§1. The Standard Hypotheses
Throughout this paper, K will be a global field: a finite algebraic extension of
Q, or a function field of transcendence degree 1 over a finite field. If v is a place of
K, let Kv be the completion of K at v, and write \x\v for canonical absolute on Kv
given by the modulus of additive Haar measure. Thus, if v is nonarchimedean and
irv is a uniformizing parameter for the maximal ideal mv of the ring of integers
Ov, then
#(Ov/mv)\ if v is archimedean and Kv = R, then \x\v is the
usual absolute value |x|; if Kv = C, then \x\v =
With this normalization, the
product formula reads
Y[ \K\V = 1 for O^KeK .
Write Kv for a fixed algebraic closure of Kv, and Cv for the completion of Kv. The
absolute value \x\v extends uniquely to an absolute value on Cv, which we continue
to denote \x\v. If Lw/Kv is a finite extension, and we fix a topological isomorphism
of Cw with Cv, then for x G Cw = Cv there are two absolute values, \x\w and \x\v.
These are related by
\x\w = i 4
^ i
Let X/K be a projective variety and let C be an invertible sheaf on X. If L/K
is an extension, we write £L = L ®x C for the sheaf on XL = X x K L induced by C
For each place v of X, put Xv = X x ^ spec(Cv). Given x G XV(CV), write Cx for
the stalk of Cv ®K C at x, and £(x) = x*(Cv SK £) for the fibre, a 1-dimensional
vector space over Cv. If / G Cx, then /(#) will denote its residue, or "value" in
C(x). We write
for the group of global sections of C®n.
Let Gal(C
/i^) be the group of continuous automorphisms a G Aut(Cv/Kv),
acting on Cv from the left; thus
Gtd(Cv/Kv) * Gzl(Kv/Kv) *
is the separable closure of Kv. We let Gal(Cv/Kv) act on X(CV) =
XV(CV) from the left by its direct action, and on the right via its inverse, so that if
a G X(CV) has coordinates ( a i , . . . , am) in some afflne patch, then
j((ai,... , am)) = (o-(ai),... , r(am)) ,
(ai,.. . , a
Gal(Cu/lfv) also acts on sections of Cv ®K C from the left: by flat base change
^cv® r(x,
given / G T(X, Cv ®
decompose / =
U with /; linearly independent
over if, and put
Write Div(X) for the group of Cartier divisors on X, and
for the group
of cycles of codimension 1. Let z : Div(X) »
be the cycle map, and if
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