On a Hasse principle for Mordell-Weil groups

0.13 MB·English

by  Grzegorz Banaszak

ON A HASSE PRINCIPLE FOR MORDELL-WEIL GROUPS
Grzegorz Banaszak
8
0
0
Abstract. In this paper we establish a Hasse principle concerning the linear de-
2
pendenceoverZof nontorsionpoints inthe Mordell-Weilgroup of an abelianvariety

n
over a number field.
a
J

7

] 1. Introduction.
T
N
Let A be an abelian variety over a number field F. Let v be a prime of O and
F
.
h let k := O /v. Let A denote the reduction of A for a prime v of good reduction
v F v
t
a and let
m
r : A(F) → A (k )
v v v
[

2 be the reduction map. Put R := EndF(A). Let Λ be a subgroup of A(F) and
v let P ∈ A(F). A natural question arises whether the condition rv(P) ∈ rv(Λ) for
4 almost all primes v of O implies that P ∈ Λ. This question was posed by W.
F
0
Gajda in 2002. The main result of this paper is the following theorem.
7
3
. Theorem 1.1. Let P1,...,Pr be elements of A(F) linearly independent over R.
2
1 Let P be a point of A(F) such that RP is a free R module. The following conditions
7 are equivalent:
0
: (1) P ∈ r ZP
v Pi=1 i
Xi (2) rv(P) ∈ Pri=1 Zrv(Pi) for almost all primes v of OF.
r
a ×
In the case of the multiplicative group F the problem analogous to W. Gajda’s
question has already been solved by 1975. Namely, A. Schinzel, [Sch, Theorem
× ×
2, p. 398], proved that for any γ ,...,γ ∈ F and β ∈ F such that β =
1 r
r γnv,i mod v for some n ,...,n ∈ Z for almost all primes v of O there
Qi=1 i i,v r,v F
are n ,...,n ∈ Z such that β = r γni. Theorem of A. Schinzel was proved
1 r Qi=1 i
again by Ch. Khare [Kh] using methods of C. Corralez-Rodriga´n˜ez and R. Schoof
[C-RS]. Ch. Khare used this theorem to prove that every family of one dimensional
strictly compatible l-adic representations comes from a Hecke character.
Theorem 1.1 strengthens the results of [BGK2], [GG] and [We]. Namely T. Weston
[We] obtainedan analogueofTheorem 1.1withcoefficients in Zfor Rcommutative.
T. Weston did not assume that P ,...,P is a basis over R, however there was
1 r
Typeset by AMS-TEX
1
2 GRZEGORZ BANASZAK
some torsion ambiguity in the statement of his result. In [BGK2] together with
W. Gajda and P. Krason´ we proved Theorem 1.1 for elliptic curves without CM
and more generally for a class of abelian varieties with End (A) = Z. We also got
F
a general result for all abelian varieties [BGK2] Theorem 2.9 in the direction of
Theorem 1.1. However in Theorem 2.9 loc. cit. the coefficients are in R and there
is also a coefficient in N associated with the point P. Recently W. Gajda and K.
G´ornisiewicz [GG] Theorem 5.1, strengthened [BGK2] Theorem 2.9 implementing
some techniques of M. Larsen and R. Schoof [LS] and showing that the coefficient
associatedwiththepoint P in[BGK2]Theorem 2.9isequal to1. The coefficients in
[GG] Theorem 5.1 are still in R. Very recently A. Perucca has proven the Theorem
5.1 of [GG] (see [Pe] Corollary 5.2) using her l-adic support problem result, see loc.
cit. At the end of this paper we prove Theorem 5.1 of [GG] by some of our methods
from the proof of Theorem 1.1.
Although not explicitely presented in our proofs, this paper makes an essential
use of results on Kummer Theory for abelian varieties, originally developed by K.
Ribet [Ri], and results of G. Faltings [Fa], J-P. Serre and J. Tate [ST], A. Weil [W],
J. Zarhin [Za] and other important results about abelian varieties. The application
of these results comes by quoting some results of [BGK1], [BGK2] and [Bar] where
Kummer Theory and results of G. Faltings, J-P. Serre, J. Tate, A. Weil and J.
Zarhin where key ingredients. This form of the exposition of our paper makes the
