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16

L-Function of an Elliptic Curve and Its Analytic

Continuation

We introduce the L-function of an elliptic curve E over a number field and derive
its elementary convergence properties. L-functions of this type were first introduced
by Hasse, and the concept was greatly extended by Weil. For this reason, they are
frequently called the Hasse–Weil L-function.
There are two sets of conjectural properties of the L-function with the second
depending on the first. The first conjecture, due to Hasse and Weil, asserts that the
L-function L E , defined as a Dirichlet series for Re(s)> 3/2 has an analytic con-
tinuation to the complex plane and satisfies a functional equation under reflection s
to 2 − s. The second conjecture, due to Birch and Swinneton-Dyer, ties up the arith-
metic of the curve E with the behavior of L E at s = 1 which is the fixed point under
s to 2 − s. This conjecture clearly depends on the first and is discussed in the next
chapter.
The Hasse–Weil conjecture has been verified for two general classes of ellip-
tic curves. First, the L-function of an elliptic curve with complex multiplication is
related to the L-function of Hecke Grossencharacters by a theorem of Deuring. In
this case, the conjecture is deduced from the analytic continuation and functional
equation of Hecke L-functions. Second, if an elliptic curve E over the rational num-
bers is the image of a map of curves X 0 (N) → E, then the L-function L E is the
Mellin transform of a modular form of weight 2 for 0 (N). In this case, the conjec-
ture follows from the functional equation of the modular form. The general case is
considered in §8 and Chapter 18.

§1. Remarks on Analytic Methods in Arithmetic

In Chapters 9, 10, and 11, analytic methods were considered to describe the complex
points on an elliptic curve. This led to elliptic functions, theta functions, and in order
to describe all elliptic curves up to isomorphism, also modular functions.
In these last two chapters, we sketch some of the flavor of the use of analytic
methods in the arithmetic of elliptic curves, that is, for elliptic curves over number
fields. The idea is to assemble the modulo p information about the elliptic curve into