Average Height Of Isogenous Abelian Varieties - Mathematisch

61 Pages·2012·0.37 MB·English

N.V.A.Aryasomayajula
Average Height Of Isogenous Abelian Varieties
Master’s thesis, defended on June 21, 2007
Thesis advisor: Dr. R.S. de Jong
Mathematisch Instituut
Universiteit Leiden
1
Contents
1 Faltings Formula for Isogenous Abelian Varieties 6
1.1 Height on Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Height on Projective Varieties . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Heights and Metrized Line Bundles . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Faltings Formula for Abelian Varieties . . . . . . . . . . . . . . . . . . . . 15
2 Average Height of Quotients of a Semi-Stable Elliptic Curve 20
2.1 Arakelov Green Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Arakelov Intersection Theory . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Average Height of Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Autissier’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Average Faltings Height of Isogenous Abelian Varieties 43
3.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Intersection Theory for Arithmetic Varieties . . . . . . . . . . . . . . . . . 46
p
3.3 Faltings Formula and the Moduli Space A . . . . . . . . . . . . . . . 50
(g;n)
3.4 Average Height Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Conclusion 58
2
Introduction
Asthetitleofthisthesissuggestswewillbestudyingtheaverage Faltingsheightformula
(which will be referred to as average height formula all throughout this thesis) of elliptic
curves and abelian varieties.
Moreprecisely, givenanelliptic curveE=K de(cid:12)nedover anumber(cid:12)eldK,letp: X
!
B bethe minimal regular model of E=K with the zero section s:B X, B = Spec(R),
!
R denoting the ring of integers of the number(cid:12)eld K, then we de(cid:12)ne the Faltings height
function of the elliptic curve E as:
1
h (E) = log#(p ! =!:R) (cid:15) log !
F [K : Q] (cid:3) X=B (cid:0) v k kv
!
vX2S1
where ! is the dualizing sheaf of the arithmetic surface X, ! a non-zero rational
X=B
section of the line bundle p ! and S denotes the set of in(cid:12)nite places of K and
(cid:3) X=B 1
(cid:15) = 1;2 according to whether K = R or C respectively.
v v (cid:24)
The Faltings height function is de(cid:12)ned analogously for abelian varieties. Let A be
K
an abelian variety de(cid:12)ned over a number (cid:12)eld K. Let (cid:25) : N(A) B be the N(cid:19)eron
!
model of A with zero section s : B N(A), where B = Spec(R), where R denotes
K
!
the ring of integers of the number (cid:12)eld K. We denote s(cid:3) g(cid:10) by ! . Then the
N(A)=R A=R
^
Faltings height of A is given as
K
1
h (A )= log#(! =R:s) (cid:15) log( s )
F K [K : Q] A=R (cid:0) v k kv
!
vX2S1
The idea of computing the average height formula has its origin in [Fa 1] where he
computes the di(cid:11)erence of height of two isogenous semi-abelian varieties. We can state
the formula for abelian varieties as:
LetA ,B betwoabelianvarietiesde(cid:12)nedoveranumber(cid:12)eldK,relatedbyanisogeny
K K
f : A B , let the map extend to f : N(A) N(B), then the following formula
K K
! !
holds:
1 1
h (B ) h (A ) = deg(f) log#(s(cid:3)! ):
F K (cid:0) F K 2 (cid:0) [K :Q] ker(f)=R
This formula will be referred to as the Faltings formula.
3
Contents
Let E=K be a semi-stable elliptic curve de(cid:12)ned over a number (cid:12)eld K. Let C denote
0
all the cyclic subgroups of order N in E: Let E denote the quotient of the elliptic curve
E by a cyclic subgroup of order N. Then the following formula known as the average
height formula for semi-stable elliptic curves holds:
1 1
0
h (E ) h (E) = logN (cid:21) ;
F F N
e (cid:0) 2 (cid:0)
N
XC (cid:16) (cid:17)
wheree denotesthenumberofcyclic subgroupsoforderN, and(cid:21) isaconstantwhich
N N
depends only on N.
