite adelic points of the unitary group. However in the special case of an automorphic representation π which arises by functoriality from an automorphic form on GLn over a totally real field we are able to show that, up to twist, this descent data preserves the π 1 isotypical component of the cohomology, and hence gives a geometric realization of rl,ι.5/.cv/. Be- cause of its geometric construction, rl,ι.5/.cv/ also makes sense in the world of variations of Hodge structures. Finally we can appeal to the fact that the Hodge p;q structure corresponding to rl,ι.5/ is regular (i.e., each h ≤ 1) to show that jtr rl,ι.5/.cv/j ≤ 1. In the case that n is even and 5 is not discrete series at any finite place,[Shin 2011] realizes twists of rl,ι.5/jGal.FC=F/ in the cohomology of the Shimura vari- eties for unitary groups of rank n C1. One takes the π 1 isotypic component of the cohomology for an unstable automorphic representation π of the unitary group, which one constructs from 5 using the theory of endoscopy. In this case our descent data relates the π 1 isotypic component of the cohomology, not to itself, but to a twist of the (π 0/1 isotypic component for a second unstable automorphic representation π 0 of the unitary group also arising from 5. (This π 0 is not even nearly equivalent to a twist of π.) This does not seem to be helpful.