Download Arithmetic and Hyperbolic Geometry

Arithmetic and Hyperbolic Geometry
Paul Vojta
Department of Mathematics, University of California, Berkeley, CA 94720, USA
We begin by recalling Faltings' theorem (née MordelPs conjecture) :
Theorem 0.1 (Faltings, 1983). Let C be a curve of genus > 1 defined over a number
field k. Then the set C(k) of k-rational points on C is finite.
Compact Riemann surfaces of genus > 1 are also characterized by the
property that they are hyperbolic, in any of the senses discussed in the next
section. Recent work has suggested that a similar link exists, formally at least,
between arithmetic properties of algebraic varieties (of any dimension) and
complex analytic properties of the corresponding complex manifold. The goal of
this talk is to discuss generalizations of Theorem 0.1 which have been motivated
by this analogy.
1. Hyperbolicity
We begin by defining several notions of hyperbolicity. Throughout this section
let X be a connected complex manifold, not necessarily compact.
Definition 1.1. The manifold X is negatively curved if there exists a (1, l)-form œ
on X and a constant K > 0 such that all holomorphic sectional curvatures of co
are < —;c.
Definition 1.2 (Kobayashi, 1970). The Poincaré distance dhyp on the open unit
disc D ç C i s the distance given infinitesimally by the form
dz Adz
(1 - N 2 ) 2 '
Then X is said to be Kobayashi hyperbolic if there exists a distance dx on X (with
dx(x,y) > 0 whenever x ^ y) such that for all holomorphic maps / : D - • X,
f*dx < dhyp.
Definition 1.3. The manifold X is Brody hyperbolic if all holomorphic maps from
C to X are constant.
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of Mathematicians, Kyoto, Japan, 1990
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Paul Vojta
Actually, the latter two definitions are valid when X is a complex space.
We have
negatively curved = > Kobayashi hyperbolic ==> Brody hyperbolic.
The converse of the second arrow holds if X is compact, but not in general.
A general reference on hyperbolicity is Lang (1987); see also Kobayashi
(1970).
2. Conjectures in Number Theory
The first conjecture relating complex analytic and arithmetic properties is the
following.
Conjecture 2.1 (Lang, 1974). A complete variety X/k has only finitely many rational
points if the corresponding complex space is Kobayashi hyperbolic.
More recently the above statement, which is qualitative in nature, has been
replaced by a more quantitative version. To describe it, we first state a theorem.
Theorem 2.2. Let D be an effective divisor on a complete irreducible nonsingular
curve C, and assume that D has no multiple points. Let K be a canonical divisor
on C, let A be an ample divisor on C, and fix e > 0. Then for almost all symbols
tl
C\ H
m(D,7) + TKCì)<8TA(l)+0(l).
(1)
Supplied with one set of definitions, this theorem is Nevanlinna's Second
Main Theorem. Indeed, let C be a compact Riemann surface, and let / : (C —> C
be a holomorphic function, regarded as an infinite collection of maps fr : JDr —> C
(here ID,, is the closed disc of radius r, r > 0). Replace the symbol "?" with r,
and interpret "almost all" to mean all r outside a set of finite Lebesgue measure.
Choose some distance function dist(P, Q) on C, and extend it to divisors by the
formula
dist(P,ZnQ • 0 = n d i s t ( P ' ß ) W ß •
Then, for any divisor D on C let
ddm
m(D, r)= [ n - log dist(f(rew), D)
2%
Jo
r
N(D,r)= £ ord z = w r/)log^;
welD,
TD(r) =
wl
m(D,r)+N(D,r).
This gives a weak form of the Second Main Theorem for curves.
With a different set of definitions, we obtain an arithmetic result which goes
by several names, depending on the genus of the curve. Let C be a curve defined
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over a number field k, and assume also that D is defined over k. We now replace
"?" by points P e C(k) \ SuppD, and interpret "almost all" to mean all but
finitely many points P. Fix a finite set S of places of k, and for all places v of
k let distp() denote some fixed distance on C in the u-adic topology, extended to
divisors as before. It is assumed that these are chosen uniformly in some sense,
as in Lang (1983). Then for any divisor D on C defined over k, let
m{D,P) = - i — £-log distro) ;
L * ^
veS
N(D,P) = - i — ^ - l o g d i s U P . D ) ;
Vf-VKts
TD(P) = m(D,P) + N(D,P)
The quantity TD is thus the Weil height of P relative to D ; in particular if D is
ample then there are only finitely many points P E C(k) of bounded height. In
the arithmetic case we will also write ho(P) for the height.
