Congruences and rationality of Stark-Heegner points Marco Adamo Seveso University of Milan Federigo Enriques department of Mathematics
[email protected] Address: Via delle Orchidee 14, Milan, Italy January 16, 2012 Abstract Let A/Q be a modular abelian variety attached to a weight 2 new modular form of level N=pM, where p is a prime and M is an integer prime to p. When K/Q is an imaginary quadratic extension the Heegner points, that are de…ned over the ring class …elds H/K, can contribute to the growth of the rank of the Selmer groups over H. When K/Q is a real quadratic …eld the theory of Stark-Heegner points provides a conjectural explanation of the growth of these ranks under suitable sign conditions on the L-function of f/K. The main result of the paper relates the growth of the Selmer groups to the conjectured rationality of the Stark-Heegner points over the expected …eld of de…nition. Keywords: Stark-Heegner points, Selmer groups, congruences between modular forms, L-functions. Subjects: Number theory MSC classi…cation: 14G05, 11G05 (Primary) 11F03, 11F67, 11F33 (Secondary)
Contents 1 Introduction
2
2 Canonical …nite subgroups attached to isogenies between abelian varieties 5 3 Descent of abelian varieties 3.1 The I-descent over a local …eld, when A has split purely toric reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Control assumptions for the I-Selmer group . . . . . . . . . . . .
1
8 9 11
4 Congruences and growth of the Selmer groups 12 4.1 Selmer structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Proof of the main result of the section . . . . . . . . . . . . . . . 16 5 Geometric lowering the level 21 5.1 Abstract setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2 Application to modular abelian varieties . . . . . . . . . . . . . . 25 6 On the local l-divisibility of Stark-Heegner points 28 6.1 The localization map . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.2 Local divisibility of Stark-Heegner points . . . . . . . . . . . . . 31 6.2.1 Stark-Heegner points and Shimura reciprocity law . . . . 31 6.2.2 Stark-Heegner points and special values of complex Lfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 7 Lifting Stark-Heegner points to Selmer classes
1
37
Introduction
Let g be a new and normalized weight two modular eigenform on 0 (N ) where N = pM is a factorization of N into coprime factors and p is a rational prime. Let Ag be the associated abelian variety, which is an optimal quotient of the Jacobian variety J0 (N ) with multiplication by an order Og in a totally real number …eld Kg . Let K=Q be a real quadratic extension of discriminant DK such that p is inert in K and the primes dividing M are split. The sign of the complex L-function of g=K is 1, so that the Birch and Swinnerton-Dyer conjecture predicts that dimKg Kg
Og
Ag (H)
[H : K]
for every narrow ring class …eld H=K of conductor prime to DK and N . The theory of Stark-Heegner points gives a conjectural explanation of this inequality. This theory was …rst proposed by H. Darmon in [7]. The theory was extended then by S. Dasgupta in [9], who gave a construction of local points p new P 2 J0 (N ) (Kp ) belonging to the p-new quotient of the Jacobian variety J0 (N ). The Stark-Heegner points on Ag are de…ned to be the image in Ag p new of the Stark-Heegner points on J0 (N ) . These points belong to Ag (Kp ). They are endowed with an action of the Galois group GH=K . Moreover they are conjecturally de…ned over the narrow ring class …eld H and are conjectured to satisfy a Reciprocity law analogous to the Shimura Reciprocity law for complex multiplication points. The conjectured rationality of Stark-Heegner points over the narrow ring class …elds is still wide open. The main known evidence for this conjecture is a rationality result over genus …elds of a real quadratic …eld, proved in [3]. However the proof does not throw light on how to approach the rationality over more general …elds. One of the ideas of this paper is to try to give a 2
cohomological approach to the rationality problem: more precisely try to show that there is a global cohomological class in the Selmer group whose restriction at p equals the Stark-Heegner point, rather than try to directly prove that the Stark-Heegner point is the restriction at p of an element of the global MordellWeil group. Since the Stark-Heegner points are expected to contribute to the rank of the Mordell-Weil group and hence to the rank of the Selmer groups when the sign of the complex L-function of g=K is 1, it is natural to put ourselver in a setting where the Selmer group is big. We are now going to in detail the main result of the paper. Let l 6= 2 be a rational prime and let L=K be a …nite l-power abelian extension contained in a narrow ring class …eld H=K. The extension L=K is a Galois dihedral extension over Q. Let l Og be a prime ideal dividing l and let F be the residue …eld at l. The main result of this paper shows that, under suitable assumptions on the prime l, the mod l-traces from H to L of StarkHeegner points with respect to the action of GH=L lift to global cohomology classes in Sell (Ag =L). In other words they belong to the image of the restriction map: Ag (Lp ) H 1 (Lp ; Ag [l]) . (1) resp : Sell (Ag =L) ! lAg (Lp ) In order to state more precisely the main result of this paper, we introduce some further notations. Write TN to denote the Hecke algebra acting on modular forms on 0 (N ). Assume that L TN is a prime of fusion in the sense of [15] arising from a congruence between the eigenform g of level N and another form f of level M . It is important to note that the signs of L (f =K; s) and L (g=K; s) are opposite, so that they are respectively +1 and 1. The special values L (f =K; 1) and L0 (g=K; 1) are expected to be frequently non-zero. Write lf Of (resp. l = lg Og ) to denote the prime ideal corresponding to L. Following [4] we will de…ne in Section 6.2.2 an element L (Af =K; 1)L , which is identi…ed as the mod lf reduction of the special value L (f =K; 1), whenever L is such a prime of fusion. Consider the triple (Ag ; Og ; l)Q that is attached above to the modular form g of level pM and any (non-zero) prime ideal l Og of residue characteristic l 6= 2. Denote by cg the conductor of the order Og and write ram (Kg =Q) to denote the set of primes of Q that ramify in Kg . Denote by (Ag ) the group of connected components of the Néron model of Ag =Qp . We make the following assumptions: the set ram (L=K) of primes of K that ramify in L is prime to lN ; l is prime to cg , ram (Kg =Q) and l - BS (Ag =K)
(Ag ; L=K) =:
(A; L=K) ,
where S = split (L=K) is the set of primes of K that are split in L and the …nite set (A; L=K) is de…ned in De…nition 2.2;
3
l is a prime appearing in the support SuppOg ( (Ag )) of the Og -module ;f ree (Ag ) prime with the set SpM; de…ned in De…nition 6.8;
the representation inertia assumption .
g
attached to Ag [l]Q is surjective and satis…es the
The main result of the paper is the following restatement of Theorem 7.3. Theorem 1.1 Let (Ag ; Og ; l)Q and L=K be as above. L := g 1 (l) is a prime of fusion, i.e. there exists a modular form f of level M which is congruent to g (modulo some prime of Q). As explained above it makes sense to consider the special value L (Af =K; 1)L . If L (Af =K; 1)L 6= 0 and Sellf (Af =K) = 0 (where lf is the maximal ideal of Of appearing in the de…nition of congruence between modular forms), Sell (Ag =L) is a rank 1 F GL=K -module generated by the mod l-trace from H to L of any Stark-Heegner point associated to the narrow ring class …eld H=K. The basic idea in the proof of the theorem is to place ourselves in a situation where: both Sell (Ag =L) and Ag (Lp ) =lAg (Lp ) are free rank 1 F GL=K -modules; Ag (Lp ) =lAg (Lp ) is generated by the traces of Stark-Heegner points; the restriction map (1) is injective. De…ne PK 2 Ag (Kp ) (resp. PL 2 Ag (Kp )) to be the trace from H to K (resp. from H to L) of a Stark-Heegner point attached to H=K. Making a suitable choice of the prime l, one can consider a related abelian variety Af , which is associated to a modular form f of weight 2 and level M obtained by lowering the level (see Theorem 5.7). The local l-divisibility of PK is related to the triviality of the element L (Af =K; 1)L : PK 6= 0 in
Ag (Kp ) , L (Af =K; 1)L 6= 0: lAg (Kp )
This is essentially obtained as a combination of the main theorem of [4] with Theorem 5.7 (see Corollary 6.12). Our assumption on the prime l implies the equality Ag (Lp ) = F GL=K P lAg (Lp ) when L (Af =K; 1)L is non-zero (see Proposition 6.5). Under certain standard control assumptions on (g; L=K), the triviality of the Selmer group Sellf (Af =K)
4
implies the triviality of the Selmer group Sellf (Af =L) (see Proposition 3.9 and Lemma 6.1). Theorem 4.2 is used to show the equivalence Sell (Ag =L) ' F GL=K
, Sellf (Af =L) = 0
and the injectivity of the restriction morphism (1). It follows that when Sellf (Af =K) = 0 the above equivalence is realized and since we assume L (Af =K; 1)L 6= 0 we …nd '
resp : Sell (Ag =L) !
