NUMBER THEORETIC BACKGROUND

13

L(X-»US)

= rR(s)d=\-^ r(S/2).

(3.2.1) F « C. For z an embedding of Fin C and N ^ 0,

defn

(3.1.3) Fnonarchimedean. For 7r a uniformizer in F,

(1 — x(7t))~l, if x ls unramified,

if / is ramified.

In every case, L is a meromorphic function of/, i.e.,

L(XDS)

is meromorphic in s,

and Z, has no zeros.

(3.2) Local abelian ^functions. We will denote by dx a Haar measure on F,

by d*x a Haar measure on F* (e.g., d*x =

||x||_1

dx) and by ^ a nontrivial additive

character of F.

Given cp and *£c, one has a "Fourier transform"

/0) = $Ax)4(xy)dx.

The local functional equation

( } lXf»irl) -^'V'*** L(X)

defines a number e(x, fi, dx)e C* which is independent of/, for/'s such that the two

sides make sense. If / i s continuous such that/(x) and/(x) are 0{e~m) as

||JC||

-»

oo, then the two sides make sense naively for x s u c n that x(*) = II*II" with 0

a 1, and each side is a meromorphic function of x O

n e

takes the same multi-

plicative Haar measure d*x on each side. The dependence of e on (J and dx comes

from the dependence of the Fourier transform f(x) on $ and dx. One finds

(3.2.2) e(x /, rdx) = re(x & dx), for r 0,

(3.2.3) e(x, 0(ax),rfx)= z(*)|| a||-i eft, 0, rfx) for a e F*.

Easy computations carried out in [Tl] and [W2] show that the function e is given

by (3.2.2), (3.2.3) and the following explicit formulas:

(3.2.4) F ^ R. Let x be the embedding of F in C and N = 0 or 1. For (p =

exp(2^r/x) and £c the usual measure, e(x~Ncos, (/, dx) = iN.

(3.2.5) F & C. Let z be an embedding of F in C and JV ^ 0. For 9^ =

exp(2;n Trc/Rz) and dx = idz A dz (= Ida db for z = a + bi), e(z~Ncos, /), dx)

= i* .

(3.2.6) F nonarchimedean. Let 0 be the ring of integers in F. Put

w(^) = the largest integer n such that

(jj{7t~n0)

= 1,

tf(^) = the (exponent of the) conductor of x (= 0 if/ is unramified, the smallest

integer m such that

z

is trivial on units = 1 (mod

izm)

if % is ramified),

c — an element of F* of valuation «(^) 4- #(/). If x is unramified,

(3-2-6.1) e(x,p,dx)=X^jedx.

(In particular, s(z, ^, dx) = l,ifjC)dx = 1, and n(p) = 0 when

z

is unramified.)