Proof of Theorem 2.2 (Algebraic Variant). Let g denote the genus of C.
Case 1. If C(k) is finite then the inequality must hold; by Faltings' theorem 0.1
this must hold if g > 1. Conversely, if g > 1 then K is ample, so taking D = 0,
A = K, and e < 1, we see that Theorem 2.2 implies a bound on the height of
rational points; hence it implies Mordell's conjecture.
Case 2. If g = 0 then Theorem 2.2 is equivalent to Roth's theorem :
Theorem (Roth, 1955). For each v e S fix av e Q. Then for all xek\
{av},
— — ]T -logmin(||x - a X , 1) < (2 + s)h(x).
(2)
' • ^ J veS
LC
Here h(x) = hA(x) with A = (9(1). For details, see Vojta (1987), Sect. 3.2.
Case 3. If g = 1, then Theorem 2.2 is equivalent to an approximation statement
on elliptic curves, derived by Lang from Roth's theorem using methods of Siegel.
See Lang (1960b) for details.
D
Thus we find that the Second Main Theorem of Nevanlinna theory translates
into the number field case, giving several known theorems. Generally speaking,
any such theorem in Nevanlinna theory should translate into a true statement
for number fields. Such statements often agree with diophantine conjectures
previously made. See also Sect. 8, as well as Vojta (1987), Chap. 4.
Again, this framework is related to hyperbolicity because proofs of Nevanlinna
theory exist using notions of curvature.
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Paul Vojta
In Nevanlinna theory, the analogues of Roth's theorem and Mordell's conjecture are both proved by a common proof. This suggested that Mordell's
conjecture should be provable by methods similar to those used in the proof of
Roth's theorem. In fact, such a proof was found in Vojta (1989) and Vojta (1990).
3. Sketch of Roth's Proof
Roth's proof proceeds by contradiction. Assume that there are infinitely many
x G k not satisfying (2). Then we choose a finite set with certain properties. These
are used to construct an auxiliary polynomial, the properties of which lead to a
contradiction.
To start, let <xi,...,am be all the conjugates of all av, v G S. We construct
a polynomial P(xi,...,xn),
of degree di in x,-, with a specified type of zero at
all points (au...,0Ci), 1 < i < m. This is viewed as a linear algebra problem,
with the coefficients of the polynomial as variables. The vanishing conditions
are set up so as to use up almost all of the degrees of freedom in choosing P ;
by the Dirichlet Box Principle, we can find such a polynomial in R[xi,...,x n ]
with bounded coefficients, where R is the ring of integers of k. This step is often
referred to as Siegel's lemma.
Next we choose a set of counterexamples to (2) such that
1 < h(xi) <••<
h(xn)
and such that, for all v G S, the numbers
- log m i n ( | | x t - a v \\v,l)
h(xt)
l<i<n
are close to each other. This involves a sphere packing argument.
Finally, for each v G S, using the Taylor expansion of P at (av,...,av), the
type of zero of P at that point, and the bound on the coefficients of P, we obtain
a bound for ||P(xi,...,x„)|| y and therefore for its norm. If the norm is too small,
we obtain a contradiction unless P(xi,..., x„) = 0. A similar argument applied to
certain derivatives of P implies that they vanish as well. Hence P has a certain
type of zero at (xi,...,x M ). But this contradicts Roth's lemma: the type of zero
of P is bounded from above, depending on the size of the coefficients of P and
the heights of the xt.
4. Vojta's Proof
A key difficulty in extending Roth's proof to the case of rational points is dealing
with the absence of the a's, which are needed to (a) provide vanishing conditions
which use up the degrees of freedom in choosing P, and (b) prove the vanishing
of P(x\,...,xn),
via the assumption that (2) fails to hold.