Let A be a principally polarised abelian variety de(cid:12)ned over a number (cid:12)eld K with
K
good reduction at a prime p. Let G denote the isotropic subgroups of order pg in A[p].
i
Let A denote the quotient abelian variety A=G . Then the following formula which
i i
would be referred to as the average height formula for abelian varieties holds:
eg
(h (A ) h (A)) = m(g;p) logp:
F i F
(cid:0) 2 (cid:0)
XC (cid:16) (cid:17)
wherethesumrunsoverallisotropicsubgroupsoforderpg inA[p], edenotesthenumber
of isotropic subgroupsof order pg in A[p], andm(g;p) denotes a constant which depends
on g, p.
We will start with the proof of Faltings formula for isogenous abelian varieties in
chapter one. We will (cid:12)rst study the classical theory of height functions in section one
and two of chapter one and then proceed to study Faltings height function in section
three. In section four we (cid:12)rst study the concept of N(cid:19)eron model and why the Faltings
stable height remains an invariant under (cid:12)eld extension. Finally using all the concepts
studied till then we prove Faltings formula for isogenous abelian varieties.
There are at least three ways of approaching the average height formula of elliptic
curves. One approach involving basic concepts of Arakelov intersection theory due to
Robin de Jong [Ro], one due to Autissier [Au] which involves the concepts of calculating
the height on the moduli space of elliptic curves, and another approach which involves
concepts from both the methods. We develop Arakelov intersection theory in section
one and two of chapter two. In section three we give the proof of average height formula
using Robin de Jong’s approach. In section four we give an outline of the approach
adopted by Autissier, and give the third proof which makes use of results from both the
approaches.
In chapter three we prove the average height formula for abelian varieties. We have
only one way of proving the average height formula for abelian varieties. It involves the
p
properties of A , the moduli space of principally polarised abelian varieties with good
g;n
reduction at p and of type (g;n). We introduce the basic notions, like the de(cid:12)nitions
4
Contents
of abelian schemes, good reduction of abelian varieties, standard sympletic pairing in
section one. In section two we study arithmetic intersection theory of arithmetic vari-
eties. In section three we look at another reformulation of Faltings formula for isogenous
abelian varieties in terms of Cartier divisors, using the intersection theory developed in
section three. In section four we prove the average height formula using all the theory
that is developed in the (cid:12)rst three sections.
In chapter four we conclude our thesis with a few remarks on the di(cid:11)erent techniques
adopted in proving the average height formula for elliptic curves, and why that only
Autissier’s approach is naturally generalised to abelian varieties.
5
1 Faltings Formula for Isogenous Abelian
Varieties
In this chapter we study the basic notions around height functions and the formula
given by Faltings for isogenous abelian varieties.
In section 1 we look at how height functions are de(cid:12)ned on projective space. In section
2 we study how height functions are de(cid:12)ned on projective varieties. In section 3 we look
at how height functions are de(cid:12)ned via metrized line bundles and how Faltings height
function is de(cid:12)ned. In section 4 we look at the proof of Faltings formula for isogenous
abelian varieties.
We closely follow Silverman’s article ’The Theory of Height Functions’ from [AG].
1.1 Height on Projective Space
In this section we see how the height function is de(cid:12)ned over a projective space. In
the next section we extend this de(cid:12)nition to projective varieties.
