Download Spencer Bloch: The proof of the Mordell Conjecture

The Proof of the Mordell Conjecture
Spencer Bloch
Andre Weil has frequently criticized the use of "conjecture" in mathematics:
conjecture said that there exist at most a finite n u m b e r
of pairs xi,y i E Q with f(xi,Yi) = 0. If f has degree d,
the genus is a n u m b e r ~ (d - 1)(d - 2)/2. (I will be
Sans cesse le math6maficien se dit: "Ce serait bien slightly more precise in a moment.) For example, bebeau" (ou: "Ce serait bien c o m m o d e " ) si telle ou fore Faltings it was not k n o w n that the equation y2 =
telle chose 6tait vrai. Parfois il le v6rifie sans trop de x5 + a has only finitely m a n y solutions in Q for every
pein; d'autres fois il ne tarde pas a se d6tromper. Si non-zero integer a.
son intuition a r6sist6 quelque temps a ses efforts, il
Subsequent generations have broadened the conjectend a parler de "conjecture", m~me si la chose a ture to include polynomials defined over any n u m b e r
peu d'importance en soi. Le plus souvent c'est pr6- field (indeed a n y "global field") and have with the
matur6. [17]
a d v e n t of abstract algebraic g e o m e t r y t e n d e d to rephrase it in terms of algebraic curves. W h a t Faltings
In this connection he refers to the Mordell conjecture: actually proves, then, is that a n y algebraic curve of
/
genus i> 2 defined over a n u m b e r field K has at most
Nous sommes moins avanc6s a l'6gard de la "con- a finite n u m b e r of K-points.
jecture de Mordell". I1 s'agit la d ' u n e question q u ' u n
Roughly speaking, the algebraic curve is the set of
arithmeticien ne p e u t gu~re m a n q u e r de se poser; all solutions of fix, y) = 0 with values in any field
on n'aper~oit d'ailleurs aucun motif s6rieux de parier containing K. In fact, this statement has to be a m e n d e d
pour ou contre.
on several c o u n t s . For o n e t h i n g , the s o l u t i o n set
m i s s e s " p o i n t s at i n f i n i t y " . To avoid h a v i n g s o m e
Probably most mathematicians w o u l d have agreed fiend stash all the goodies out at infinity where we
with Weil (certainly I would have), until earlier last c a n n o t get at t h e m (cf. The Hitchhiker's Guide to the
year, w h e n a G e r m a n mathematician, Gerd Faltings, Galaxy) we i n t r o d u c e projective space. Projective nproved the Mordell conjecture, opening thereby a new space over a field k is the set of lines t h r o u g h the origin
chapter in n u m b e r theory. In fact, his paper also es- in a vector space of dimension n + 1. If we fix a basis
tablishes two other important conjectures, due to Tate of the vector space we can specify a line, and hence a
a n d Shafarevich, a n d these a c h i e v e m e n t s m a y well point of projective space, by the coordinates (x0. . . . .
prove to be equally significant. I w a n t to sketch briefly Xn) of a non-zero point on the line. If F is a homogew h a t these three conjectures say and w h a t the ingre- neous polynomial in X o, X 1. . . . .
X n, we say F vand i e n t s are in F a l t i n g s ' proof. To m a i n t a i n the " G "
ishes at the projective point if it vanishes along the
rating appropriate to a (mathematical) "family maga- corresponding line.
zine" for as long as possible, there will of course be
For example, let F(X, Y, Z) be the unique homogeno question of giving details. I will also not enter into neous polynomial of degree d, where d is the degree
the history of the conjecture other t h a n to note it was of f(x, y), such that F(x, y, 1) = fix, y). The locus of
first posed by Mordell in [7], and that the analogue for zeroes of F in projective 2-space will then contain the
curves over a function field was proved i n d e p e n d e n t l y solutions of f(x, y) = 0, but there will be other points
by Manin in the U.S.S.R. and by Grauert [6], [12].
corresponding to lines in the plane Z = 0 along which
F = 0. For example, if fix, y) = y2 _ x3 _ 1, t h e n
F(X, Y, Z) = Y2Z - X 3 - Z 3, a n d there is one exThe Mordell Conjecture
tra point at infinity, n a m e l y X = Z = 0, Y = 1 (cf.
fig. 1).