Ag (Lp ) = F GL=K P . lAg (Lp )
We note that the main result of [3] specialized to the trivial character deals with the rationality the non-torsion part of the Stark-Heegner point PK , giving an unconditional result. On the other hand, when specialized to the case L = K, our main result deals with the rationality of the torsion part of the Stark-Heegner point PK , relating it with the descent of the abelian variety Af obtained by lowering the level. When L 6= K it also o¤ers non-trivial instances of the Reciprocity law, of course depending on the behavior of the special value L (Af =K; 1)L and the rank of Sellf (Af =K). Note that our L (Af =K; 1)L is the element denoted by L (E=K; 1)(p) in [4] when Af = E is an elliptic curve. This is identi…ed with an element of a localization L := L SS of L , with = (J) the group of the connected component at p of the Néron model of J over the maximal unrami…ed extension of Qp . Here J is the torus de…ned in [9] and admitting an isogeny with two p new copies of the p-new quotient J0 (N ) . The set S is enlarged in order to identify
L
with
p new
J0 (N )
2 L
. This is explained in Section 6.2.2. Then,
in order to apply the main result of [4] we need to be careful to choose the prime p new l in such a way that the morphism from J0 (N ) to the optimal quotient Ag p new
induces an isomorphism between J0 (N ) and L in Corollary 6.12. We …nally note that the implication
(Ag )l . This is done
L (Af =K; 1)L 6= 0 ) Sellf (Af =K) = 0 is discussed in [4, Section 4].
2
Canonical …nite subgroups attached to isogenies between abelian varieties
Now we recall from [11, ph. 5] the de…nition of certain canonical …nite groups attached to an abelian variety A = AK over a local …eld K (and more generally attached to isogenies between abelian varieties). We will write T (A) (resp. X (A) and (A)) to denote the torus (resp. the character group and the group 5
of connected components) of the …ber Ak of a Néron model AO of A over the ring of integers O of K (here k is the residue …eld). When f is a morphism between abelian varieties we will write fO to denote the morphism between the Néron models and T (f ) (resp. X (f ) and (f )) to denote the induced morphism. By de…nition X := D T , where D is the Cartier duality. We give the following variant of [11, Lemma 2.2.1]. The proposition holds for an arbitrary valuation ring O with fraction …eld K, i.e. it is not necessary to assume O to be the ring of integer of the complete local …eld K. Proposition 2.1 Let : A ! B be an isogeny between abelian varieties over the …eld K. Then the following facts are equivalent: (a)
0 O
(b)
0 O
(c)
0 O
is ‡at on A0O ; 0 A0O = BO (topologically);
is quasi-…nite on A0O ;
(d) char (k) - # ker
or A has semistable reduction.
Moreover, if the equivalent conditions are satis…ed by , then ker 0O is quasi-…nite separated and ‡at over O. When = n for a positive integer n > 0 put A0O [n] := ker n0O . If the equivalent conditions are satis…ed by n, they are satis…ed by n for every integer 0 and the multiplication by n in A0O n +1 uniquely factors through A0O [n ], giving rise to a faithfully ‡at O-morphism: p : A0O n
+1
! A0O [n ]
When G=O is a …nite ‡at group scheme over O (resp. k), we say that G is multiplicative if and only if its Cartier dual D (G) is étale. Under the henselian assumption on O (that is satis…ed in our setting where O is a complete discrete valuation ring) the closed …ber functor gives an equivalence between the category of …nite ‡at étale (resp. multiplicative) group schemes over O and k. Whenever G=O (resp. G=k) is a …nite ‡at group scheme over O (resp. k) we will write G Get to denote the maximal étale quotient; by de…nition it makes sense to consider the maximal multiplicative subgroup Gm ,! G too, i.e. the Cartier dual of the maximal étale quotient satis…es the dual maximality property. Both formations commute with local henselian base change and in particular with taking the closed …ber. We now recall the construction of the so-called …nite part attached to a quasi…nite separated scheme over a henselian base: for further details see [11, 2.2.2], [6, Lemma 8.3] and [10, Theorem 18.5.11 (c)] for the proof of the existence. Let G=O be a quasi-…nite separated O-scheme of …nite presentation over a henselian valuation ring O. There is a unique decomposition G = Gf t G0 into disjoint O-schemes such that: 6
Gf =O is …nite over O; G0k =
(hence G0 = G0K and Gk = Gfk ).
The above decomposition is canonical and Gf =O is canonically characterized as the maximal O-subscheme of G which is O-…nite. The formation of Gf =O is compatible with henselian local base change and with …ber products over O. In particular, when G=O is an O-group scheme, Gf =O is canonically an O-subgroup scheme and the maximal subgroup scheme which is O-…nite. This discussion shows that, to a quasi-…nite separated O-group scheme G=O, we can attach the canonical …ltration: Gm
0
Gf
G.
Moreover, taking the closed …ber, the above …ltration becomes (write G?k to ? denote G? k = (Gk ) ): 0 Gm Gfk = Gk . k Under one of the equivalent conditions of Proposition 2.1, the above discussion applies to G = ker 0O . Under the perfectness assumption on k, T (A) is characterized as the maximal subtorus of A0k . Thus we can consider the int clusion ker 0k := ker T (f ) ker 0k , that factors through the inclusion m ker 0k ker 0k . Since we remarked that the closed …ber functor is an equivalence between the category of …nite ‡at multiplicative group schemes over t m O and k, we can de…ne ker 0O ker 0k as the unique lift of the inclusion 0 t k
m
ker 0k (note that ker 0O k = ker 0k ). This discussion shows that, to the isogeny : A ! B we attached the canonical …ltration: ker
0
0 t O
ker
ker
0 m O
ker
0 f O
0 O
ker
.
Moreover, taking the closed …ber, the above …ltration becomes: 0
ker
0 t k
ker
0 m k
ker
0 f k
= ker
0 k
.
Finally, by de…nition, the following …ltration on ker 0O K = ker ( ) is obtained by taking the generic …bers from the above …ltration on ker 0O : 0
t
ker ( )
m
ker ( )
f
ker ( )
ker ( ) .
It is clear by construction that the functoriality of the above …ltrations has to be understood in the category whose objects are the isogenies : A ! B between abelian varieties over K and whose morphisms are the obvious commutative squares. We introduce useful notations to be in force for the remainder of the paper.
7
De…nition 2.2 Given the tower of …eld extensions L=K=k we write: ram (L=K) to denote the set of primes of K that ramify in L=K; ram (L=K=k) to denote the set of primes of K that ramify in L=K and are inert or rami…ed over k; split (L=K) to denote the set of primes of K that split in L=K. Let A=K be an abelian variety over K and let be a set of primes of K. Denote by v the group of connected components of the Néron model of A=Kv and by kv the residue …eld of Kv . We will use the following notations: Bad (A=K) to denote the set of primes of K that are of bad reduction for A=K and do not appear in ; Q B (A=K) := v2= # v (and Bv0 (A=K) when it is reduced to the singleton fv0 g); Q (A; L=K) = v2ram(L=K) #A (kv ).
