NUMBER THEORETIC BACKGROUND
25
Notice that the representations obtained from motives via Hodge theory are
very special, in that the p and q are integers.
Finally, define L{Ma, s) and e(Mff, s) to be the L- and s-factors associated to the
representation V(Ma) as in §3. For a table making these explicit see [D6, 5.3].
(4.5) Fglobal. Let Fbe a global field and M a motive with complex multiplication
by E, defined over F. For each place v of F, let Mv denote the restriction of M to
Fv. Let o'.E -* C. The product L(Mff, s) = J\v L{MVf(n s) converges in a right half-
plane. It is conjectured that it is meromorphic in the whole 5-plane and satisfies the
functional equation
UM„ s) = e(M„ s) L(M*9 1 - s)
with M* = Hom(M, Q) and e(Mff9 s) = Uve(MVtff9 s).
In the function field case this conjecture has been proved by Deligne. Let q be
the number of elements in the constant field k of F. Grothendieck proved that for
any given A-adie representation V of GF which is unramified at all but a finite num-
ber of places v, the corresponding L-function L(V, s) = UVL(VV, s) is a rational
function of
q~$
(even a polynomial if
VG
= 0 and VQ = 0, where G is the geometric
Galois group, i.e., the kernel of the map of GF to Gk), and satisfies a functional
equation of the form L(V, s) = s(V, s)L(V*, 1 s) with an e which is a monomial
in q~s of degree TiV[k(v):k] a(Vv). Later, Deligne showed that Grothendieck's
e(F, s) is equal to the product of the local e(Vv, s)y$ if V = VXQ is a member of a
family (Vx)xe^ of A-adic representations of GF for some infinite set of places
of a number field E, and the family is compatible in the following weak sense: for
each X, fj, e if there is a finite set S of places of F such that for v £ S, the represen-
tations Vx and Vp are unramified at v and the characteristic polynomals of 0V
acting on Vx and V^ have coefficients in E and are equal. Deligne's method is to
prove that Grothendieck's e is congruent to the product of the local e's modulo X
for all A e if and is therefore equal to that product. By (4.3) any A-adic representa-
tion coming from /-adic cohomology, i.e., from a motive, is a member of a system
which is weakly compatible in the above sense.
When dim(F) = 2, then by Jacquet-Langlands (resp. Weil), Springer Lecture
Notes 114 (resp. 189), these results show that L(V, s) comes from an automorphic
representation of (resp. modular form on) GL2(^4F). On the other hand, Drinfeld
has recently shown that automorphic representations of GL2 give rise to systems
of /-adic representations occurring as constituents in tensor products of those com-
ing from 1-dimensional /-adic cohomology, hence from motives. Thus for GL2
over function fields, the equivalence between motives, compatible systems of /-
adic representations, and automorphic representations is pretty well established.
In this connection it should be mentioned that Zarhin [Z] has proved the isogeny
theorem over function fields: if two abelian varieties A and B over a global function
field Fgive isomorphic /-adic representations, then they are isogenous; more pre-
cisely,
Ql ® HomF(^, B) = HomGF(VtA), V£B)).
Over number fields our knowledge is not nearly so advanced. For Artin motives
of rank 2, Langlands has made a beginning with the theory of base change (see
the remarks (3.5.5)). For elliptic curves M over Q, it is not even known whether
Previous Page Next Page