As originally formulated, the conjecture said that any
We can give a more precise formula for the genus
irreducible polynomial in 2 variables with rational coefoff:
ficients, having " g e n u s " greater than or, equal to two,
genus = (d - 1)(d - 2)/2 - Z Vp
has at most a finite number of solutions. Writing fix, y)
for the polynomial a n d Q for the rational numbers, the where the s u m is taken over all projective points with
THE MATHEMATICAL INTELLIGENCER VOL. 6, NO. 2 9 1984 Springer-Verlag New York
41
Figure 1
has a solution it will have an infinite number. To see
this we argue geometrically. Let P be a point in the
solution set off, and let L be a line not passing t h r o u g h
P (cf. fig. 1). For Q a point on L with coordinates in
our given field K, the line P Q usually meets the solution set in a second point R. As Q runs t h r o u g h the
infinite set of K-points of L, the collection of points R
is an infinite set of K-solutions of f. For example, applying this procedure to the equation X2 q- y2 _ 1 gives
the familiar parameterization
t2 - 1
2t
x - t2 + 1
Y - t 2 + !"
Figure 2
~x(p ) = -~(p)
OF
OF
= -~-(p)
= 0
and Vp i> 1 for all p. Such points are called singular
points of the zero locus. For example, the Fermat polynomial x n + yn _ 1 has no singular points; its genus
is (n - 1)(n - 2)/2.
On the other h a n d y2 = x2(x + 1) has a double point
at the origin, a n d the corresponding projective curve
F = Y2Z - X2(X + Z) = 0 has these partial derivatives:
aF -
-3X 2 -
2XZ, ~
aF = y 2 _
= 2 y z , aZ
X2,
which all vanish at [0, 0, 1]. So the genus of this curve
is not 1 but 0. Singular curves, like y2 = x 2 (x + 1)
have smaller genus t h a n their degree w o u l d suggest.
Since Mordell's conjecture is formulated in terms of
genus, it will be necessary to keep track of h o w singular are the curves that we introduce.
Curves of G e n u s 0 and 1
What h a p p e n s w h e n degree f ~< 3 so the conjecture
doesn't apply? For d = 1, f = ax + by + c clearly has
an infinite n u m b e r of solutions. For d = 2, f m a y have
no solutions (e.g., x2 + y2 + 1 w h e n K = Q) but if it
42
THE MATHEMATICAL INTELLIGENCER VOL. 6, NO. 2, 1984
W h e n F is non-singular of degree 3, the set of solutions forms a group, a so-called elliptic curve. The line
t h r o u g h points P and Q meets the zero locus in a third
point R which equals - P - Q for the group law. If
one fixes a suitable origin P0, the procedure for multiplying a point R by 2 is given in fig. 2. Assuming the
original point was not of finite order for the group law,
iteration will yield an infinite set of solutions. One consequence of Faltings' work is that for F non-singular
of degree/> 4, no such geometric m e t h o d of generating
points is possible. However, there are higher dimensional algebraic varieties, called abelian varieties,
which do have similar group laws on them. It is the
study of these abelian varieties that form the heart of
Faltings' proof.
H o w Shafarevich's Conjecture Comes In
One key step in the proof was taken by A. N. Parshin
[10], w h o s h o w e d that the Mordell conjecture would
follow from a conjecture of Shafarevich about good
reduction for curves. Suppose for example that our
h o m o g e n e o u s polynomial F is non-singular (i.e., has
no singular points) and K = Q. By clearing denominators we m a y suppose the coefficients of F are integers w i t h no c o m m o n factors, and we m a y then consider the equation m o d p for some prime n u m b e r p. If
the partial derivatives have no c o m m o n zero rood p,
Figure 4 curve of genus 3
Figure 3
then the equation F(X, Y, Z) = 0 (mod p) gives a nonsingular curve over the field Z/pZ, a n d the original
curve is said to have good reduction at p. For example
the Fermat curve X 3 + y3 + Z 3 does not have good
reduction at the prime 3. A more substantial example
is the curve Y2Z = X 3 - 17Z 3. It too is nonsingular as
a curve over Q, but it becomes singular w h e n reduced
m o d u l o either 2, 3, or 17.