3
Descent of abelian varieties
We collect results that will be useful in the subsequent sections. In order to better state the results it is convenient to introduce the following terminology. De…nition 3.1 Let (A; R; I) = (A; R; I)K be a triple, where (A; R) = (A; R)K is an abelian variety with multiplication (de…ned over K) by the commutative unital ring R and I R is an invertible ideal. Then we say that this triple is: slightly regular, when the completion of R at I is a (one-dimensional) regular ring, i.e. the direct sum of discrete valuations rings; regular, when the completion of R at I \ Z = Zn is a (one-dimensional) regular ring, i.e. the direct sum of discrete valuations rings; unrami…ed, when the completion of R at I\Z = Zn is a (one-dimensional) absolutely unrami…ed ring. For an integral domain O, let Fr (O) be its fraction …eld. De…nition 3.2 An abelian variety with real multiplication over K is a couple (A; O) = (A; O)K where A=K is an abelian variety over K and O is an order in the totally real number …eld F := Fr (O) such that [F : Q] = dim (A). In the following lemma we let K=Qp be a local …eld.
8
Lemma 3.3 If (A; O)K has real multiplication by the order O in the totally real …eld F and reduction is semistable, then it is either purely toric or good. In purely toric reduction case, the torus becomes trivial after an at most quadratic unrami…ed …eld extension, Q Z X (A) is a F -vector space of dimension 1 and hence X (A) is an invertible O-module, whenever O = OF is the ring of integers of F . Assume that K=Qp is a local …eld and that A=K has split purely toric reduction. Let pf be the order of the residue …eld of K. When (A; O)K has real multiplication, A [l] is a two-dimensional F-vector space for every prime ideal l of O. There is a canonical exact sequence: 0!F
l
! A [l] ! F ! 0.
(2)
Lemma 3.4 Suppose that A=K has split purely toric reduction and real multiplication by O. For every prime ideal l (of O) of characteristic 6= p the following implications hold: if the l-torsion sequence is split (over K), then A [l] is an unrami…ed GK module and the Frobenius F robkK acts on A [l] with the two eigenvalues pf and 1; if A [l] is an unrami…ed GK -module, the Frobenius F robkK acts on A [l] with the eigenvalues pf and 1 and pf 6= 1 (mod l), then the l-torsion sequence is split (over K). Proof. The proof is easy (see [18, Lemma 2.31] for details).
3.1
The I-descent over a local …eld, when A has split purely toric reduction
Here we work with a triple (A; R; I) = (A; R; I)K de…ned over a local …eld K assuming that A=K has purely toric reduction and we de…ne n 2 N by the formula nZ = Z\R. Consider the canonical morphism between exact sequences of GK -modules: 0
!
0
!
X (At ) # I 1 R X (At )
! !
X (A) I
1
Gm K # Gm K R X (A)
! !
A K # I 1 RA K
!
0
!
0
Taking the kernel-cokernel exact sequence, it gives the following canonical short exact sequence, called the I-torsion sequence: 0 ! X (A)
Gm K
[I] ! A [I] !
I 1 R
R
X At ! 0.
Here, the zero on the right follows from the fact that X (A) Gm K is an I 1 n-divisible group and hence R R X (A) Gm K = 0. The zero on the left 9
follows from the fact that the I-torsion of X (At ) is contained in the n-torsion, that is trivial. Note that the I-torsion sequence shows that t
Gm K
A [I] = X (A)
[I] :
Lemma 3.5 Let (A; R) = (A; R)K be de…ned over a local …eld K, suppose that A=K has split purely toric reduction and let I R be an invertible ideal. Then the image SK of the I-Kummer map coincides with t
SK = Im H 1 K; A [I]
! H 1 (K; A [I]) :
Proof. This is well known, see for example [12, Proof of Lemma 8]. Lemma 3.6 Let (A; R; I)K and (B; R; I)K be two slightly regular triples de…ned over a local …eld K, suppose that A=K and B=K have purely toric reduction and let A ! B be an R-linear morphism of abelian varieties (over K) inducing an ' isomorphism A [I] ! B [I] over K. Then the I-torsion sequences are canonically identi…ed. Proof. The proof is easy and is left to the reader (see [18, Lemma 2.21]). In the following proposition let p be the residue characteristic of K=Qp , let pf be the order of the residue …eld kK and write F robkK to denote the Frobenius. Proposition 3.7 Let (A; R; I)K be a slightly regular triple de…ned over a local …eld K=Qp and suppose that A=K has split purely toric reduction. Furthermore assume the residue characteristic p of K to be coprime with n and pf 6= 1 (mod n). Then the specialization map into the group of connected components gives a canonical isomorphism A (K)
R
I 1 = R
(A)
R
I 1 R
and the I-Kummer map factors through a canonical inclusion 0!
(A)
R
I 1 ! H 1 (K; A [I]) : R
Proof. It is easy to show that, if (A; R; I)K is a slightly regular triple, the I-torsion sequence takes the following form (see [18, Lemma 2.20]): 0!
I 1 R
R
X (A)
n
! A [I] !
I 1 R
R
X (A)
n
R
X At ! 0.
By Lemma 3.5 we know SK = Im H 1 K;
I 1 R
10
! H 1 (K; A [I]) :
Since GK acts trivially on X (A) , p is coprime with n and pf 6= 1 (mod n) (so that H 1 (K; n ) = Z=nZ canonically): H 1 K;
I 1 R
R
X (A)
=
n
=
I 1 R I 1 R
R
X (A)
R
X (A) :
H 1 (K;
n)
Hence, taking the long exact sequence of the I-torsion sequence yields the following exact sequence, describing the image SK of the I-Kummer map: I 1 R
R
X At !
I 1 R
R
X (A) ! H 1 (K; A [I]) :
The …rst arrow is obtained as follows; recall the exact sequence describing the group of connected components of [11, Theorem 11.5]: 0 ! X At ! X (A) !
(A) ! 0. 1
Tensoring this exact sequence over R with IR gives the …rst arrow above. 1 1 Hence A (K) R IR = (A) R IR and the image of the I-Kummer map factors through the canonical inclusion: 0!
3.2
(A)
R
I 1 ! H 1 (K; A [I]) . R
Control assumptions for the I-Selmer group
Here we work with a triple (A; O; l) = (A; O; l)k with real multiplication by the order O, de…ned over a …eld k, under the étale assumption and supposing that l 6= 0 is a prime ideal with residue …eld F = Ol of characteristic l 6= 2. Then A [l] is a two-dimensional F-vector space and it is completely characterized, as a k-scheme, by the F [Gk ]-module structure on A [l], or equivalently by the representation l : Gk ! Aut (A [l]) = GL2 (F) . We consider a tower L=K=k such that L=K is an extension whose order is the power of a single odd prime q and L=k is solvable. In the applications we have in mind q = l and L=k is a dihedral extension, but the assumption is not necessary. Lemma 3.8 If K=k is abelian, (A; O) = (A; O)k is de…ned over k and l is surjective then A [l] = A [l]k is an absolutely irreducible Gk -module and we have A [l] (L) = 0.
11
Proof. Since
l
is surjective, the induced morphism of F-algebras F [ l ] : F [Gk ] ! M2 (F)
is surjective (use l 6= 2) and this means that A [l]k is an absolutely irreducible Gk -module. Then A [l] (L) A [l] is an F [Gk ]-submodule and the claim follows once we exclude the case A [l] (L) = A [l]. But in this case k (A [l]) L and hence we …nd that Gk(A[l])=k = Im ( l ) = GL2 (F) is a quotient of the solvable group GL=k , which is not possible. Proposition 3.9 Assume that the following conditions are satis…ed: ram (L=K) \ BadS (A=K) = , ram (L=K) \ fv j qg = , q - BS (A=K) l
(A; L=K) (where S = split (L=K)),
is surjective.
Then A [l] = A [l]k is an absolutely irreducible Gk -module, A [l] (L) = 0 and the restriction gives a canonical identi…cation GL=K
Sell (A=K) = Sell (A=L)
.
Proof. For a proof when A = E is an elliptic curve see [1, 2.12-2.19] or [2, Lemma 1.4]. In the general case it holds the same formalism (see [18, Lemma 2.38 - Proposition 2.43] for details).