proofs of our results concise and transparent for the reader.
2. Proof of Theorem 1.1. Let L/F be an extension of number fields and let w
denote a prime ideal in O over a prime v of good reduction, such that v does not
L
divide l. It follows by [BGK1], Lemma 2.13 that the reduction map
r : A(L) → A (k )
w l w w l
is injective, where G denotes the l-torsion part of an abelian group G. The main
l
ingredients in the proof of the Lemma 2.13 loc. cit. are [ST], Theorem 1 and Weil
conjectures for abelian varieties, proven by Weil [W]. For additional information
about the injectivity of the reduction map
r : A(L) → A (k )
w tor w w
seealso[K]p. 501-502. Wewilluseseveraltimesinthispaperthefollowingresultof
S. Baran´czuk which is a refinement of the Theorem 3.1 of [BGK2] and Proposition
2.2 of [BGK3].
Lemma 2.1. ([Bar], Th. 5.1) Let l be a prime number. Let m ,...,m ∈ N∪{0}
1 s
and let m := max{m ,...,m }. Let L/F be a finite extension and let Q ,...,Q ∈
1 s 1 s
A(L) be independent over R. There is a family of primes w of O of positive density
L
such that r (Q ) has order lmi in A (k ) for all 1 ≤ i ≤ s,
w i w w l
ON A HASSE PRINCIPLE 3
Corollary 2.2. Let m ∈ N. Let Q ,...,Q ∈ A(F) be independent over R and
1 s
let T ,...T ∈ A[lm]. Let L := F(A[lm]). There is a family of primes w of O of
1 s L
positive density such that for the prime v of O below w :
F
(1) r (T ),...,r (T ) ∈ A (k ),
w 1 w s v v
(2) r (Q ) = r (T ) in A (k ) for all 1 ≤ i ≤ s.
v i w i v v l
Proof. Observe that the points Q −T ,...,Q −T are linearly independent over
1 1 s s
R in A(L). Indeed if:
s
Xβi(Qi −Ti) = 0
i=1
for some β ,...,β ∈ R, then
1 s
s
XlmβiQi = 0.
i=1
Hence lmβ = 0 for all 1 ≤ i ≤ s. Since R is a torsion free abelian group it
i
shows that β = ··· = β = 0. It follows by Lemma 2.1 that there is a family of
1 s
primes w of O of positive density such that r (Q − T ) = 0 in A (k ) . Since
L w i i w w l
Q ,...,Q ∈ A(F), it follows that r (Q −T ) = r (Q )−r (T ) = r (Q )−r (T )
1 s w i i w i w i v i w i
for the prime v of O below w. Hence we get r (T ) = r (Q ) ∈ A (k ) for all
F w i v i v v l
1 ≤ i ≤ s.
Proof Theorem 1.1. It is enough to prove that (2) implies (1). By Theorem 2.9
[BGK2] there is an a ∈ N and elements α ,...,α ∈ R such that:
1 r
r
(2.3) aP = X αiPi.
i=1
Step 1. Assume that α ∈ Z for all 1 ≤ i ≤ r. We will show (cf. the proof of
i
Theorem 3.12 of [BGK2]) that P ∈ r ZP . Let lk be the largest power of l that
Pi=1 i
divides a. Lemma 2.1 shows that for any 1 ≤ i ≤ r there are infinitely many primes
v such that rv(P1) = ··· = rv(Pi−1) = rv(Pi+1) = ··· = rv(Pr) = 0 and rv(Pi)
has order equal to lk in A (k ) . By (2.3) we get ar (P) = α r (P ) Moreover by
v v l v i v i
assumption (2) of the theorem, r (P) = β r (P ) for some β ∈ Z. Hence
v i v i i
(α −aβ )r (P ) = 0
i i v i
in A (k ) . This implies that lk divides α for all 1 ≤ i ≤ r. So By (2.3) we get
v v l i
r
a α
i
(2.4) lkP = X lk Pi +T,
i=1
for some T ∈ A(F)[lk]. Again, by Lemma 2.1 there are infinitely many primes v
in O such that r (P ) = 0 in A (k ) for all 1 ≤ i ≤ r. In addition r (P) ∈
F v i v v l v
4 GRZEGORZ BANASZAK
r Zr (P ) for almost all v. So (2.4) implies that r (T) = 0, for infinitely many
Pi=1 v i v
primes v. This contradicts the injectivity of r , unless T = 0. Hence
v
r
a α
i
(2.5) lkP = X lk Pi.
i=1
Repeating the above argument for primes dividing a shows that condition (1)
lk
holds.
Step 2. Assume α ∈/ Z for some i. Observe that α is an endomorphism of the
i i
Riemann lattice L, such that A(C) ∼= Cg/L. To make the notation simple, we
will denote again by α the endomorphism α ⊗ 1 acting on T (A) ∼= L ⊗ Z . Let