Shafarevich conjectured [13] that for a fixed genus g
and a fixed finite set S of primes in a n u m b e r field K
there were at m o s t a finite n u m b e r of isomorphism
classes of curves of g e n u s g defined over K with good
reduction outside S. For example, as applied to the
collection of elliptic curves y2 = X ( X - 1)(X - X)(K E
Z) the conjecture follows from finiteness of the set of
X such that X a n d X - 1 are divisible only by primes
of S.
Suppose C is such a curve with g / > 2 a n d P is a Kpoint of C. Parshin s h o w e d that after a finite extension
K' of K one could find a n o t h e r curve C' with genus g'
and good reduction outside S' together with a map
C'---*C which is ramified only over P. (A m a p of curves
is ramified at a point P if the n u m b e r of points in the
inverse image of P is strictly less than the degree of
the mapping.) Further, K', g', and S d e p e n d only on
g and S, not on P. Shafarevich's conjecture then says
there are only a finite n u m b e r of such C'. S u p p o s e n o w
C has an infinite n u m b e r of K-points Pr, r = 1, 2, 3 . . . . .
An infinite n u m b e r of the c o r r e s p o n d i n g C' r would
n e c e s s a r i l y be i s o m o r p h i c to a c o m m o n c u r v e C "
w h i c h w o u l d a d m i t a n infinite n u m b e r of d i s t i n c t
maps to C. By classical Riemann surface theory, the
set of maps from one surface of g e n u s / > 2 to another
is finite, h o w e v e r , so the set of K-points of C is finite.
The Passage to the Jacobian
Subtle q u e s t i o n s a b o u t c u r v e s are o f t e n p r o f i t a b l y
treated b y " p a s s i n g to the Jacobian", which numbertheoretically m e a n s generalizing from elliptic curves to
abelian varieties. Indeed, in the present case, Shafarevich formulated and Faltings proves a stronger version of Shafarevich's conjecture, dealing with abelian
varieties.
To explain, I n e e d to recall a bit more about Riemann
surfaces (viz. compact complex manifolds of dimension 1). The curves we h a v e b e e n discussing give rise
to Riemann surfaces b y choosing an e m b e d d i n g of the
coefficient field K in the complex n u m b e r s and t h e n
looking at all complex points. The genus g of the curve
equals the n u m b e r of " h o l e s " in the surface (fig. 4). A
h o l o m o r p h i c differential 1-form on an o p e n set U in
the complex plane is an expression f(z)dz w h e r e z is
the complex p a r a m e t e r a n d f is a holomorphic function. If p is a path in U we m a y c o m p u t e the integral
Yff(z) dz. A Riemann surface is covered by such o p e n
sets U i, a n d w e m a y d e f i n e a h o l o m o r p h i 4 1 - f o r m
on the surface to be a collection of these on the U i
which agree on U i n Uj. The space V of holomorphic
1-forms has dimension equal the genus of the surface.
A p a t h p on the surface defines an element Sp in the
dual space V* to V by integration, and the collection
of such functionals for p r u n n i n g over all closed paths
forms a lattice L of maximal rank in V*. The quotient
J = V*/L is a complex torus called the Jacobian of the
surface. There is a natural divisor 0 on J, so called
because its inverse image in V* is defined by the vanishing of the Riemann theta function. It turns out that
0 d e t e r m i n e s a polarization, i.e., t h e r e is an e m b e d ding of J in projective space pn and a h y p e r p l a n e section H such that H n J = a multiple of 0. The Torelli
t h e o r e m says t h a t the pair J, 0 a c t u a l l y d e t e r m i n e
the curve.