4
Congruences and growth of the Selmer groups
Let g be a (weight two) eigenform of level N (and trivial character). Let TN be the Hecke algebra acting on modular forms on 0 (N ) and let Og be the order generated by the Fourier coe¢ cients an (g) of g. Consider the ring homomorphism g
: TN Og OQ Tn 7! an (g) :
The ideal Ig := ker ( g ) is a saturated ideal, i.e. TINg is torsion free, since the image of g is torsion free and the quotient Ag := J0 (N ) =Ig J0 (N ) is an optimal quotient of J0 (N ) with real multiplication by Og . This means that the order Og acts on Ag via homomorphisms de…ned over Q, the fraction …eld Kg := Fr (Og ) of Og is a totally real number …eld and [Kg : Q] = dim (A). Write cg to denote the conductor of Og in the maximal order OKg . We recall that we have (let p be a rational prime) reduction of Ag =Q at p =
good, p-N non-good semistable, p k N and g new at p
12
(3)
Now assume N = pM (and p k N ) and let f be an eigenform of level M . We say that f and g are congruent modulo a prime l of Q (which does not divide pM ) if the following relation holds in OQ : an (f )
an (g) mod l
(4)
for almost all n. The congruence relation allows us to consider the triple (Af ; Of ; lf )Q (resp. (Ag ; Og ; lg )Q ), where lf := Of \ l (resp. lg := Og \ l). Consider the residual representation f (resp. g ) of f (resp. g) at the prime lf (resp. lg ), i.e. the Ff [GQ ]-module (resp. Fg [GQ ]-module) A [lf ] (resp. A [lg ]) where Ff := Of =lf (resp. Fg := Og =lg ). Denote by cf (resp. cg ) the conductor of Of (resp. Og ). De…nition 4.1 We say that Af [lf ] (resp. Ag [lg ]) satis…es the inertia assumption if, for every q k M , the dimension of the inertia invariants of Af [lf ]Qq is 1 and, for any other prime dividing M , the dimension of the inertia invariants is 0. The following theorem is a reformulation in the language of modular forms of Theorem 4.7, to be proved in the subsequent subsections. The proof is strongly based on the techniques developed in [12], combined with standard control results on the Selmer groups and the Mazur-Rubin theory of local constant (generalized in [19] from elliptic curves to abelian varieties with real multiplication). The reader is invited to give a look at Section 4.1 for the notations concerning Selmer groups and may …nd useful to read Section 4.2 until the statement of Theorem 4.7 before reading the proof. Theorem 4.2 Let (Af ; Of ; lf )Q and (Ag ; Og ; lg )Q be two triples obtained from two modular forms f and g of level M and pM which are congruent modulo some (odd) prime l of OQ of characteristic l and such that f is new at p. Suppose that l is prime to cf , cg , ram (Kf =Q) and ram (Kg =Q). Let K=Q be a quadratic extension such that the discriminant D = DK of K=Q is coprime with pM , the primes dividing M split, and the prime p is inert in K. Consider a …nite abelian l-power extension dihedral over Q such that ram (L=K) is coprime with lpM . Suppose one of the GQ -representations f or g attached to Af [lf ] or Ag [lg ] to be absolutely irreducible and one between Af [lf ] or Ag [lg ] to satisfy the inertia assumption. Furthermore assume that p2 6= 1 (mod l). If L = K one has for (1; 2) = (f; g) or (g; f ) dimF2 Sell2 (A2 =K) = dimF1 Sell1 (A1 =K) and if dimF2 Sell2 (A2 =K)
1
dimF1 Sell1 (A1 =L) one has
[v ]
(v )
Sell2 (A2 =K) = Sell2 0 (A2 =K) and Sell1 (A1 =K) = Sell1 0 (A1 =K) ;
13
so that dimF2 Sell2 (A2 =K) = 1 and [v ] Sell2 (A2 =K) = Sell2 0 (A2 =K)
, Sell1 (A1 =K) = 0:
If the above equivalence is satis…ed, the GQ -representation 2 attached to A2 [l2 ]k is surjective and l - (Ai ; L=K) (for i = 1 or 2 is the same) we have [ (v0 )]
Sell2 (A2 =L) = Sell2
(A2 =L) ' F2 GL=K
and Sell1 (A1 =L) = 0
and the restriction morphism resp : Sell2 (A2 =L) !
l2 1 R2
R2
A2 (Lp ) =
is an isomorphism of F2 GL=K -modules.
L
w2 (p)
l2 1 R2
R2
A (Lw )
Proof. Letting Ff , Fg and F be the residue …elds at the primes lf , lg and l, by the Eichler-Shimura relation and (4), the GQ -modules F Ff Af [lf ] and F Fg Ag [lg ] have isomorphic semisimpli…cations. If one between Af [lf ] or Ag [lg ] is absolutely irreducible as a GQ -module the same is true for F Ff Af [lf ] or F Fg Ag [lg ] and hence they are isomorphic GQ -modules by the Brauer-Nesbitt principle. In other words (Af ; Of ; lf )Q and (Ag ; Og ; lg )Q are congruent triples (in the sense of De…nition 4.6) and hence a fortiori they are congruent over K. We now discuss the reduction of Af =K and Ag =K. We have described the reduction of Af and Ag over Q in (3). To describe the reduction over K simply note that, since the primes of K dividing M splits, if v j q j M is a prime of K, then Kv = Qq and the reduction of Af =Kv (resp. Ag =Kv ) is the same as the reduction of Af =Qq (resp. Ag =Qq ). If v0 is the unique prime of K dividing p, since Af =Qp (resp. Ag =Qp ) has semistable reduction, the same is true for Af =Kv0 (resp. Ag =Kv0 ) and, since Ag =Qq has non-good semistable reduction, Ag =Kv0 has purely toric split reduction by Lemma 3.3. Finally, if v - pM , they both have good reduction. By de…nition of congruence between modular forms, the prime l involved in the congruence relation is coprime with the level of the two modular forms. Thus assumption II) of Theorem 4.7 is satis…ed. The assumption I) is satis…ed since we assume l to be prime to cf , cg , ram (Kf =Q) and ram (Kg =Q), so that (Af ; Of ; lf ) and (Ag ; Og ; lg ) are regular and unrami…ed triples. See De…nition 3.1 for the meaning of regular and unrami…ed triples. The assumption III) is satis…ed thanks to [12, Lemma 3 and Lemma 4], since the primes dividing M splits in K. The assumption IV ) is satis…ed since it is well known that the congruence relation (4) implies (see [15, (2:2)]) 2
ap (f )
2
(p + 1)
mod l.
Then we …nd ap (f ) (p + 1) which implies, together with the EichlerShimura relation, that F robp2 acts on the unrami…ed module Af [lf ] with distinct (since p2 6= 1 (mod l)) eigenvalues p2 and 1. Then, Theorem 4.7 applies, 14
thanks to the fact that ram (L=K) is coprime with lpM , to the prime v0 = pOK , that splits completely in the extension L=K, because p is inert in K by assumption and L=Q is a dihedral extension.
4.1
Selmer structures
Assume that V is a …nite dimensional F [GK ]-module, where K is a number …eld and F is a …nite dimensional …eld. De…nition 4.3 A Selmer structure on V is a collection S = fSv g of F-vector subspaces Sv H 1 (Kv ; V ) for every place v of K such that, for almost all v, 1 one has Sv = H Kvur ; V Iv (where Iv is the inertia group at v). Given the Selmer structures S = fSv g, T = fTv g and a …nite set of K there are the following possible operations:
of places
The Selmer structure S + T is de…ned by taking the subspace Sv + Tv for each v; The Selmer structure S \ T is de…ned by taking the subspace Sv \ Tv for each v; [ ]
The relaxed Selmer structure S [ ] is de…ned by taking Sv [ ] for each v 2 S and Sv = Sv otherwise;
( )
The strict Selmer structure S ( ) is de…ned by taking Sv ( ) v 2 S and Sv = Sv otherwise.