Let us denote the set of places over the number (cid:12)eld K by M , the set of non-
K
archimedean places by Mf and the set of archimedean places by M1. For v
M1; a denotes (cid:28)(a) (cid:15)vKwhere (cid:28) : K C is an embedding associatedKto the place2v
K k kv j j !
and (cid:15) = 1 or 2 according to whether (cid:28) is a real or a complex embedding respectively
v
for a K. For v Mf a denotes a [Kv:Qv], where a is the usual v-adic absolute
2 2 K k kv j jv j jv
value on K;K denotes the completion of the (cid:12)eld K with respect to v, for a K.
v
2
1.1 De(cid:12)nition. A height function is de(cid:12)ned as a function from the points of a
projective space Pn(K) de(cid:12)ned over a number (cid:12)eld K to the (cid:12)eld of real numbers:
H :Pn(K) R
K
!
where H (P) = max x ;::::::::::::; x
K fk (cid:14)kv k nkvg
v2YMK
for all P Pn(K):
2
The height function does not dependon the choice of homogeneous coordinates of the
projective space Pn(K), it is well-de(cid:12)ned, butthe height function H evidently depends
K
on the number (cid:12)eld K .
6
1 Faltings Formula for Isogenous Abelian Varieties
For a (cid:12)nite extension L=K, we know that [L : K ] = [L : K], where the sum is
w v
over the places w M lying over a given v M . Hence we can relate the height
L K
2 P2
function H de(cid:12)ned over Pn(K) with H de(cid:12)ned over Pn(L):
K L
H (P) = H (P)[L:K]:
L K
We can always choose homogeneous coordinates for a point P Pn(K) with some
2
x = 1. Hence H (P) which does not depend on the choice of coordinates is always
i K
(cid:21)
1 for all points P Pn(K).
2
We saw that our height function H depends on the number (cid:12)eld K. So we now
K
de(cid:12)ne what is called the absolute height function which is independent of the (cid:12)eld of
de(cid:12)nition.
1.2 De(cid:12)nition. The absolute height function is a function from Pn(Q(cid:22)) to the (cid:12)eld
of real numbers :
H : Pn(Q(cid:22)) R;
!
H(P) = H (P)1=[K:Q]
K
where K is any number (cid:12)eld such that P Pn(Q(cid:22):).
2
EXAMPLE: Let P Pn(Q). Let P = (x : x : :::: : x ) where x Z and
(cid:14) 1 n i
2 2
gcd(x ;x ;:::;x )=1. Then
(cid:14) 1 n
H(P) = max x ; x ;::::; x :
(cid:14) 1 n
fj j j j j jg
Then it is easy to see that the set P Pn(Q) : H(P) C for some constant C is
f 2 (cid:20) g
(cid:12)nite.
If P = (x : :::: :x ) Pn(Q(cid:22)) , then we can de(cid:12)ne Q(P) as
(cid:14) n
2
Q(P)= Q(x =x ::::: : x =x ) for some x = 0
(cid:14) i n i i
6
Now we state a very important property of the absolute height function H.
Theorem 1.3 (Finiteness Theorem) Let C and d be constants. Then
S = P Pn(Q(cid:22)): H(P) C and [Q(P) :Q] d
2 (cid:20) (cid:20)
is a (cid:12)nite set. (cid:8) (cid:9)
Proof. We (cid:12)rst prove that the set
S0 = P P1(Q(cid:22)) :H(P) C0 and [Q(x) :Q]= d0
2 (cid:20)
n o
7
1 Faltings Formula for Isogenous Abelian Varieties
is (cid:12)nite.
Let P0 P1(Q(cid:22)) and[Q(x): Q]= d. Let x1;::::;xd be the conjugates of x over Q and let
2
1 = s ,.....s be the elementary symmetric polynomials in x1;:::::;xd. Then each s is in
(cid:14) d j
Q,and x is a root of the polynomial.
d d
F(X) = ( 1)js Xd(cid:0)j = (X x(i)) Q[X]:
j
(cid:0) (cid:0) 2
j=0 i=1
X Y
Now using the triangle inequality , one easily checks that H([1;s ]) c H([1;x])j for
j j
(cid:20)
certain constants c which do not depend on x. Hence we can see that there are (cid:12)nitely
j
manysetsof s ’s, hence(cid:12)nitelymanypossibilitiesforthepolynomialF(X), andsoonly
j
(cid:12)nitely many possibilities for x.
