Number Theory: An Introduction to Mathematics

5 Further Results and Conjectures 571

which is now an integer, has minimal absolute value. (It has been proved by Tate that
we then have|∆|>1.) The discussion which follows presupposes thatWis chosen in
this way so that, in particular, discriminant means ‘minimal discriminant’. We say that
such aWis aminimal modelfor the elliptic curve.
For any primep,letWpbe the cubic curve defined over the finite fieldFpby the
polynomial

Y^2 + ̃a 1 XY+ ̃a 3 Y−(X^3 + ̃a 2 X^2 + ̃a 4 X+ ̃a 6 ),

wherea ̃j∈aj+pZ.Ifp
∆the cubic curveWpis non-singular, but ifp|∆thenWp
has a unique singular point. The singular point(x 0 ,y 0 )ofWpis acuspif, on replacing
XandYbyx 0 +Xandy 0 +Y, we obtain a polynomial of the form

c(aX+bY)^2 +···,

wherea,b,c∈Fpand the unwritten terms are of degree>2. Otherwise, the singular
point is anode.
For any primep,letNpdenote the number ofFp-points ofWp, including the point
at infinityO, and put

cp=p+ 1 −Np.

It was conjectured by Artin (1924), and proved by Hasse (1934), that

|cp|≤ 2 p^1 /^2 ifp
∆.

Since 2p^1 /^2 is not an integer, this inequality says that the quadratic polynomial

1 −cpT+pT^2

has conjugate complex rootsγp,γ ̄pof absolute valuep−^1 /^2 or, if we putT=p−s,
that the zeros of

1 −cpp−s+p^1 −^2 s

lie on the lineRs= 1 /2. Thus it is an analogue of the Riemann hypothesis on the zeros
ofζ(s), but differs from it by having been proved. (As mentioned in§5 of Chapter IX,
Hasse’s result was considerably generalized by Weil (1948) and Deligne (1974).)
TheL-functionof the original elliptic curveW is defined by

L(s)=L(s,W):=

∏

p|∆

( 1 −cpp−s)−^1

∏

p
∆

( 1 −cpp−s+p^1 −^2 s)−^1.

The first product on the right side has only finitely many factors. The infinite second
product is convergent forRs> 3 /2, since

1 −cpp−s+p^1 −^2 s=(p^1 /^2 −s−p^1 /^2 γp)(p^1 /^2 −s−p^1 /^2 γ ̄p)