Millenium Prize Award [讨论和分享]

~HeBe~_@

Birch and Swinnerton-Dyer ConjectureMathematicians have always been fascinated by the problemof describing all solutions in whole numbers x,y,z toalgebraic equations like
    x2 + y2 =z2
Euclid gave the complete solution for that equation, but for morecomplicated equations this becomes extremely difficult. Indeed, in1970 Yu. V. Matiyasevich showed that Hilbert's tenth problem isunsolvable, i.e., there is no general method for determining when suchequations have a solution in whole numbers. But in special cases onecan hope to say something. When the solutions are the points of anabelian variety, the Birch and Swinnerton-Dyer conjecture asserts thatthe size of the group of rational points is related to the behavior ofan associated zeta function ζ(s) near the point s=1.  Inparticular this amazing conjecture asserts that if ζ(1) is equalto 0, then there are an infinite number of rational points(solutions), and conversely, if ζ(1) is not equal to 0, then thereis only a finite number of such points.

~HeBe~_@

THE BIRCH AND SWINNERTON-DYER CONJECTURE
ANDREW WILES
A polynomial relation f(x, y) = 0 in two variables defines a curve C0. If the
coefficients of the polynomial are rational numbers, then one can ask for solutions
of the equation f(x, y) = 0 with x, y 2 Q, in other words for rational points on
the curve. If we consider a non-singular projective model C of the curve, then
topologically it is classified by its genus, and we call this the genus of C0 also. Note
that C0(Q) and C(Q) are either both finite or both infinite. Mordell conjectured,
and in 1983 Faltings proved, the following deep result.
Theorem ([9]). If the genus of C0 is greater than or equal to 2, then C0(Q) is
finite.
As yet the proof is not effective so that one does not possess an algorithm for
finding the rational points. (There is an effective bound on the number of solutions
but that does not help much with finding them.)
The case of genus zero curves is much easier and was treated in detail by Hilbert
and Hurwitz [12]. They explicitly reduce to the cases of linear and quadratic equations.
The former case is easy and the latter is resolved by the criterion of Legendre.
In particular, for a non-singular projective model C we find that C(Q) is non-empty
if and only if C has p-adic points for all primes p, and this in turn is determined
by a finite number of congruences. If C(Q) is non-empty, then C is parametrized
by rational functions and there are infinitely many rational points.
The most elusive case is that of genus 1. There may or may not be rational
solutions and no method is known for determining which is the case for any given
curve. Moreover when there are rational solutions there may or may not be infinitely
many. If a non-singular projective model C has a rational point, then C(Q) has a
natural structure as an abelian group with this point as the identity element. In
this case we call C an elliptic curve over Q. (For a history of the development of
this idea see [19].) In 1922 Mordell [15] proved that this group is finitely generated,
thus fulfilling an implicit assumption of Poincar´e.
Theorem. If C is an elliptic curve over Q, then
C(Q) ' Zr  C(Q)tors
for some integer r  0, where C(Q)tors is a finite abelian group.
The integer r is called the rank of C. It is zero if and only if C(Q) is finite. We
can find an affine model for the curve in Weierstrass form
C : y2 = x3 + ax + b
with a, b 2 Z. We let
denote the discriminant of the cubic and set
Np := #{solutions of y2  x3 + ax + b mod p},
ap := p − Np.


Then we can define the incomplete L-series of C (incomplete because we omit the
Euler factors for primes p|2
) by
L(C, s) := Yp-2

(1 − app−s + p1−2s)−1.
We view this as a function of the complex variable s and this Euler product is then
known to converge for Re(s) > 3/2. A conjecture going back to Hasse (see the
commentary on 1952(d) in [26]) predicted that L(C, s) should have a holomorphic
continuation as a function of s to the whole complex plane. This has now been
proved ([25], [24], [1]). We can now state the millenium prize problem:
Conjecture (Birch and Swinnerton-Dyer). The Taylor expansion of L(C, s) at
s = 1 has the form
L(C, s) = c(s − 1)r + higher order terms
with c 6= 0 and r = rank(C(Q)).
In particular this conjecture asserts that L(C, 1) = 0 , C(Q) is infinite.
Remarks. 1. There is a refined version of this conjecture. In this version one has
to define Euler factors at primes p|2
to obtain the completed L-series, L(C, s).
The conjecture then predicts that L(C, s)  c(s − 1)r with
c = |XC|R1w1 Yp|2