This difficulty is overcome by means of intersection theory; in particular, we
make use of the fact that the diagonal divisor A on C x C is a divisor with
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negative self-intersection. Also let Fi (resp. F2) be a divisor on C x C coming
from a divisor F of degree 1 on the first (resp. second) factor. Then let
A' =
A-F1-F2;
f
Yr = A + a1Fi+a2F2,
where a\ G Q are chosen so that a\/a2 = r and 7,.2 is positive but small. Then, by
Riemann-Roch, high multiples of Yr will have relatively few global sections; this
is how we use up our degrees of freedom in this case. (Here r is a large rational
number.)
The proof replaces many of the classical arguments involving polynomials and
absolute values with the language of arithmetic intersection theory, as developed
by Arakelov, Gillet, and Soulé. Therefore, we work on an arithmetic scheme W
of dimension 3 corresponding to C x C, and replace the Siegel's lemma argument
with a use of the Riemann-Roch theorem of Gillet and Soulé, to construct a
section y of a multiple of Yr with certain arithmetical properties.
Next we choose points, only two this time, such that
1 < h(Pi) < h(P2)
and satisfying a certain sphere packing condition again - this time in the MordellWeil group of the Jacobian of C. Let X be the arithmetic surface corresponding
to C; for / = 1,2 let Ei be the (Arakelov) divisor on X corresponding to P„ and
let E be the curve on W corresponding to (P\,P2). Then this sphere packing
condition implies that (Fi . E2) on X will be small, and therefore (E . A) on W
will be small. Then (E . Yr) will be small and in fact negative. Thus y must be
zero on E ; similarly certain derivatives must vanish, giving a lower bound for the
type of zero as before. Again, this produces a contradiction.
As is the case with Roth's theorem, this proof - as well as all proofs derived
from it - is ineffective; i.e., it can give a bound on the number of rational points
but not on their heights. Thus it is often true that one cannot prove that a given
set of rational points is the set of all rational points.
5. Extensions Due to Faltings
In Faltings (1990), Vojta's methods were generalized, giving two new theorems.
Theorem 5.1 (Conjectured in Lang (1960a)). Let X be a complete subvariety of
an abelian variety A, and assume that X does not contain any translates of any
nontrivial abelian subvarieties of A. Let k be a number field over which X and A
are defined. Then X(k) is finite.
Theorem 5.2. Let A be an abelian variety, and let Y ç A be a subvariety. Let k
be a number field over which A and Y are defined, let v be a place of k, and let
disty(x, Y) be a v-adic distance on A. Fix e > 0, and fix a height function h(x) on
A relative to some ample divisor. Then the set ofxG A(k) \ Y such that
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Paul Vojta
— logdisty(x, Y) > sh(x)
is finite.
Corollary 5.3 (Conjectured in Lang (1960a)). Let H be an ample divisor on A.
Then the set of integral points on A(k) relative to H is finite.
In proving these results, Faltings adds the following ingredients to Vojta's
work.
1. Use of the Poincaré divisor on A x A in place of A'.
2* JJse of the product of A with itself n times instead of just two. The divisor
he considers is again a combination of divisors from each factor, together with
Poincaré divisors associated to the z-th and (i + l)st factors, 1 <> i < n.
3. A new zero estimate (the "product theorem") is needed to deal with
technical issues concerning the use of n factors.
4. Instead of proving separately an upper bound on the type of zero of the
section y at (Pi,...,P n ), he builds this upper bound into the construction of y,
using Minkowski's theorem on successive minima. Also, the use of the Gillet-Soulé
Riemann-Roch theorem is replaced by more elementary manipulations.
6. Recent Work of Bombieri
A few months ago Bombieri reworked Vojta's proof, obtaining a more elementary
proof using only the classical theory of heights, Siegel's lemma, and a few
necessary results from algebraic geometry over a field; see Bombieri (1990). In
particular, he also eliminated the use of the Gillet-Soulé Riemann-Roch theorem,
but in a manner independent of the extensions due to Faltings.
Bombieri uses the same divisor Y as in Sect. 4, but he expresses it as a
difference of two very ample divisors, as follows.
First choose a positive integer s such that
B := sFi + sF2 - Ä
is very ample; write
: C x C -» P™
for the corresponding, embedding. Likewise choose a divisor F of degree \ ox^C
and a positive integer N such that NF is very ample, giving an embedding
4>B
<ßNF : C -• IP*.
This gives another embedding
xp : C x C -> Pw x F".