The construction of the Jacobian can be carried out
algebraically and yields an abelian variety J defined over
the same field K as the original curve. An abelian variety is a closed, c o n n e c t e d subvariety of projective
space w h i c h has a g r o u p law. To say it is defined over
K m e a n s that it is the zero set of a collection of polynomials all of w h o s e coefficients are in K. Over the
THE MATHEMATICAL INTELL1GENCER VOL. 6, NO. 2, 1984
43
complex numbers such a thing is necessarily a complex
toms, but not every complex torus can be embedded
in projective space as the zero set of polynomials. The
simplest abelian varieties are those of dimension one;
they are the elliptic curves already mentioned above.
The Shafarevich conjecture for abelian varieties says
that the set of principally polarized abelian varieties of
dimension g defined over K with good reduction outside a given finite set of places S is finite. It is this
result Faltings actually proves. One sees from the Torelli theorem that it implies the Shafarevich conjecture
for curves, which by the Parshin argument implies the
Mordell conjecture.
Height
There is a classical method, due to Fermat, for proving
that some numerical problems have no solutions, and
Faltings' work is in part a generalization of that
method. Fermat showed that there are no solutions in
non-zero integers to x4 + y4 = z 2, by associating a
height to any triple (p, q, r) satisfying this equation-for example, Irl. The method of infinite descent then
allowed him to show that if a triple (p, q, r) with positive height lies on the curve then so does a new triple
(p, q, ~) which has smaller height, 0 < I~1< Irl. It follows
that there are no integer points on x4 + y4 = z2 other
than (0, 0, 0).
How does Faltings generalize this? As before, suppose V is the vector space of holomorphic 1-forms on
a projective curve. If v 1, v2. . . . . w5 are a basis for the
coordinate functions on V*, the differential dvl^dV2 ^
^dvg is invariant u n d e r translations and so descends to a g-form w on J. In fact w can be defined
algebraically for any g-dimensional abelian variety and
is canonical up to multiplication by a scalar in K. Faltings associates to an abelian variety over K a numerical
invariant h(A) called the height. This is a sum of local
terms associated to prime ideas of K (non-archimedean
places) as well as real or complex completions (archimedean places). For a prime ideal p of K the theory of
the Neron model gives a preferred way to reduce the
variety A modulo p. If the differential w vanishes (respectively has a pole) when reduced mod p, the local
term is an appropriate multiple of -log(Np) (resp.
+ log(Np)), where Np is the number of elements in the
finite field at p. (When K = Q, Np = p.) For each
embedding of K into R or C, Faltings takes a suitable
multiple of the log volume of A, 1/21og (fA W^W) 9 He
adds these local invariants to get the height. By the
product formula in number theory, h(A) is independent of the choice of w.
The notion of height in diophantine geometry is
more commonly applied to points in projective space.
Suppose for example x is a point in pn defined over
the rationals, corresponding to a line in Q~ + 1. Let x0,
9
44
.
.
THE MATHEMATICAL INTELLIGENCER VOL. 6, NO. 2, 1984
X1
Xn be coordinates of a point on the line such
that the x r are all integers with no common factors,
and define
.
.
.
.
.
h(x) = max {log IXrl}.
There is an analogous construction when Q is replaced
by any number field K. If the number field K and a
constant c are fixed, there are at most a finite number
of points of height ~ c.
Faltings relates the two notions of height by means
of the Siegel moduli space ~,9"~gparametrizing principally polarized abelian varieties of dimension g. He
notes that ~)Lg can be embedded as a locally closed
subvariety of a projective space in such a way that if
x corresponds to an abelian variety A, then h(A) and
h(x) coincide up to a fixed non-zero multiple plus an
error term having at worst logarithmic growth near
infinity on ~ g. (Infinity makes sense here because ~Tt g
is not complete. Abelian varieties can degenerate.) It
follows that there exist at most a finite number of gdimensional principally polarized semi-stable abelian
varieties of height ~< c. I will refer to this as the principle of bounded height. It is the main idea of the
proof9
Isogeny
The next step is to study how heights behave under
isogeny. By definition, an isogeny of abelian varieties is a
surjective homomorphism of algebraic groups with finite
kernel. The classical case is a map between elliptic
curves f: A--~B. This lifts to a map on the covering
spaces f: C---~C which satisfies f(z + ~) = f(z) whenever
lies in the lattice of periods A of A (A ~- C/A). Accordingly the derivative of f is periodic and holomorphic, and is therefore a constant, say R by Liouville's
theorem 9 Consequently A is mapped into A', the lattice of periods for B, as a subgroup of finite index, so
it has a finite kernel.
