60 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMSSL(2, Z), n = @n v is the associated representation of GL(2, A), the corresponding
Hecke eigenvalue ap is , with the notation of§ 2.2:
ap = p^112 (t1 + t2)·
According to Arthur we have either lti l = 1 or lt1I = p^1 /^2 , lt2I = p-^1 /^2 and
lapl = p + 1 (impossible for cusp forms by Hecke's estimate).
We note that in Arthur's paper [4b] this conjecture is stated for Ac,dis rather
than Ac (see loc. cit, § 8). This does not matter because Langlands's theory implies
that the orthogonal complement of Ac,dis in Ac is composed of subspaces induced
from parabolic subgroups; if P = MN C G is such a group, M imbeds in G [Sb]; if
the conjecture is true for AM,dis it will be true for the corresponding part of Ac.
(In the next lecture it will be important to be able to work with the whole of Ac).
We would like to state a watered-down - but still as deep -version of Conjec-
ture 2. Let t E T be the parameter associated to n P. Recall that T ~ ( <C x Y. This
isomorphism if of course not canonical, but any other is given by an automorphism
of the lattice zr of characters of f , i.e., is of the form
(z1,... ) Zr) f--+ (Z1, ... ' Zr ) ' z i =II z;ij ' (aij) E GL(r, Z).
j
Conjecture 3. - If 1rp is a local unramified component of an automorphic n, 1rpis holobaric, i.e. for any isomorphism T ~ (<exy
tw = (ti,... , t,.) with ltil = p~ , Wi E Z.
By the previous remark this is well-defined; of course Conjecture 3 follows from
Conjecture 2.Before we say something about the ramified components, we would like to give
some motivation for the Conjecture, in particular for the "holobaric" property and
the intervention of SL(2). This comes from Shimura varieties - the reader who is
interested in spectral questions and does not like algebraic geometry can skip this.
Assume for instance that G = GSp(g, Q) is the symplectic (similitude) group,
that n occurs in Ac,cusp and that n 00 is a cohomological representation of G(~)
(see [9]). Then there is a cohomology space H•(n 00 ) ; if K 00 = GU(g) is maximal
compact (mod center) in G(~) and if KC G(A. 1 ) is compact-open, we can consider
the complex algebraic variety
SK= G(Q)\G(A)/K 00 K
and H· ( 7r 00) @ n1J is a SU bspace of H· ( s K) q. The variety s K is defined over Q)
and this subspace can be defined over «Jle (or some extension of «Jle) and relized in
etale cohomology; it then carries an action of Gal(Q/Q). We denote it by He.
A fundamental conjecture of Langlands is, roughly, that the eigenvalues of
frobp in Ht,3 are monomials in the eigenvalues of tw,p· (This is not quite correct
because of "L -indistinguistability". For the precise formulation see [36, 7b]). Thus
Conjecture 3 is just of version of purity in algebraic geometry ; in this context it
can b e checked that the full Conjecture 2 (with SL(2) !) is a reformulation of
Lefschetz's theory on SK -in particular SL(2) here is "Lefschetz's SL(2)", see [4b,
36]. The extraordinary import of Arthur's conjectures is that it imparts the same
(^3) For p such that the representation of Gal(Q/«Jl) is unramified when restricted to Gal(Qp/Qp) ...