= H 1 (Kv ; V )
= f0g for each
The Selmer structures are naturally ordered by componentwise inclusion of the local conditions and one write S T to denote this relation. Given a Selmer structure, the associated Selmer group is de…ned by setting HS1 (K; V ) := ker H 1 (K; V ) ! so that S
T implies by de…nition HS1 (K; V )
Q H 1 (Kv ; V ) v Sv
;
HT1 (K; V ). It is clear that:
1 HS\T (K; V ) = HS1 (K; V ) \ HT1 (K; V ) ; 1 1 HS (K; V ) + HT1 (K; V ) HS+T (K; V ) :
If S is a Selmer structure and a given …nite set of places, we will write 1;[ ] 1;( ) HS (K; V ) (resp. HS (K; V )) to denote the Selmer group associated to the Selmer structure S [ ] (resp. S ( ) ). Assume V to be endowed with a skew-symmetric, GK -equivariant and perfect pairing V V ! F (1) : 15
'
The above data is equivalent to the data of an isomorphism V ! V thus ' inducing H 1 (Kv ; V ) ! H 1 (Kv ; V ) and the local Tate duality gives, for every place v of the …eld K, a perfect symmetric pairing h ; iv : H 1 (Kv ; V )
'
H 1 (Kv ; V ) ! H 2 (Kv ; F (1)) ! F:
De…nition 4.4 The dual Selmer structure S = fSv g is de…ned by taking the orthogonal complement Sv of Sv for the local pairing h ; iv and the Selmer structure S is said to be selfdual (for the given pairing) whenever S = S . Here is a parity lemma: Lemma 4.5 Suppose S and T to be selfdual Selmer structures and …nite set of primes such that Sv = Tv for every v 2 = . Then 1 dimF HS+T (K; M ) 1 dimF HS+T (K; M )
dimF HS1 (K; M )
1 dimF HS\T (K; M ) =
P
v2
dimF
dimF HS1 (K; M ) + HT1 (K; M ) P dimF HT1 (K; M ) v2 dimF
to be a
Sv Sv \ Tv
;
mod 2; Sv Sv \ Tv
mod 2:
Proof. See [13, Proposition 1.3 and Theorem 1.4].
4.2
Proof of the main result of the section
Fix the following data: two triples (Ai ; Ri ; li ) = (Ai ; Ri ; li )K for i = 1; 2 of abelian varieties Ai , with real multiplication by an order in a number …eld Ri End (Ai ), the entire data being de…ned over a number …eld K=Q. When (Ai ; Ri ; li ) is a (slightly) regular triple, li is an invertible ideal and the li Kummer sequences for the descent take the form: 0 ! Ai [li ] ! Ai ! li
1
Ri
Ai ! 0.
In the following discussion let li be prime ideals. De…nition 4.6 We say that the two triples are congruent if there exists an order R in a number …eld such that Ri R and a prime l of R such that li = l \ Ri and R R R1 A1 [l1 ] ' R2 A2 [l2 ] over K: l l Let l 6= 2 be the rational prime divided by l and suppose the above data to be subject to the following conditions: I) The two triples (Ai ; Ri ; li ) are (slightly) regular.
16
II) The two abelian varieties have bad reduction at the same primes of K, except at one prime v0 of K, where one of the two abelian varieties has good reduction and the other has purely toric split reduction and they both have good reduction at the primes dividing l. In other words Bad (Ai =K) = Bad (Aj =K) t fv0 g ; Bad (Ai =K) \ fv j lg = and the reduction of Ai =K at v0 is purely toric split. III) l is coprime with Bv0 (Ai =K) for i = 1; 2. IV ) pf 6= 1 mod l, where pf is the order of the residue …eld of the extension I Kv0 =Qp and the Frobenius F robpf 2 GKv0 acts on one of the Ai [li ] v0 s with eigenvalues 1 and pf (here Iv0 is the inertia group at v0 ). Set F := Ri R.
R l ,
Fi :=
Ri li
and regard Fi as a sub…eld of F via the inclusion
Following the arguments in [12] we give a proof of the following: Theorem 4.7 Let L=K be a …nite abelian l-power extension where v0 splits completely and let (v0 ) be the set of primes of L dividing v0 , so that # (v0 ) = [L : K]. Assume ram (L=K) \ Bad (Ai =K) = for i = 1; 2 and ram (L=K) \ fv j lg = . Then: If L = K one has dimF2 Sell2 (A2 =K) = dimF1 Sell1 (A1 =K) and, if for example dimF2 Sell2 (A2 =K)
1
dimF1 Sell1 (A1 =L), one has
[v ]
(v )
Sell2 (A2 =K) = Sell2 0 (A2 =K) and Sell1 (A1 =K) = Sell1 0 (A1 =K) ; so that dimF2 Sell2 (A2 =K) = 1 and [v ] Sell2 (A2 =K) = Sell2 0 (A2 =K)
, Sell1 (A1 =K) = 0:
If the above equivalence is satis…ed, K=k is quadratic, L=k is dihedral, the triple (A2 ; R2 ; l2 ) = (A2 ; R2 ; l2 )k is de…ned over k, regular, unrami…ed, the Gk representation 2 attached to A2 [l2 ]k is surjective and l - (Ai ; L=K) (for i = 1 or 2, it is the same), we have [ (v0 )]
Sell2 (A2 =L) = Sell2
(A2 =L) ' F2 GL=K
and Sell1 (A1 =L) = 0
and the restriction morphism resv0 : Sell2 (A2 =L) !
l2 1 R2
R2
A2 (Lv0 ) =
is an isomorphism of F2 GL=K -modules. 17
L
w2 (v0 )
l2 1 R2
R2
A (Lw )
Let (A; R) be a couple de…ned over a local …eld K, with ring of integers OK and residue …eld k and denote by K the group of connected components of the Néron model of A over OK . The …rst step is to give criteria to identify the local conditions SK de…ning the l-Selmer group, i.e. the image of l 1 R A (K) in H 1 (K; A [l]). We distinguish three cases (the assertions do not depend on any assumption on R, but l is assumed to be invertible): (a) lOK = OK (i.e. l is invertible in OK ) and
K
[l] = 0, implying
1 SK = Hur (K; A [l]) (see [12, Lemma 6]);
(b) A has good reduction, implying SK = Hf1l (OK ; A [l]) (see [12, Lemma 7]); (c) A has split purely toric reduction, implying SK = Im H 1 (K; F (see Lemma 3.5).