wp/|C(Q)tors|2.
Here |XC| is the order of the Tate–Shafarevich group of the elliptic curve C, a
group which is not known in general to be finite although it is conjectured to be so.
It counts the number of equivalence classes of homogeneous spaces of C which have
points in all local fields. The term R1 is an r×r determinant whose matrix entries
are given by a height pairing applied to a system of generators of C(Q)/C(Q)tors.
The wp’s are elementary local factors and w1 is a simple multiple of the real period
of C. For a precise definition of these factors see [20] or [22]. It is hoped that a
proof of the conjecture would also yield a proof of the finiteness of XC.
2. The conjecture can also be stated over any number field as well as for abelian
varieties, see [20]. Since the original conjecture was stated, much more elaborate
conjectures concerning special values of L-functions have appeared, due to Tate,
Lichtenbaum, Deligne, Bloch, Beilinson and others, see [21], [3] and [2]. In particular,
these relate the ranks of groups of algebraic cycles to the order of vanishing
(or the order of poles) of suitable L-functions.
3. There is an analogous conjecture for elliptic curves over function fields. It has
been proved in this case by Artin and Tate [20] that the L-series has a zero of order
at least r, but the conjecture itself remains unproved. In the function field case it
is now known to be equivalent to the finiteness of the Tate–Shafarevich group, [20],
[17, Corollary 9.7].
4. A proof of the conjecture in the stronger form would give an effective means
of finding generators for the group of rational points. Actually, one only needs
the integrality of the term XC in the expression for L(C, s) above, without any
interpretation as the order of the Tate–Shafarevich group. This was shown by
Manin [16] subject to the condition that the elliptic curves were modular, a property
which is now known for all elliptic curves by [25], [24], [1]. (A modular elliptic curve
is one that occurs as a factor of the Jacobian of a modular curve.)

1. Early History
Problems on curves of genus 1 feature prominently in Diophantus’ Arithmetica.
It is easy to see that a straight line meets an elliptic curve in three points (counting
multiplicity) so that if two of the points are rational then so is the third.1 In
particular, if a tangent is taken at a rational point, then it meets the curve again
in a rational point. Diophantus implicitly used this method to obtain a second
solution from a first. He did not iterate this process, however, and it was Fermat
who first realized that one can sometimes obtain infinitely many solutions in this
way. Fermat also introduced a method of ‘descent’ that sometimes permits one to
show that the number of solutions is finite or even zero.
One very old problem concerned with rational points on elliptic curves is the
congruent number problem. One way of stating it is to ask which rational integers
can occur as the areas of right-angled triangles with rational length sides. Such
integers are called congruent numbers. For example, Fibonacci was challenged in
the court of Frederic II with the problem for n = 5, and he succeeded in finding
such a triangle. He claimed, moreover, that there was no such triangle for n = 1,
but the proof was fallacious and the first correct proof was given by Fermat. The
problem dates back to Arab manuscripts of the 10th century (for the history see
[27, Chapter 1, §VII] and [7, Chapter XVI]). It is closely related to the problem of
determining the rational points on the curve Cn : y2 = x3 − n2x. Indeed,
Cn(Q) is infinite () n is a congruent number.
Assuming the Birch and Swinnerton-Dyer conjecture (or even the weaker statement
that Cn(Q) is infinite , L(Cn, 1) = 0) one can show that any n  5, 6, 7 mod 8
is a congruent number, and, moreover, Tunnell has shown, again assuming the
conjecture, that for n odd and square-free
n is a congruent number ()
#{x, y, z 2 Z: 2x2 + y2 + 8z2 = n}
= 2 × #{x, y, z 2 Z: 2x2 + y2 + 32z2 = n},
with a similar criterion if n is even [23]. Tunnell proved the implication left to right
unconditionally with the help of the main theorem of [5] described below.
2. Recent History
It was the 1901 paper of Poincar´e that started the modern theory of rational
points on curves and that first raised questions about the minimal number of generators
of C(Q). The conjecture itself was first stated in the form we have given
in the early 1960s (see [4]). It was found experimentally using one of the early
EDSAC computers at Cambridge. The first general result proved was for elliptic
curves with complex multiplication. The curves with complex multiplication fall
into a finite number of families including {y2 = x3 − Dx} and {y2 = x3 − k} for
varying D, k 6= 0. This theorem was proved in 1976 and is due to Coates and Wiles
[5]. It states that if C is a curve with complex multiplication and L(C, 1) 6= 0, then
C(Q) is finite. In 1983 Gross and Zagier showed that if C is a modular elliptic
curve and L(C, 1) = 0 but L0(C, 1) 6= 0, then an earlier construction of Heegner
actually gives a rational point of infinite order. Using new ideas together with this
result, Kolyvagin showed in 1990 that for modular elliptic curves, if L(C, 1) 6= 0
1This was apparently first explicitly pointed out by Newton.