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312 16. L-Function of an Elliptic Curve and Its Analytic Continuation

√
|α(p)|=|β(p)|≤ p
holds for elliptic curves, 13(1.2) and 13(1.6) by Hasse, and also for general smooth
curves

√
|α j (p)|≤ p
by Weil, see 13(2.5). Hence we have the following convergence result as a direct
application of 11(6.6).

∗
(2.3) Proposition. The Hasse–Weil L-function L (s) for a curve C over Q con-
C
verges for Re(s)> 1 + 1/2 = 3/2.
(2.4) Remark. When Q is replaced by a number field K, we consider products over
all but a finite set S of valuations (or places) v of K, and the finite field F p is replaced
where q v is the number of elements in the residue class field of v. The crude
by F q v
Hasse–Weil L-function takes the form
1

∗
L (s) = ,
C −s −s
v (1 − α 1 (p)q v )...(1 − α 2g (p)q v )
and there is a similar expression for the Hasse–Weil zeta function. This is an Euler
product of degree 2g · [K : Q] since all v with q v = p n(v) for given p combine to
give the p-Euler factor in the product. Thus by the Riemann hypothesis, 13(1.2) and
13(2.5), we see that 11(6.6) applies to show that 2.3 is true for curves over a number
field.

(2.5) Remark. For a discussion of the proof by Deligne of the Riemann hypothesis
for smooth projective varieties, together with references, see Katz [1976]. In the last
section of his paper, Katz considers the Hasse–Weil zeta function for varieties.

§3. Hasse–Weil L-Function and the Functional Equation

Now we concentrate on the case of an elliptic curve E over a number field k.For
many considerations k can be taken to be the rational numbers. We know that for all
valuations v of k such that v( E ) = 0, the curve E has an elliptic curve E v for re-
duction over k(v), the residue class field of v. In the previous section, we considered
a quadratic polynomial f v (T ) that we defined using E v , giving an Euler factor in the
∗
L-function L (s) of E. Next we introduce the Euler factors for the other primes v
E
where E has bad reduction.
,let E v,0 (k(v))
Let E v denote the reduction of E at any v, a curve over k(v) = F q v
be the group of nonsingular points, and let N v = #E v,0 (k(v)) be the number of points
in this group curve.
(3.1) Notations. For an elliptic curve E over a number field k and a non-Archimedean
valuation v of k,wedenoteby f v (T ) the polynomial: