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306 15. Elliptic Curves over Global Fields and -Adic Representations

Faltings is part of his proof of the Mordell conjecture, which he uses in the proofs
of his basic finiteness theorems on abelian varieties and curves of higher genus. One
such result that Faltings generalizes is the following.
ˇ
(7.2) Theorem (Safareviˇ c). Let S be a finite set of places of a number field K. The
number of isomorphism classes of elliptic curves over K with good reduction at all
v outside S is finite.
We give an argument due to Tate which reduces the result to Siegel’s theorem on
the finiteness of integral points on curves which was mentioned in the Introduction.
Faltings proves two generalizations to abelian varieties, one of which was conjec-
ˇ
tured by Safareviˇ c on the finiteness of isogeny classes of abelian varieties with good
reduction outside a finite set S.
Returning to the reduction of the assertion of (7.2) to Siegel’s theorem, we first
extend S if necessary to include all places dividing 2 and 3 and so that R, the ring
of S-integers in K, is principal. Recall that c in K is an element of R if and only if
v(c) = 0 for all v outside S. For each E over K we can, by (1.3), choose a minimal
model of the form
2 3
y = 4x − ux − v
3
2
∗
with E = u −27v in R , the group of units in the ring R.If E has good reduction
outside S, then v( E ) = 0 for all places outside of S. Now, for a given E, the class
∗ 12
E in R /(R ) is well defined by the isomorphism class of E from 3(3.2). This
∗
∗ 12
∗
group R /(R ) is finite by the Dirichlet unit theorem for number fields. Thus there
is a finite set D in R such that any curve E over K with good reduction outside of
∗
2
2
3
3
S can be written in the form y = 4x − ux − v with d = u − 27v in D.
For a given d, the affine elliptic curve
3
2
27v = u − d
2
has only a finite number of integral points (u,v) on it, i.e., points (u,v) in R . This
is a generalization of a theorem of Siegel, see Lang [1962, Chapter VII].
We close with the statements of Faltings’ two generalizations of (7.2) to abelian
varieties. For the proofs, see his basic paper Endlichkeitss¨ atze f¨ ur abelsche Variet¨ aten
¨ uber Zahlk¨ orpern [Faltings, 1983].
(7.3) Theorem (Satz 5). Let S be a finite set of places in an algebraic number field
K. In a given dimension, there are only finitely many isogeny classes of abelian
varieties over K with good reduction outside S.
(7.4) Theorem (Satz 6). Let S be a finite set of places in an algebraic number field
K and d > 0. In a given dimension n there are only finitely many isomorphism
classes of d-fold polarized abelian varieties over K with good reduction outside S.