Milne. .Abelian.Varieties.(1998).[sharethefiles.com]

[ Pobierz całość w formacie PDF ]

h(A) to A, called the Faltings height of A. The Faltings heights of two isogenous
abelian varieties are related, and Faltings proved:
Theorem 19.6. Let A be an abelian variety with semistable reduction over a num-
ber field k. T he set
{h(B)| B is isogenous to A}
is finite.
There is natural notion of the height of a point in Pn(k), namely, if P =(a0 : � � � :
an), then
H(P ) =�v max(|ai|v).
i
Here the v s run through all primes of k (including the archimedean primes) and | � |v
denotes the normalized valuation corresponding to v. N ote that
�v max(|cai|v) =(�v max(|ai|v))(�v |c|v) =�v max(|ai|v)
i i i
because the product formula shows that �v |c|v = 1. Therefore H(P ) is independent
of the choice of a representative for P . When k = Q , we can represent P by an
n-tuple (a0 : ... : an) with the ai relatively prime integers. Then maxi(|ai|p) =1 for
all prime numbers p, and so the formula for the height becomes
H(P ) =max |ai| (usual absolute value).
i
A fundamental property of heights is that, for any integer N,
Card{P " Pn(k) | H(P ) d" N}
is finite. When k = Q, this is obvious.
Using heights on projective space, it is possible to attach another height to an
abelian variety. There is a variety V (the Siegel modular variety) over Q that
parametrizes isomorphism classes of principally polarized abelian varieties of a fixed
dimension g. It has a canonical class of embeddings into projective space
V �! Pn.
An abelian variety A over k corresponds to a point v(A) in V (k), and we define the
modular height of A to be
H(A) =H(v(A)).
We know that the set of isomorphism classes of principally polarized abelian varieties
over k of fixed dimension and bounded modular height is finite.
Note that if we ignore the principally polarized in the last statement, and the
semistable in the last theorem, then they will imply Finiteness I once we relate
ABELIAN VARIETIES 75
the two notions of height. Both heights are continuous functions on the Siegel
modular variety, which has a canonical compactification. If the difference of the two
functions h and H extended to the compact variety, then it would be bounded, and
we would have proved Finiteness I. Unfortunately, the proof is not that easy, and the
hardest part of Faltings s paper is the study of the singularities of the functions as
they approach the boundary. One thing that makes this especially difficult is that,
in order to control the contributions at the finite primes, this has to be done over Z,
i.e., one has to work with a compactification of the Siegel modular scheme over Z.
References. The original source is:
Faltings, G., Endlichkeitss�tze f�r Abelsche Variet�ten �ber Zahlk�rpern, Invent.
Math. 73 (1983), 349-366; Erratum, ibid. 1984, 75, p381. (There is a translation:
Finiteness Theorems for Abelian Varieties over Number Fields, in Arithmetic Ge-
ometry pp 9 27.)
Mathematically, this is a wonderful paper; unfortunately, the exposition, as in all
of Faltings s papers, is poor.
The following books contain background material for the proof:
Serre: Lectures on the Mordell-Weil theorem, Vieweg, 1989.
Arithmetic Geometry (ed. Cornell and Silverman), Springer, 1986 (cited as Arith-
metic Geometry).
There are two published seminars expanding on the paper:
Faltings, G., Grunewald, F., Schappacher, N., Stuhler, U., and W�stholz, G., Ra-
tional Points (Seminar Bonn/Wuppertal 1983/84), Vieweg 1984.
Szpiro, L., et al. S�minaire sur les Pinceaux Arithm�tique: La Conjecture de
Mordell, Ast�risque 127, 1985.
Although it is sketchy in some parts, the first is the best introduction to Faltings s
paper. In the second seminar, the proofs are very reliable and complete, and they
improve many of the results, but the seminar is very difficult to read.
There are two Bourbaki talks:
Szpiro, L., La Conjecture de Mordell, S�minaire Bourbaki, 1983/84.
Deligne, P., Preuve des conjectures de Tate et Shafarevitch, ibid.
There is a summary of part of the theory in:
Lang, S., Number Theory III, Springer, 1991, Chapter IV.
Faltings s proofs depend heavily on the theory of N�ron models of abelian varieties
and the compactification of Siegel modular varieties over Z. Recently books have
appeared on these two topics:
Bosch, S., L�tkebohmert, W., and Raynaud, M., N�ron Models, Springer, 1990.
Chai, Ching-Li and Faltings, G., Degeneration of Abelian Varieties, Springer, 1990.
76 J.S. MILNE
20. N�ron models; Semistable Reduction
Let R be a discrete valuation ring with field of fractions K and residue field k. Let
� be a prime element of R, so that k = R/(�). We wish to study the reduction14
of an elliptic curve E over K. For simplicity, we assume p =2, 3. Then E can be
described by an equation
df
2
Y = X3 + aX + b, " =4a3 +27b2 =0.
By making the substitutions X �! X/c2, Y �! Y/c3, we can transform the equation
to
2
Y = X3 + ac4X + bc6,
and this is essentially the only way we can transform the equation. A minimal equation
for E is an equation of this form with a, b " R for which ord(") is a minimum. A
minimal equation is unique up to a transformation of the form
(a, b) �! (ac4, bc6), c " R�.
Choose a minimal equation for E, and let E0 be the curve over k defined by the
equation mod (�). There are three cases:
(a) E0 is nonsingular, and is therefore is an elliptic curve. This occurs when
ord(") = 0. In this case, we say that E has good reduction.
(b) E0 has a node. This occurs when �|" but does not divide both a and b. In this
df
case E0(k)nonsing = E0(k)-{node} is isomorphic to k� as an algebraic group (or
becomes so after a quadratic extension of k), and E is said to have multiplicative
reduction.
(c) E0 has a cusp. This occurs when � divides both a and b (and hence also ").
In this case E0(k)nonsing is isomorphic to k+, and E is said to have additive
reduction.
The curve E is said to have semistable reduction when either (a) or (b) occurs.
Now suppose we extend the field from K to L, [L : K]
valuation ring S with field of fractions L such that S )" K = R. When we pass from
K to L, the minimal equation of E remains minimal in cases (a) and (b), but it may
change in case (c). For a suitable choice of L, case (c) will become either case (a) or
case (b). In other words, if E has good reduction (or multiplicative) reduction over
K, then the reduction stays good (or multiplicative) over every finite extension L;
if E has additive reduction, then the reduction can stay additive or it may become
good or multiplicative over an extension L, and for a suitable extension it will become [ Pobierz całość w formacie PDF ]

Pokrewne