i i l l
P(t) := det(tIdL − αi) ∈ Z[t], be the characteristic polynomial of αi acting on
L. Let K be the splitting field of P(t) over Q. We take l such that it splits in
K and l does not divide primes of bad reduction. Since P(t) has all roots on
O and is also the characteristic polynomial of α on T (A), we see that P(t) has
K i l
all roots in Z by the assumption on l. If P(t) has at least two different roots
l
in O , we easily find a vector u ∈ T (A) which is not an eigenvector of α on
K l i
T (A). If P(t) has a single root λ ∈ O then P(t) = (t − λ)2g and we must
l K
have λ ∈ Z because we are in characteristic 0. Hence P(t) = (t − λ)2g is the
characteristic polynomial of α as an endomorphism of L. Since α ∈/ Z we find
i i
easily u ∈ L such that u is not an eigenvector of α acting on T (A). In any case
i l
there is u ∈ T (A) which is not an eigenvector of α acting on T (A). Rescaling if
l i l
necessary, we can assume that u is not divisible by l in T (A). Hence for m ∈ N
l
and m big enough we can see that the coset u+lmT (A) is not an eigenvector of
l
α acting on T (A)/lmT (A). Indeed, if α u ≡ c u mod lmT (A) for c ∈ Z/lm for
i l l i m l m
each m ∈ N, then c u ≡ c u mod lmT (A). Because u is not divisible by l in
m+1 m l
T (A), this implies that c ≡ c mod lm for each m ∈ N. But this contradicts
l m+1 m
the fact that u is not an eigenvector of α acting on T (A). Consider the natural
i l
isomorphism of Galois and R modules T (A)/lmT (A) ∼= A[lm]. We put T ∈ A[lm]
l l
to be the image of the coset u+lmT (A) via this isomorphism. Put L := F(A[lm]).
l
By Corollary 2.2 we choose a prime v below a prime w of O such that
L
(i) r (T) ∈ A (k ) ,
w v v l
(ii) r (P ) = 0 for all j 6= i and r (P ) = r (T) in A (k ) .
v j v i w v v l
From (2.3) and (ii) we get ar (P) = α r (P ) = α r (T) in A (k ) . Hence for
v i v i i w v v l
the prime w in O over v we get in A (k ) the following equality:
L w w l
(2.4) ar (P) = α r (P ) = α r (T)
w i w i i w
By assumption (2) and (ii) there is d ∈ Z, such that ar (P) = adr (P ) =
v v i
adr (T) in A (k ) . Hence, for the prime w in O over v, we get in A (k )
w v v l L w w l
the following equality:
(2.5) ar (P) = adr (P ) = adr (T).
w w i w
ON A HASSE PRINCIPLE 5
Since r is injective, the equalities (2.4) and (2.5) give:
w
α T = adT in A[lm].
i
But this contradicts the fact that T is not an eigenvector of α acting on A[lm]. It
i
proves that α ∈ Z for all 1 ≤ i ≤ r, but this case has been taken care of already in
i
step 1 of this proof. (cid:3)
Corollary 2.6. Let A be a simple abelian variety. Let P ,...,P be elements
1 r
of A(F) linearly independent over R. Let P be a nontorsion point of A(F). The
following conditions are equivalent:
(1) P ∈ r ZP
Pi=1 i
(2) r (P) ∈ r Zr (P ) for almost all primes v of O .
v Pi=1 v i F
Proof. This is an immediate consequence of Theorem 1.1. Indeed, for a nontorsion
point P the R-module RP is a free R-module since D = R ⊗Z Q is a division
algebra because A is simple. (cid:3)
Corollary 2.7. Let A be a simple abelian variety. Let P and Q be nontorsion
elements of A(F). The following conditions are equivalent:
(1) P = mQ for some m ∈ Z,
(2) r (P) = m r (Q) for some m ∈ Z for almost all primes v of O .
v v v v F
Proof. This is an immediate consequence of Corollary 2.6 because RP and RQ are
free R-modules since A is simple. (cid:3)
The following proposition is the Theorem 5.1 of [GG]. We give here a new proof of
this theorem using some methods of the proof of Theorem 1.1.
Proposition 2.8. Let A be an abelian variety over F. Let P ,...,P be elements
1 r