It then follows from the Enriques-Severi-Zariski lemma (Zariski 1952) that for
all sufficiently large integers öi,ö2, the map
F (Pw x P 1 , ß(Su S2)) ^T(CxC,
öiNFi + Ö2NF2)
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is surjective. Thus, writing
dY =
d1NF]+d2NF2-dB,
we find that for any sections s of d Y and sf of dB, the product ssf is the restriction
to C x C of a global section on P" x P". Thus, it is possible to revert to use of
polynomials in this proof.
Since he is working with polynomials over a field, the arguments in Vojta's
proof regarding analytic torsion and cohomology overfinitefieldsare unnecessary.
Also, to obtain the upper bound on the type of zero of the section at (P\,P2),
Bombieri uses suitably chosen projections from C to P 1 ; thus he is able to use
the same lemma as Roth uses.
This latter simplification, together with the ideas of Faltings concerning use
of Siegel's lemma in place of the Gillet-Soulé Riemann-Roch theorem, remove
the obstacles to combining this proof with Roth's original proof. Thus we have
a unified proof of the number field version of Theorem 2.2.
Of course, combining the arithmetic proof with the proof in the case of
Nevanlinna theory remains a distant goal.
7. Algebraic Points of Bounded Degree
The unified proof mentioned above can be further extended to give a generalization of Theorem 2.2 which contains Wirsing's generalization, Wirsing (1971), of
Roth's theorem to algebraic points of bounded degree.
Before stating the theorem, however, let X be an arithmetic surface corresponding to C, and for all points P G C(k) let Ep denote the corresponding
Arakelov divisor on X. Then let
_
da{F}
-
(E.E+œx/B)
[fe(P):Q] '
where OJX/B is the relative dualizing sheaf.
Theorem 7.1. Let C, k, D, K, A, and e be as in Theorem 2.2. Also fix v G TL, v > 0.
Then for all points P G C(k) with [k(P) : k] < v, we have
m(D,P) + hK(P) < da(P) + ehA(P) + 0(1).
This is a weak form of the following conjecture.
Conjecture 7.2. Under the same conditions as Theorem 7.1, we have
m(D,P) + hK(P) <; d(P)+shA(P) + 0(1),
where
j(Pi
1 J
. = log|^/c(P)/Ql
"
[*(P):Q1 '
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Paul Vojta
This conjecture is very strong; in particular it implies the abc conjecture:
Conjecture 7.3 (Masser-Oesterlé). Given c > 0 there exists a constant C = C(e) > 0
such that for all relatively prime integers a, b, and c with a + b + c = 0, we have
mzx(\a\,\b\,\c\)<C-Y[p1+E.
p\abc
This conjecture, in turn, implies the "asymptotic Fermât conjecture:"
Conjecture 7.4. For all sufficiently large integers n, the only rational solutions to
the equation
Xn + Yn + Z n = 0
are the trivial ones, i.e., XYZ = 0.
8. Open Problems
In addition to the conjectures mentioned just above, the following problems are
still unresolved at this time.
1. (Lang, 1965) Replace the error term shA(P) in the arithmetic variant of
Theorem 2.2 with something sharper, e.g., (1 + s)loghA(P).
2. (Vojta, 1987) Prove a formula similar to (1) for rational points on varieties
of higher dimension. In that case, however, one would first have to exclude a
proper Zariski-closed subset Z from the set of points P. See also Schmidt (1975)
and Vojta (1987), Example 3.5.1 for a discussion of a partial answer to this
question, where the variety is projective space and the divisor D is a collection
of hyperplanes.
3. (Lang, 1960b, p. 29) Generalize Theorem 5.1 to the case where X may
contain the translate of an abelian subvariety of A. In this case the translated
abelian subvarieties may contain infinitely many rational points, but it is conjectured that the rational points are not Zariski dense, unless X itself is a translated
abelian subvariety of A. In Nevanlinna theory the analogue of this conjecture is
called Bloch's conjecture; it was proved independently by Kawamata (1980) and
Green and Griffiths (1980), both using work of Ochiai (1977). (In loc. cit., Lang
actually makes the stronger conjecture that the rational points are contained in
finitely many translated sub-abelian varieties contained in X; this conjecture is a
consequence of the above conjecture, by the Kawamata structure theorem.)
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