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V = Specm(R), then M corresponds to the R-module M =“(V, M), and if v ”! m,
then M(v) =M/mM. Note that, for any open subset U of V containing v, there is
a canonical map “(U, M) ’!M(v).
ABELIAN VARIETIES 101
Proposition 26.6. Let V be a complete geometrically connected variety over a
field k, and let M be a free sheaf of finite rank on V . For any v " V (k), the map
“(V, M) ’!M(v) is an isomorphism.
Proof. For M = OV , “(V, M) =k (the only functions regular on the whole of a
complete variety are the constant functions), and the map is the identity map k ’! k.
By assumption MH"(OV )n for some n, and so the statement is obvious.
Proposition 26.7. Let A be an abelian variety of dimension g over a field k. T he
canonical maps
“(A, &!1) ’! &!1(0), “(A, &!g) ’! &!g(0)
are isomorphisms.
Proof. By 0 we mean the zero element of A. For the proof, combine the last two
results.
Now let A be an abelian variety over a number field K, and let R be the ring
of integers in K. Recall from §20 that there is a canonical extension of A to a
smooth group scheme A over Spec R (the Néron model). The sheaf &!g of (relative)
A/R
differential g-forms on A is a locally free sheaf of OA-modules of rank 1 (it becomes
free of rank 1 when restricted to each fibre, but is not free on the whole of A). There
is a section s : Spec R ’!A whose image in each fibre is the zero element. Define
M = s"&!g . It is a locally free sheaf of rank 1 on Spec R, and it can therefore be
A/R
regarded as a projective R-module of rank 1. We have
M —"R K =&!g (0) = “(A, &!g )
A/K A/K
the first equality simply says that &!g restricted to the zero section of A and then
A/R
to the generic fibre, is equal to &!g restricted to the generic fibre, and then to the
A/R
zero section; the second equality is (26.7).
Let v be an infinite prime of K. We have to define a norm on M —"K Kv. But
M —"K Kv =“(AK , &!g ), and we can set
v
AKv /Kv
1
g
2
i
É v = É '" É .
¯
2 al
A(Kv )
al
Note that Kv = C. Now (M, ( · v)) is a normed R-module, and we define the
Faltings height of A,
H(A) =H(M).
We can make this more explicit by using the expression (26.2.1) for H(M). Choose
a holomorphic differential g-form É on A/K this will be our m. It is well-defined
up to multiplication by an element of K×. For a finite prime v, we have a N éron
differential g-form Év for A/Kv (well-defined up to multiplication by a unit in Rv),
and we have
1
H(A) =
g µ /2
v
É i
· É '" É
¯
É
al
v " v| "
2 A(Kv )
v
102 J.S. MILNE
For any infinite prime v, choose an isomorphism
al
± : Cg/› ’! A(Kv )
such that ±"(É) =dz1 '" dz2 '" . . . '" dzg; then the contribution of the prime v is
µv
2
(volume of a fundamental domain for ›) .
This is all very explicit when A is an elliptic curve. In this case, Év is the differential
corresponding to the Weierstrass minimal equation (see above, and Silverman 1986,
VII.1). There is an algorithm for finding the Faltings height of an elliptic curve, which
has surely been implemented for curves over Q (put in the coefficients; out comes the
height).
Define
1
h(A) = log H(A).
[K : Q]
If L is a finite extension of K, it is not necessarily true that h(AL) =h(A) because
the Néron minimal model may change (Weierstrass minimal equation in the case of
elliptic curves). However, if A has semistable reduction everywhere, then h(A) is
invariant under finite field extensions. We define the stable Faltings height of A,
hF (A) =h(AL)
where L is any finite field extension of K such that AL has stable reduction at all
primes of L (see 20.3).
27. The Modular Height.
Heights on projective space. (Serre, 1989, §2). Let K be a number field, and let
P =(x0 : . . . : xn) " Pn(K). The height of P is defined to be
H(P ) = max |xi|v.
0d"id"n
v
Define
1
h(P ) = log H(P ).
[K : Q]
(Warning: Serre puts the factor [K : Q] into H(P ).)
Proposition 27.1. For any number C, there are only finitely many points P of
Pn(K) with H(P ) d" C.
Note that an embedding ± : V ’! Pn of an algebraic variety into Pn defines on it a
height function, H(P ) =H(±(P )).
Proposition 27.2. Let ±1 and ±2 be two embedding of V into Pn such that
±-1(hyperplane)
1 2
±2 on V differ by a bounded amount.
In other words, given a variety V and a very ample divisor on V , we get a height
function on V (K), well defined up to a bounded function.
ABELIAN VARIETIES 103
The Siegel modular variety. For any field L, let Mg,d(L) be the set of isomorphism
classes of pairs (A, ») with A an abelian variety over L of dimension g and » a
polarization of A of degree d.
Theorem 27.3. There exists a unique algebraic variety Mg,d over C and a bijec-
tion j : Mg,d(C) ’! Mg,d(C) such that:
(a) for every point P " Mg,d, there is an open neighbourhood U of P and a family
A of polarized abelian varieties over U such that the fibre AQ represents j-1(Q)
for all Q " Mg,d;
(b) for any variety T over C, and family A of polarized abelian varieties over T of
dimension g and degree d, the map T ’! Mg,d, t ’! j(At), is regular (i.e., is a
morphism of algebraic varieties).
Proof. Uniqueness: Let (M , j ) be a second pair, and consider the map j æ%
j : Mg,d(C) ’! M (C). To prove that this is regular, it suffices to prove that it is
regular in a neighbourhood of each point P of Mg,d. But given P , we can find a
neighbourhood U of P as in (a), and condition (b) for M implies that (j æ% j)|U is
regular. Similarly, its inverse is regular.
Existence: This is difficult. Siegel constructed Mg,d as a complex manifold, and
Satake and others showed about 1958 that it was an algebraic variety. See E. Freitag,
Siegelsche Modulfunktionen, Springer, 1983.
The variety in the theorem is called the Siegel modular variety.
Example 27.4. The j-invariant defines a bijection
{elliptic curves over C}/H" = M1,1(C) ’! M1,1(C), M1,1 = A1.
(See MF §8.)
Note that the automorphisms of C act on Mg,d(C).
Let V be a variety over C, and suppose that there is given a model V0 of V over Q
(AG §9). Then the automorphisms of C act on V0(C) =V (C).
Theorem 27.5. There exists a unique model of Mg,d over Q such the bijection
j : Mg,d(C) ’! Mg,d(C) commutes with the two actions of Aut(C) noted above.
Write Mg,d again for this model. For each field L ƒ" Q, there is a well-defined map
j : Mg,d(L) ’! Mg,d(L)
that is functorial in L and is an isomorphism whenever L is algebraically closed.
Proof. This is not difficult16, given (27.3).
Example 27.6. The model of M1,1 over Q is just A1 again. The fact that j
commutes with the actions of Aut(C) simply means that, for any automorphism à of
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