Isogeny is often the "right" notion of equivalence
for abelian varieties. For example, Poincar6's complete
reducibility theorem allows one, given an isogeny, P,
from A to B with image A', to replace A and B by
isogenous varieties A' x A, and A' x B,, so that f
becomes the composite A' x A,---~A'--~A' x B, of a
projection and an injection. Consequently many problems concerning abelian varieties are profitably tackled
by first considering the variety and its isogeneous images as equivalent, and Faltings' paper proceeds in just
this way. So we must look at how height varies under
isogeny.
The phenomenon of interest is possible ramification
of an isogeny on reduction mod p. Consider as an example the one dimensional abelian variety defined by
X3 + y3 _ 1 = 0. If we take (1, 0) as origin for the
THE MATHEMATICAL INTELLIGENCER VOL. 6, NO, 2, 1984
45
This m a p p i n g has a vertical t a n g e n t over the point (1,
0) w h e n we reduce m o d 2 (i.e., set 2 = 0 in the formula). The change of height u n d e r isogeny is given by
one half the log of the degree of the isogeny minus a
correction term which measures this sort of ramification.
Tate's Conjecture
To explain the application of this formula, which will
lead us to Tate's conjecture, I m u s t talk a bit about the
arithmetic of abelian varieties. Consider the kernel, nA,
of multiplication by an integer n on an abelian variety
A defined over a field K. Recalling that the complex
points of A look like Cg/L for a lattice L, we see that as
an abstract group ,A looks like (Z/nZ) 2g. But ,.4 has
additional structure. The coordinates of the finite set
nA of points are algebraic over K but need not themselves lie in K. Thus the galois group G of the algebraic
closure of K acts on hA, giving a representation of G
into GL2g(Z/nZ ). It is convenient to fix a prime t and
consider simultaneously all ~4 as n runs through all
powers o f t . Using multiplication bye to join them,
n A--~nA, we can take the inverse limit to obtain the
Tare module, denoted by Tt(A ). It is isomorphic as an
abstract group to Z 2g, where Z e is the t-adic completion of the integers. But G also acts on Tr(A ), giving
a representation ~
GL2g(Z e). To gain f u r t h e r perspective on T t (A), again think of the complex points
of A as Cg/L. Then nA is isomorphic to L/nL, and T t (A)
to L Q Z t . Since L corresponds in a natural w a y to the
first h o m o l o g y of Cg/L, we can think of T t (A) as being
the first h o m o l o g y of A with Z t-coefficients.
If B is another abelian variety, one can ask to compare the group Hom(A,B) of mappings of abelian varieties over K with the corresponding group of G-mapp i n g s Horn c(Tt(A),Te(B)), w h e r e the latter is the
group of h o m o m o r p h i s m s which commute with the
action of G. An important conjecture of Tate (that he
proved for K a finite field) was that the galois representation on Te (A) was semi-simple, and that
H o m (A,B)QZ t = Homc(V r (A),T t (B))
(isomorphism of groups)
group law, then multiplication by - 2 can be described
geometrically as associating to a point P on the curve
the third point of intersection of the tangent to P with
the curve (fig. 2). The tangent line to the curve at a
point (x, y) is defined by x2X + y2y _ 1 = 0, so one
deduces for multiplication by - 2 the coordinate transformation
(_x4 - 2x,_y4 -- 2 y / .
