Number Theory: An Introduction to Mathematics

574 XIII Connections with Number Theory

HW-conjecture but also, for sufficiently many Dirichlet charactersχ,the‘twisted’
L-functions

L(s,χ)=

∑

n≥ 1

χ(n)cnn−s

have analogous properties.
The definition of modular elliptic curvecan be given a more intuitive form: the
elliptic curveCa,bis modular if there exist non-constant functionsX = f(τ),Y =
g(τ)which are holomorphic in the upper half-plane, which are invariant under
Γ 0 (N),i.e.

f((aτ+b)/(cτ+d))=f(τ), g((aτ+b)/(cτ+d))=g(τ)

for every

A=

(

ab

cd

)

∈Γ 0 (N),

and which parametrizeCa,b:

g^2 (τ)=f^3 (τ)+af(τ)+b.

The significance of modular elliptic curves is that one can apply to them the
extensive analytic theory of modular forms. For example, through the work of Kolyva-
gin (1990), together with results of Gross and Zagier (1986) and others, it is known that
(as theBSD-conjecture predicts) a modular elliptic curve has rank 0 if itsL-function
does not vanish ats=1, and has rank 1 if itsL-function has a simple zero ats=1.
The third conjecture, stated rather roughly by Taniyama (1955) and more precisely
by Weil (1967), is simply this:

TW-Conjecture:Every elliptic curve over the fieldQof rational numbers is modular.

The name of Shimura is often also attached to this conjecture, since he certainly
contributed to its ultimate formulation. Shimura (1971) further showed that any elliptic
curve which admits complex multiplication is modular. A big step forward was made
by Wiles (1995) who, with assistance from Taylor, showed that any semi-stable elliptic
curve is modular. A complete proof of theTW-conjecture, due to Diamond and others,
has recently been announced by Darmon (1999). Thus all the results which had previ-
ously been established for modular ellipticcurves actually hold for all elliptic curves
overQ.
It should be mentioned that there is also a ‘Riemann hypothesis’ for elliptic curves
overQ, namely that all zeros of theL-function in the critical strip 1/ 2 <Rs< 3 / 2
lie on the lineRs=1.
Mordell’s theorem was extended from elliptic curves overQto abelian varieties
over any algebraic number field by Weil (1928). Many other results in the arithmetic
of elliptic curves have been similarly extended. The topic is too vast to be considered
here, but it should be said that our exposition for the prototype case is not always in
the most appropriate form for such generalizations.