of A(F) linearly independent over R. Let P be a point of A(F) such that RP is a
free R module. The following conditions are equivalent:
r
(1) P ∈ RP
Pi=1 i
r
(2) r (P) ∈ Rr (P ) for almost all primes v of O .
v Pi=1 v i F
Proof. Again we need to prove that (2) implies (1). Let us assume (2). By [BGK2],
Theorem 2.9 there is an a ∈ N and elements α ,...,α ∈ R such that equality (2.3)
1 r
holds. Let l be a prime number such that lk||a for some k > 0. Put L := F(A[lk])
and take arbitrary T ∈ A[lk]. By Corollary 2.2 we can choose a prime v below a
prime w of O such that
L
(i) r (T) ∈ A (k ) ,
w v v l
(ii) r (P ) = 0 for all j 6= i and r (P ) = r (T) in A (k ) .
v j v i w v v l
6 GRZEGORZ BANASZAK
From (2.3) and (ii) we get ar (P) = α r (P ) = α r (T) in A (k ) . Hence we
v i v i i w v v l
get the following equality in A (k ) :
w w l
(2.9) ar (P) = α r (P ) = α r (T)
w i w i i w
By assumption (2) and (ii) there is δ ∈ R, such that ar (P) = aδr (P ) =
v v i
aδr (T) = 0 in A (k ) . Hence we get the following equality in A (k ) :
w v v l w w l
(2.10) ar (P) = aδr (P ) = aδr (T) = 0.
w w i w
By injecivity of r , the equalities (2.9) and (2.10) imply:
w
α T = 0 in A[lk].
i
This shows that α maps to zero in End (A[lk]). By [Za] Corollary 5.4.5, cf. the
i G
F
proof of Lemma 2.2 of [BGK2], we have a natural isomorphism:
R/lkR ∼= End (A[lk]).
G
F
Hence we proved that α ∈ lkR for all 1 ≤ i ≤ r. So
i
r
a
′
(2.11) lkP = XβiPi +T ,
i=1
where β ∈ R for all 1 ≤ i ≤ r and T′ ∈ A(F)[lk]. By Lemma 2.1 there are infinitely
i
many primes v in O such that r (P ) = 0 in A (k ) for all 1 ≤ i ≤ r. In addition
F v i v v l
r ′
r (P) ∈ Rr (P ) for almost all v. So (2.11) implies that r (T ) = 0, for
v Pi=1 v i v
′
infinitely many primes v. Hence T = 0 by the injectivity of r (see [BGK1] Lemma
v
2.13). Hence
r
a
(2.12) P = XβiPi.
lk
i=1
Repeating the above argument for primes dividing a finishes the proof of the
lk
proposition. (cid:3)
3. Remark on Mordell-Weil R systems. Let R be a ring with identity. In
the paper [BGK1] the Mordell-Weil R systems have been defined. In [BGK2] we
investigated Mordell-Weil R systems satisfying certain natural axioms A − A
1 3
and B − B . We also assumed that R was a free Z-module. Let us consider
1 4
Mordell-Weil R systems which are associated to families of l-adic representations
ρ : G → GL(T ) such that ρ (G ) contains an open image of homotheties. Since
l F l l F
Theorem 2.9 of [BGK2] and Theorem 5.1 of [Bar] were proven for Mordell-Weil
R systems, then Proposition 2.8 and its proof generalize for the Mordell-Weil R
ON A HASSE PRINCIPLE 7
systems. This shows that Theorem 2.9 of [BGK2], which is stated for Mordell-Weil
R systems, holds with a = 1. Let us also assume that there is a free Z-module
L such that R ⊂ EndZ(L) and for each l there is an isomorphism L ⊗ Zl ∼= Tl
such that the action of R on T comes from its action on L. Abelian varieties are
l
principal examples of Mordell-Weil R systems satisfying all the requirements stated
above with R = End (A). Then Theorem 1.1 generalizes also for Mordell-Weil R
F
systems satisfying the above assumptions because we can apply again Theorem 2.9
of [BGK2] and Theorem 5.1 of [Bar].
Acknowledgements: The research was partially financed by the research grant N
N201 1739 33 of the Polish Ministry of Science and Education and by the Marie
Curie Research Training Network ”Arithmetic Algebraic Geometry” MRTN-CT-
2003-504917.
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8 GRZEGORZ BANASZAK
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Department of Mathematics, Adam Mickiewicz University, Poznan´, Poland
E-mail address: [email protected]