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

The functions L E/K (s) and # E/K (s) are defined as Dirichlet series and Dirichlet
series multiplied with elementary factors which converge to holomorphic functions
on the half plane

3
Re(s)> .
2
This leads to one of the main conjectures of the subject.
(3.4) Conjecture (Hasse–Weil). Let E be an elliptic curve over a number field K.
The modified Hasse–Weil L-function # E/K (s) has an analytic continuation to the
entire complex plane as an analytic function, and it satisfies the functional equation

# E/K (2 − s) = w# E/K (s)

with w =±1.
If # E/K has an analytic continuation to the complex plane, then so will L E/K (s)
n
and further from the gamma factor (s) and its poles we see that L E/K (s) will have
zeros of order at least n at all the negative integers. The order of the possible zero of
# E/K (s) at s = 1 is related to the sign w = 1or −1 in the functional equation. If
w =−1, then # E/K (1) =−# E/K (1) and # E/K (s) has a zero at s = 1. In fact, the
parity of ord s=1 # E/K (s) is even for w = 1 and odd for w =−1.
(3.5) Remark. Elliptic curves over Q will be of special interest, and in the case of
E/Q, the modified Hasse–Weil L-function becomes
−s
# E (s) = N s/2 (2π) (s)L E (s).
This conjecture has been established in two cases. First, if E/K has complex mul-
tiplication, then the Hasse–Weil L-function is related to the L-functions of type
L(s,χ), where χ is a Hecke Grossencharacter. The functional equation for L-
functions with Hecke Grossencharacters implies the functional equation for elliptic
curves. The object of this chapter is primarily to explain this result. Second, if E is
an elliptic curve over the rational numbers which is the image of a certain modular
curve related to its conductor, then L E (s) is the Mellin transform of a modular from
and the functional equation follows from this fact. These topics are taken up in the
last two sections.

In the next section we review the functional equation of the classical zeta and L-
functions and lead into the theory of algebraic Hecke Gr¨ ossencharakters, see Hecke,
Mathematische Werke, No. 14. We define algebraic Hecke characters as extensions
of algebraic group characters over Q to algebraic valued characters on an id` ele class
group. The Gr¨ ossencharakters in the sense of Hecke, which we are interested in, are
Archimedean modifications defined on the id` ele class group. From this perspective,
there are also -adic modifications related to -adic representations and the Hecke
characters coming from elliptic curves with complex multiplication.