(x,y) ~
46
\ 1 - 2x 3 1 - 2y3]
THE MATHEMATICAL INTELLIGENCER VOL. 6, NO. 2, 1984
a s s u m i n g K finitely generated over the prime field.
This important result fits into the general framework
of characterizing w h e n a m a p b e t w e e n the homology
of two objects is induced by an actual geometric m a p
b e t w e e n the objects.
Suppose W C T t (A) is a s u b m o d u l e which is stable
u n d e r the galois group G. We can write W as an inverse limit of finite groups W, C A and define A n to
be the quotient A/W n. A key step in Tate's a r g u m e n t
was t h a t u n d e r a certain t e c h n i c a l c o n d i t i o n concerning polarizations an infinite n u m b e r of the A , were
isomorphic. Then if, say, A n and Amn are isomorphic,
he c o n s t r u c t s a g e o m e t r i c m a p f r o m A to itself by
taking a chain of m a p s
A--~Amn = An--->A.
But it turns out that Faltings' formula for the behavior
of heights u n d e r isogeny, together with a theorem of
Tate on p-divisible groups, applied to the A n as above
(but n o w defined over a n u m b e r field) gives a uniform
b o u n d for h(An). The principle of b o u n d e d height enables him to deduce that an infinite n u m b e r of the A n
are isomorphic, a n d n o w Tate's m e t h o d a n d arguments due to Zarhin [18] lead to a proof of the Tate
conjecture.
Back to the Shafarevich Conjecture
Faltings n o w h a d to prove the Shafarevich conjecture
that there exist at most a finite n u m b e r of principally
polarized abelian varieties of dimension g defined over
a n u m b e r field K with good reduction outside a given
finite set of primes S. Consider first the problem of
.showing there exist at most a finite n u m b e r of isogney
classes of g dimensional abelian varieties with good
reduction outside S. (I.e., we identify two abelian varieties if there is an isogeny from one to the other.)
Since we have the Tate conjecture, this a m o u n t s to
s h o w i n g there are o n l y a finite n u m b e r of isom o r p h i s m classes a m o n g the Tt (A)(~)Q; since the representations are semisimple, it sufficies to s h o w they
give rise to only a finite n u m b e r of different trace functions on the galois group.
To do this, Faltings shows that there is a finite subset
of G c o n s i s t i n g of Frobenius elements s u c h t h a t the
image of this finite subset in the e n d o m o r p h i s m ring
of Tt (A) generates the same subring as all of G. Further, the finite set d e p e n d s only on the primer , the
dimension of A, a n d the set S; it is otherwise independent of A. This reduces one to showing that for each
given Frobenius e l e m e n t , the trace of its action on
T t (A) can take on only finitely m a n y values. But now
one can apply the well-known Weft conjectures, which
say that the trace is an integer (i.e., in Z) with a precisely b o u n d e d absolute value.
The proof of the Shafarevich conjecture again invokes the principle of b o u n d e d height. Given a set of
g - d i m e n s i o n a l principally polarized abelian varieties
over K, with good reduction outside S, it will suffice
to s h o w the heights are bounded. By the result of the
previous paragraph, we m a y assume the varieties are
all isogenous. The a r g u m e n t is completed by applying
the variation of height formula, together with a theorem of R a y n a u d on galois actions on points of group
schemes and another application of the Weil conjectures for abelian varieties.
References
(I have included those papers which appear in Falting's bibliography, together with a few others of historical interest.)
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2. Baily, W. L. and Borel, A. (1966) Compactification of
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Ann. of Math. 84: 442-528.
3. Deligne, P. and Mumford, D. (1969) The irreducibility of
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et Arakelov, Asterisque 64: 169-202.
15. Tate, J. (1967) p-divisible groups, in Proceedings of a
conference on local fields (Driebergen, 1966), 158-183.
Berlin: Springer-Verlag.
16. Tate, J. (1966) Endomorphisms of abelian varieties over
finite fields, Inv. Math 2: 134-144.
17. Weil, A. (1980) Collected Works, Vol. III, p. 454. Berlin:
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Department of Mathematics
University of Chicago
Chicago, IL 60637
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