Now choose homogeneous coordinates for P = (x :::: : x ) with some x = 1. Let us
(cid:14) n i
denote Q(P) by K. Then
H (P) = max x ;::::: x maxH (x )
K f k (cid:14)kv k nkvg(cid:21) j K j
v2YMK
Hence H(P) H((1 : x )) for all j. Now P0=[1,x ] Pn(Q(cid:22)) with H(P0) C , and
j j
(cid:21) 2 (cid:20)
[Q(x) : Q] d. But from the above argument there are only (cid:12)nitely many possibilities
(cid:20)
for such an x . Hence the set S is (cid:12)nite. (cid:3)
i
In practice the logarithm of the absolute function is more widely used. Hence
h(P) = logH(P)
will be referred to as the ’height function’ from now on.
1.2 Height on Projective Varieties
In the last section we saw how the height function is de(cid:12)ned over projective spaces.
In this section we see how height functions are de(cid:12)ned on projective varieties de(cid:12)ned
over Q(cid:22).
In order to de(cid:12)ne a height function on V, a projective variety of dimension n over
Q(cid:22), we take a map from V into projective space and use the height function from the
previous section.
2.1 De(cid:12)nition. Let F : V Pn be a morphism. The (logarithmic) height on V
!
relative to F is de(cid:12)ned by
hF : V(Q(cid:22)) R; hF(P) = h(F(P)):
!
8
1 Faltings Formula for Isogenous Abelian Varieties
It is well-known that any morphism from the projective variety V, F : V Pn into
!
the projective space Pn is associated to aninvertible sheaf(or line bundle)on V, namely
the pull-back of the twisting sheaf, F(cid:3)O (1).
P
In fact we know that if X is any scheme over A, and (cid:30) : X Pn an A-morphism of
! A
X to Pn, then ‘ = (cid:30)(cid:3)(O(1)) is an invertible sheaf on X, and the global sections s ,
A (cid:14)
s ,.....,s where s =(cid:30)(cid:3)(x ) generate the sheaf ‘. Conversely any invertible sheaf ‘ and
1 n i i
global generating sections s determine a morphism (cid:30) : X Pn. This statement has
i ! A
been proved in [Ha] chapter two proposition 7.1. Naturally many di(cid:11)erent maps give
rise to the same sheaf. Hence we can expect the height function to be essentially same
for the morphisms which determine the same sheaf. The following proposition says that
they are essentially the same.
0
2.2 De(cid:12)nition. Two height functions h and h de(cid:12)ned on a projective variety V are
said to be equivalent if h h0 is bounded as P ranges over V(Q(cid:22)).
(cid:0)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
2.3 Theorem. Let
F :V PnandG : V Pm
! !
be two maps of V such that F(cid:3)OPn(1) (cid:24)= G(cid:3)OPm(1). Then hF and hG are equivalent.
Proof. Let E be a divisor in the linear system of ‘. (That is E 0 and ‘ (E)).
V
(cid:21) (cid:25)O
Then on the complement of E, we can write F = [f ;f ;:::f ] and G = [g ;:::g ] with
(cid:14) 1 n (cid:14) n
rational functions f and g such that
i j
(f ) and (g ) such that
i j
0 0
(f ) =D E and (g ) = D E for divisors D D 0:
i i (cid:0) j j (cid:0) i j (cid:21)
We are guarenteed of such a divisors from arguments from [Ha] II.7.8.1.
We know that theF hasnobasepoints onV whichmeansthat the D ’s have nopoint
i
in common. Let K be a common (cid:12)eld of de(cid:12)nition for V, f , f ,...,f and g ,....,g .
(cid:14) 1 n (cid:14) m
Nowpickanyj andlookattheideal = (f =g ;::::::f =g )inthering =K[f =g ;::::::f =g ].