l)
! H 1 (K; A [l])
Write V to denote one of the two isomorphic L-schemes F Fi Ai [li ]L and write Si;w to denote the local condition at w de…ned by the image under the Kummer map of li 1 Ri Ai (Lw ) in H 1 (Lw ; Ai [li ]). Using (a c) one can proceed to the identi…cation of the local conditions Si;w de…ning the li -Selmer group, with K = Lw . The images under the Kummer map at the prime w 2 = (v0 ) [ f1g coincide. If w - l then (a) applies to Ai =Lw for i = 1; 2, thanks to III) and the assumption that ram (L=K) is coprime with the set Bad (Ai =K). In fact if w j v and v 2 = Bad (Ai =K) is a prime of good reduction, Ai =Lw has good reduction too and i;w [l] = 0 (where i;w denotes the group of connected components of the Néron model of ALw ). If w j v and v 2 Bad (Ai =K) then v 2 = ram (L=K) (by assumption) and hence the extension Lw =Kv is unrami…ed. It follows that the Néron model of ALw is the base change of the Néron model of AKv and hence # i;v = # i;w (and III) applies, giving i;w [li ] = 0). Thus: 1 Si;w = Hur (Lw ; Ai [li ]) ) F
Fi Si;w
1 = Hur (Lw ; V ) :
If w j l then (b) applies thanks to II), because if w j v this assumption implies that v 2 = Bad (Ai =K) and hence Ai =Lw has good reduction too. Thus: Si;w = Hf1l (OLw ; Ai [li ]) ) F
Fi Si;w
= Hf1l (OLw ; V ) :
The images under the Kummer map at the prime w = 1 coincide. Since l is odd and V is an F-vector space with char (F) = l, we …nd H 1 (Lw ; V ) = 0; 18
for Lw = C; R. The images under the Kummer map at the primes w 2 (v0 ) are "transverse". We now proceed to the identi…cation of the local conditions at the primes w 2 (v0 ). Suppose, to …x ideas, that A1 =Kv0 has purely toric split reduction, while A2 =Kv0 has good reduction (this is possible thanks to II)). First of all note that, since A2 =Kv0 has good reduction and l 6= p, A2 [l2 ] =Kv0 is unrami…ed, I so that the same is true for V and then for A1 [l1 ] V . Thus Ai [li ] v0 = Ai [li ] for both modules and, since the Frobenius F robpf acts on one of them with eigenvalues 1 and pf , which are distinct since pf 6= 1 (mod l), it also acts with I the same distinct eigenvalues on V and then on A1 [l1 ] v0 = A1 [l1 ]. Thus, by Lemma 3.4, A1 [l1 ] splits into the sum of the 1-eigenspace and the pf -eigenspace: the latter it is the copy of F1 (1) contained in A1 [l1 ], while the former project isomorphically onto F1 and it is the space of invariants. Thus we …nd that the l-torsion sequence (2) splits and hence there is a canonical decomposition: V = F (1)
F:
By (c) (…rst equality) and the canonical splitting of the l1 -torsion sequence (second equality), we …nd: S1;w = Im H 1 (Lw ; F1 (1)) ! H 1 (Lw ; A [l]) = H 1 (Lw ; F1 (1)) : Applying F
F1
gives F1 S1;w
F
= H 1 (Lw ; F (1)) ' F:
Since the abelian variety A2 =Lw has good reduction, applies (in fact w - l, so that lOKw = OKw ). Hence
2;w
[l] = 0 and (a)
1 S2;w = Hur (Lw ; A2 [l2 ])
and tensoring with F F
gives
F2
F2
1 S2;w = Hur (Lw ; V ) = H 1 (Lw ; F) ' F:
Application of the parity lemma. Consider the sets of local conditions F Fi Si := fF since F Fi commutes with taking cohomology, HF1
Fi S i
(L; V ) = F
Fi Selli
Fi Si;w g
and note that,
(Ai =L) :
Since the relaxed and the restricted conditions at (v0 ) (associated to F Fi Si ) are obtained from the relaxed and the restricted conditions at (v0 ) (associated to Si ) by scalar extension, it is also true that 1;[ (v0 )] Fi S i
(L; V ) = F
[ (v0 )] Fi Selli
(Ai =L) ;
1;( (v0 )) Fi S i
(L; V )
( (v0 )) Fi Selli
(Ai =L) :
HF HF
= F 19
F
Moreover, we remark that the explicit description of the local conditions Fi Si tells us that: 1;[ (v0 )] Fi S i
(L; V ) = HF1
1;( (v0 )) Fi S i
(L; V )
HF HF
= HF1
F1 S1 +F
F2 S 2
F1 S1 \F
F2 S 2
(L; V ) ; (L; V ) :
( (v ))
By de…nition Sell2 0 (A2 =L) is the kernel of resv0 , so that the injectivity of resv0 , which is claimed in the second part of the theorem, is automatic 1;( (v )) 1;( (v )) ( (v )) when HF F S10 (L; V ) = HF F S20 (L; V ) (or equivalently Sell1 0 (A1 =L)) 1 2 is trivial; the surjectivity then will follow by comparing the F2 -dimensions of L ( (v )) Sell2 0 (A2 =L) and w2 (v0 ) S2;w (which is the F2 -dimension of the corresponding objects obtained by base change to F). In other words, this discussion shows that, in order to prove the theorem, it is su¢ cient to consider (both in the assumption and the claims) the corresponding statements for the cohomology of V . The …rst step is to apply the parity Lemma 4.5 to V and hence we need to check that the local conditions de…ning the Selmer groups are selfdual for a given pairing on V . Since every triple with real multiplication is polarizable ([19, Section 4]), there is for i = 1 or 2, say i = 1, a perfect alternating GK -equivariant pairing on A1 [l1 ], which can be extended by F-linearity to a perfect alternating GK -equivariant pairing: V
V ! F (1) :
The Bloch-Kato generalization of local Tate duality for Tl1 (A1 ) (see [5, Proposition 3.8]) shows that the conditions F F1 S1 are selfdual and the same is true for the conditions F F2 S2;w for all w - v, because F F1 S1;w = F F2 S2;w at these places. The fact that F F2 S2;w is selfdual at the places w 2 (v0 ) too (for the pairing on V induced from the Weil pairing on A1 [l1 ]) follows from the fact that VLw = F (1) F and F (1), F are selfdual for the Weil pairing on V . For all w 2 (v0 ) we know F F2 S2;w \ F F1 S1;w = 0 while F F2 S2;w is one-dimensional. Hence the parity Lemma 4.5 gives: dimF
HF1 HF1
dimF HF1 dimF HF1
Fi S2 +F
Fi S 1
Fi S2 \F
Fi S 1
Fi S2 +F Fi S 2
(L; V ) (L; V )
Fi S 1
(L; V )
= # (v0 ) = [L : K] ;
(L; V )
HF1
dimF HF1
Fi S 2 Fi S 1
(L; V ) + HF1
(L; V )
Fi S 1
(L; V ) mod 2;
[L : K] mod 2:
(5)
Now the theorem, when L = K, easily follows from the above equations. For the second part of the theorem …rst note that the Weil pairing on V gives an identi…cation F F1 A1 [l1 ] ' F F1 At1 [l1 ] (over K) and hence (A1 ; L=K) = (At1 ; L=K); since (A2 ; R2 ; l2 ) is polarizable by [19, Section 4], it is also true that (A2 ; L=K) = (At2 ; L=K) and since F F1 A1 [l1 ] ' F F2 A2 [l2 ], the prime l does not divide (A1 ; L=K) if and only if it does not divide (A2 ; L=K). 20
Since the Gk -representation 2 is surjective, A2 [l2 ] (L) = 0 by Lemma 3.8 and hence A1 [l1 ] (L) = 0 (since F F1 A1 [l1 ] ' F F2 A2 [l2 ]). Now the assumption l - (A1 ; L=K) together with III) implies, by Proposition 3.9: G
Sell1 (A1 =L) = Sell1 (A1 =K) = 0. We deduce that Sell1 (A1 =L) = 0 essentially by [17, Chapter IX, Section 1, Lemma 2] (and hence HF1 F S1 \F F S2 (L; V ) = 0 too). Hence the …rst equation 1 2 of (5) gives: dimF HF1
F2 S 2
(L; V )
dimF HF1
F2 S2 +F
F2 S 1
(L; V ) = [L : K] :
Since A2 [l2 ] (K) = 0 (again by Lemma 3.8) we can apply [19, Second Corollary of the Introduction]. Together with the above inequality it gives 1;[ (v )] HF1 F S2 (L; V ) = HF F S02 (L; V ) in the dihedral setting. The fact that resv0 2 2 is an isomorphism follows from the fact that resv0 is an inclusion (since the L ( (v )) group Sell1 0 (A1 =L) is trivial) and the F-dimension of w2 (v0 ) F F2 S2;w is [L : K], since F F2 S2;w is one-dimensional by its explicit description.
5 5.1
Geometric lowering the level Abstract setting
In the following discussion we will focus on an abelian variety J=K over a global (or local) …eld K, having the structure of a T-module, T EndK (J) being a unital commutative subring. Moreover we understand that, during the following discussion, all the other abelian varieties and morphisms between them are de…ned over K. As usual, by a saturated ideal I T we mean and ideal I T such that TI is torsion free. By a T-closed immersion we mean a closed immersion A ,! J (de…ned over K) which is T-stable, while by a T-optimal quotient we mean a surjection (de…ned over K) with connected kernel (i.e. whose kernel is an abelian variety) which is T-stable. We remark that, for a closed immersion (resp. an optimal quotient) to be a T-closed immersion (resp. a T-optimal quotient) it is su¢ cient that it is T-stable up to isogeny and then A (resp. the quotient variety of J) inherits a unique T-action making the inclusion (resp. the quotient map) T-equivariant. If Ai with i = 1; 2 are two abelian varieties (over K) endowed with a T-action via T ! EndK (J) of unital rings, there is a canonical T-action on their duals Ati = Ext1 (A; Gm ); a T-homomorphism f : A1 ! A2 (resp. T-isogeny or Tpolarization when A2 = At1 ) is a homomorphism (resp. isogeny or polarization) which is T-equivariant. Proposition 5.1 The duals At ,! J t of T-optimal quotients J A are Tclosed immersions. The duals J t At of T-closed immersions A ,! J are T-optimal quotients. 21
There is a bijection between T-stable quotients (resp. T-stable morphisms with …nite kernel) up to isogeny and T-optimal quotients (resp. T-closed immersions). Proof. This is well known, for example see [6, Propoposition 3.3] for the …rst statement. Fix a T-polarization : J ! J t (a principal polarization in the applications). Let f : A ! J be a T-homomorphism with …nite kernel (a closed immersion in the applications) and consider the dual morphism f t : J t ! At . Then the composition ft
f
: A ! J ! J t ! At is a T-isogeny (being a T-polarization) and we can consider the T-module object (f ) := ker . When f is a closed immersion, we will also write (A) and when f is a closed immersion and has been …xed, we will simply write (A). Then (f ) is a …nite group scheme over K and a T-module object: we can de…ne its T-annihilator in the usual way and then its support SuppT ( (f )) as being the subset of those primes of T containing the T-annihilator. When (f ) is étale (for example when char (K) = 0) the de…nition is the usual one. Lemma 5.2 Let O be a henselian valuation ring with …eld of fraction K and perfect residue …eld k. Let : A ! B be an isogeny between abelian varieties over K with semistable reduction. There is a canonical diagram: ker
( ) ,! co ker X
t
ker .