(cid:14) j n j (cid:14) j n j
0 I R
Since (f /g )=D -D and the D ’s have no point in common, it follows that I is the unit
i j i j i
ideal. Suppose not, then there will be a maximal ideal of containing . Since
M R 0I
Spec(R) is isomorphic to an open subset of V containing the complement of D , will
j M
0
correspond to a point P of V not in D such that (f =g )(P) = 0 for all i. But then P
j i j
will lie in the support of all D ,yielding a contradiction.
i
Hence we can (cid:12)nda polynomial (cid:30) (T ;::::::T ) K[T ;:::T ] having no constant term
j (cid:14) n (cid:14) n
2
such that
(cid:30) (f =g ;::::::f =g ) = 1
j (cid:14) j n j
(cid:14)
9
1 Faltings Formula for Isogenous Abelian Varieties
Taking the v-adic absolute value and using the triangle inequality, one easily (cid:12)nds a
0
constant C =C (v;F;G;(cid:30) ) 0 such that for all P in the complement of D ,
1 1 j (cid:21) j
max f =g (P) ;::::: f =g (P) C
j (cid:14) j jv j n j jv (cid:21) 1
We can choose C = 1 for all but (cid:12)nitely many v, independent of P. (Note that it may
1 (cid:8) (cid:9)
be necessary to extend K so that P(K).)
Next multiply through by g (P) . Then the equality also holds for g (P) = 0, so
j v j
j j
taking the maximum over j yields:
max f ;::::: f C max g (P) ;::::: g (P)
fj (cid:14)jv j njvg(cid:21) 2 fj (cid:14) jv j m jvg
for a constant C =C (v,F,G) 0, where P ranges over the complement of E and C =1
2 2 2
(cid:21)
for all but (cid:12)nitely many v. Now raise to the [K : Q ] power, multiply over all v M
v v k
2
and take the [K :Q] th root. This gives
H(F(P)) C H(G(P));
3
(cid:21)
with C =C (F;G) >0 , as P ranges over the complement of E in V.
3 3
Next , since ‘ has no base points , we can choose (cid:12)nitely many divisors E ,...,E in the
1 r
linear system for ‘ so that the E ’s have trivial intersection. In this way we obtain the
i
above inequality on all of V. Taking logarithms gives one of the desired bounds, and
the others followed by symmetry. (cid:3)
Now that we have the desired equivalence we try to relate Pic(V), the group of equiv-
alence classes of invertible sheaves and the group of height functions modulo constant
functions .
Let us denote the group of functions h :V R mod O(1) by (V).
f ! g H
2.4 De(cid:12)nition. Let ‘ be a sheaf without base points on V. The height function
associated to‘istheclassoffunctionsh‘ (V)obtainedbytaking theheight function
2 H
hF for any map F associated to ‘. (From theorem 1.2 h‘ is well de(cid:12)ned.)
Proposition 2.5 Let ‘ and be base point-free sheaves on V. Then h‘(cid:10)M and
M
h‘+hM are equivalent.
Proof. Let F = [f ;::::f ] and [g ;::::g ] be maps associated to ‘ and respectively.
(cid:14) n (cid:14) m
(cid:0) (cid:1) M
Then
T =[::::;f g ::::] :V Pnm+n+m:
i j 0(cid:20)i(cid:20)n;0(cid:20)j(cid:20)m !
is the map associated to the invertible sheaf ‘ . This map is obtained by composing
(cid:10)M
the diagonal morphism D : V V V with the composition of the map F:V V
! (cid:2) (cid:2) !
Pn Pm with the Segre embedding S : Pn Pm Pnm+n+m:
(cid:2) (cid:2) !
T = S (F G) D
(cid:14) (cid:2) (cid:14)
Since max ::; f g :: = max ::; f :: max ::; g ::
j i jjv f j ijv g j jjv
we have
(cid:8) (cid:9) (cid:8) (cid:9)
hT(p)= hF(P) +hG(P)+O(1) for all P in V (cid:3)
10