Here canonical means in the category whose objects are the isogenies between abelian varieties over K and whose morphisms are the obvious commutative squares. Proof. The Grothendieck description of the group of connected components in the semistable reduction case yields the following commutative diagram with exact rows (see [11, Theorem 11.5]): 0 0
X
! !
t
X (At ) # X (B t )
! X( ) !
X (A) # X (B)
! ( ) !
(A) # (B)
!
0
!
0
Since is an isogeny and the above functors are additive, the vertical arrows in the above diagram are isogenies too: in particular, since X ( ) takes value in the category of free Z-modules, X (') must be injective. Hence the kernelcokernel exact sequence takes the form: ( ) ! co ker X
t
By de…nition X t := D T co ker X t = D ker T t .
t
0 ! ker
! co ker X ( ) ! co ker
( )!0
, so that there is a canonical identi…cation
22
t;0 O
Now recall the canonical inclusion ker
t
t;0 O
ker
in the discust
sion following Proposition 2.1: thus the closed …ber of ker t;0 is ker T t O and, since the Cartier duality commutes with base changes, the closed …ber of D
ker
t;0 O
t
inclusion ker
t
is D ker T t;0 O
t
ker t;0 O
nite K-group scheme ker
t;0 O t K
t
= co ker X
. Taking generic …bers of the
, we obtain the canonical inclusion of the …into ker
t
. By the Cartier duality we …nd the
faithfully ‡at morphism (the Cartier duality is exact on …nite schemes over a base): ker Since ker
t;0 O
t
= D ker
D
t;0 O
ker
t K
t
is multiplicative, the Cartier dual D
nite étale O-group scheme. We know that the closed …ber of D D ker T
t
= co ker X
t
and the generic …ber is D
t
t;0 O
ker
is a …-
ker
ker
t;0 O
t
t;0 O
is
t K
(again
the Cartier duality commutes with base changes). Since taking the …ber of a …nite étale O-group scheme does not depend on the choice of the point on the base, there is an identi…cation of GKur =K -modules D
ker
The claim follows.
t;0 O
t
K
= co ker X
t
.
Let now p be a …xed (…nite) prime of K and write ( ) (resp. X ( )) to denote the connected component (resp. character group) functor at p, i.e. consider the local …eld K = Kp (completion of K at p). Proposition 5.3 With the above notations, there are inclusions: SuppT (ker (f )) SuppT ( (f )) SuppT co ker ft SuppT ( (f )) Proof. By functoriality ker canonical diagram: ker
(f )
(f ) ker
ker ( )
( ) and we get, from Lemma 5.2, the co ker X
t
ker .
Since f and are T-stable, the ring T acts on them and hence the arrows in the above diagram are T-stable. The …rst inclusion follows. Since t = f t t f , by functoriality, there is a canonical epimorphism co ker
t
co ker
ft .
The kernel-cokernel exact sequence induced by the morphism between the short exact sequences describing the group of connected components, obtained with 23
t
in place of as in the proof of Lemma 5.2, shows the existence of a canonical t epimorphism co ker X t co ker . Thus the primes in the support of t co ker (f ) are contained in the support of co ker X t . To get the second inclusion it is su¢ cient to show that the support of co ker X t is contained in the support of co ker X t and then apply again Lemma 5.2. But consider an exact sequence: h 0 ! M ! N ! co ker h ! 0, where h is a morphism between T-modules. Taking the Z-dual we get the following exact sequence h
N ! M ! Ext1Z (co ker h; Z) , so that co ker h Ext1Z (co ker h; Z). If t 2 T belongs to the annihilator of co ker h it will induce the zero morphism on Ext1Z (co ker h; Z) and hence on co ker h too. Thus SuppT (co ker h ) SuppT (co ker h) and the application of this remark to h = X t gives the claim. Now assume f to be a closed immersion and form the exact sequence: f
0 ! A ! J ! B ! 0. By Proposition 5.1 the dual sequence 0 ! B t ! J t ! At ! 0 is exact too. Let TA (resp. TAt , TB or TB t ) be the maximal quotient of T acting on A (resp. At , B or B t ). Note that, since ft
f
A ! J ! J t ! At t
t
Bt ! J t ! J ! B are T-equivariant isogenies, TA = TAt and TB = TB t . Remark 5.4 Up to the identi…cation of A (resp. B t ) with its image in J (resp. J t ) via f (resp. t ), the following equality holds: (A) = A \
1
Bt .
In particular, when is a principal T-polarization, the primes in the support SuppT ( (A)) of (A) appear in the support of the T-modules TA and TB . Proof. This is easy. Assume now that there is an exact sequence as above where A = J p old has good reduction and B = J p new has purely toric reduction (at p). Since every morphism from an abelian variety with good reduction (at p) to an abelian 24
variety with purely toric reduction (at p) is trivial (see [18, Corollary 2.3]), it is easily checked that J p old (resp. J p new ) is characterized like the maximal subvariety (resp. maximal quotient) of J having good (resp. purely toric) reduction (at p). This maximality property shows that T acts on J p old (resp. J p new ) through a quotient Tp old (resp. Tp new ) that we call the p-old (resp. p-new) quotient. De…nition 5.5 By a prime of fusion we mean a prime in the support of both the T-modules Tp new and Tp old . The following remark is now a consequence of Remark 5.4: Remark 5.6 When : J ! J t is a principal T-polarization, the primes in the support SuppT Jip new of Jip new are primes of fusion.
5.2
Application to modular abelian varieties
By a modular form g of level N , except when di¤erently stated, we mean a modular form of level N , weight two and trivial character, which is an eigenform for all the Hecke operators Tn belonging to TN , the Hecke algebra of level N . We will often distinguish between the Hecke operators corresponding to the primes not dividing N and the primes dividing N : as usual we will write Tl 2 TN for l - N and Ul 2 TN for l j N . We are going to explain how the abstract setting of Section 5.1 applies to modular forms. Let p k N be a prime exactly dividing N = pM . In [15, Proof of Theorem 3.8] is explained how the results of Raynaud (see [14]) and Deligne-Rapoport (see [8]) combine together to produce an exact sequence 0 ! A ! J0 (pM ) ! Q ! 0. More precisely there are degeneracy maps (as de…ned for example in [16]): fi : X0 (pM )
X0 (M ) i = 1; 2.
The map f2 is the composition of f1 with the Atkin-Lehner involution Wp on X0 (pM ). They induce, by contravariant functoriality (i.e. Pic functoriality): 2
f : J0 (M ) ! J0 (pM ) . Dually they induce, by covariant functoriality (Albanese functoriality): 2
f : J0 (pM ) ! J0 (M ) . The Jacobian variety J0 (M ) has good reduction at p and the same is true for p new A := Im f . On the other hand J0 (pM ) := Q has purely toric reduction. Thus A ! J0 (pM ) (resp. J0 (pM ) ! Q) is a maximal good subvariety (resp. maximal purely toric quotient). In particular it makes sense to consider the pnew quotient TpN new , the p-old quotient TpN old and the derived notion of prime of fusion, that coincides with the Ribet de…nition (see [15] or [16]). 25
Let f and g be two modular forms of level M and pM , g being p-new. Denote by Of (resp. Og ) the order generated by the Fourier coe¢ cients of f (resp. g) and set Kf := Fr (Of ) (resp. Kg := Fr (Og )). The geometric meaning of the p-new assumption is that the Eichler-Shimura construction of Ag factors thought the p-new quotient of J0 (pM ): p new
; TppMnew
(J0 (pM ) ; TpM )Q ! J0 (pM )
Q
! (Ag ; Og )Q :
Assume that f and g are congruent modulo a prime l of Q (which does not divide pM ). As explained at the beginning of Section 4 the congruence relation allows us to consider the triple (Af ; Of ; lf )Q (resp. (Ag ; Og ; lg )Q ), where lf := Of \ l (resp. lg := Og \ l). Note that f is no longer an eigenform for the Up 2 TpM operator, when considered as a level pM form, so that one is led to introduce the so-called p-stabilized eigenform f # . More precisely, as in the proof of Theorem 4.2, (4) implies X2
ap (f ) X + p
X2
Since g is p-new, there is a unique unit root Then f # (q) := f (q)
ap (g) X + p mod l. p
p
(f ) mod l of the above polynomial.
p f (q p ) (f )
is a level pM mod l eigenform with eigenvalue 1 for the Up 2 TpM p (f ) = operator and an eigenform with eigenvalue an (f ) for the Tn 2 TpM operators (resp. the Un 2 TpM operators), for every n such that p - n. It follows that, O O setting Ff := lff (resp. Fg := lgg ) the forms f # and g de…ne homomorphisms (let g be the mod lg reduction of g obtained sending Tn to an (g)): f#
: TpM
Ff
F;
g
: TpM
Fg
F:
If f and g are congruent modulo a prime l of Q, L := g 1 (0) = f #1 (0) is a prime of fusion and conversely, if L TpM is a prime of fusion, there exist f and g as above and a prime l of Q such that f and g are congruent modulo l (see [16, 2.]). Hence the notion of prime of fusion and congruence between modular forms are essentially equivalent. Note that, in this case, the images Lp new and Lp old of L in TpN new and TpN old are non-trivial. In the following theorem we work with the special …ber at p of the modular p new Jacobian J0 (pM ). In fact J0 (pM ) , as well as any optimal quotient obtained from a p-new modular form of level pM , is a split purely toric abelian variety over the quadratic unrami…ed extension Kp =Qp . We set T := TpM . Theorem 5.7 Let : J0 (pM ) Ag = A be an optimal quotient, g being a p-new modular form of level pM . If l 2 SuppOg ( (A)) is coprime with cg , 2pM and p2 1 and it is such that the residual representation l is irreducible then: 26
L := g 1 (l) is a prime of fusion, i.e. there exists a modular form f of level M which is congruent to g (modulo some prime of Q); the quotient map p new
T
p new
L 1 : T
p new
: J0 (pM )
! Ag induces an isomorphism:
p new
J0 (pM )
T
L 1 ' ! T
(Ag )
Og
l 1 ; Og
the specialization map into the group of connected components gives an 1 1 isomorphism Ag (Kp ) Og lOg = (A) Og lOg and the local l-Kummer map factors through a canonical inclusion 0!
(A)
l 1 ! H 1 (Kp ; A [I]) . Og
Og
p new
The same holds for J0 (pM ) and its local L-Kummer map (as well as its factorization through the group of connected components) is identi…ed with the l-Kummer map of A. p new
Proof. Write A = Ag , J p new = J0 (pM ) , J = J0 (pM ) and consider them as T-module objects: they de…ne regular triples at L, since L is coprime with cg . The p-new assumption means that the optimal quotient uniquely factors through J p new A. By de…nition A is an optimal quotient J A and it is well known that taking torsion from an optimal quotient gives a faithfully ‡at morphism J [L] A [L] (see [6, Proof of Proposition 3.3]). Since J ! J p new is an optimal quotient we …nd the faithfully ‡at morphism J [L] J p new [L] which gives, followed by J p new [L] ! A [L], the faithfully ‡at morphism J [L] A [L]. It follows that J p new [L] ! A [L] is faithfully ‡at too. Since the residue characteristic of L is coprime with 2pM and l is irreducible, we know from [16, Theorem 5.2 (b)] that J [L] ' A [L] abstractly. In particular they have the same order and the chain of faithfully ‡at morphisms (i.e. epimorphisms in the category of GQ -modules) J [L]
Jp
new
[L]
A [L]
has to be a chain of isomorphisms. In particular the morphism J p new A induces an isomorphism J p new [L] ' A [L]. Lemma 3.6 implies that the L-torsion exact sequences of J p new and A are identi…ed, so that the same happens for the long exact sequences obtained by taking the GKp -invariants. Since (J p new ; T; L) and (A; T; L) are regular triples, the reductions are split purely toric (over Kp ) and L is coprime with p2 1, the third and the second points of the claim follow from Lemma 3.5 and Proposition 3.7. Since (A) is a …nite group, to say that L 2 SuppT ( (A)) is the same as 1 1 to say that (A) T LT is non-trivial and in this case, since ( p new ) T LT 1 is an isomorphism, we know that (J p new ) T LT is non-trivial, i.e. L 2 SuppT ( (J p new )). But l is irreducible so that, by [16, Theorem 5.2 (c)], L is 27
not an Eisenstein prime; since the primes in the support of primes (thanks to [16, Theorem 3.12]), we deduce that: L 2 SuppT co ker
J ! Jp
new
(J) are Eisenstein
.
Now the …rst point of the claim follows, in light of Remark 5.6, from the second inclusion of Proposition 5.3.
6
On the local l-divisibility of Stark-Heegner points
Let be a …nite abelian l-torsion group and let R be a noetherian commutative ring whose residue …elds are all of characteristic l. Let I be the augmentation ideal of the augmented R-algebra R [ ] and we write N to denote the norm operator. Lemma 6.1 The augmentation ideal I is contained in the Jacobson ideal and in particular R [ ] is a local ring when R is a local ring. Moreover the following facts hold: (a) if M is a …nite R [ ]-module, M = 0 implies M = 0; (b) if ' : M ! N is a morphism of R [ ]-modules, where M is …nite and ' : M ! N is injective, ' is injective too. Proof. (b) follows from (a) and the proof of (a) is easily reduced to [17, Chapter IX, Pharagraph 1, Lemma 2] (see [18, Lemma A.7]). Lemma 6.2 Let M be an R [ ]-module and let R be a …nite ring. Suppose that Re1 + ::: + Ren
M
is a free R-submodule where fei : i = 1; :::; ng is a basis. If there exist eei 2 M such that N (e ei ) = ei , then M 0 := R [ ] ee1 + ::: + R [ ] een
M
is a free R [ ]-submodule where fe ei : i = 1; :::; ng is a basis and M 0 = Re1 + ::: + Ren : Proof. This is easily deduced from Lemma 6.1 (see [18, Lemma A.8]).
6.1
The localization map
Now let H=L=K be a tower of …nite abelian …eld extensions, let G = GH=K be the Galois group of H=K and assume L=K to be an abelian l-power extension. Given an R-module and H=L=K de…ne L , where = : L := 2GL=K 28
This R-module can be also described as the set of maps from GL=K to and it is a GL=K -module by setting (
f ) ( ) := f (
) for ; 2 GL=K :
There are natural diagonal inclusions :
L=K
!
L,
H=L
:
H=K
:
!
H;
!
H
and also a diagonal inclusion
which is obtained, for
2 G, by the formula (let H=L f
It is clear that
L
=
H=L
L=K
( )=
H=K
and
H=L
(
in GL=K ):
( ) := f ( ) : L=K .
GL=K , L L)
be the image of
=
Then H=K
( )=
GH=K H
GH=L : H
Let S be a set endowed with a GH=K -action contained in an R-module M and suppose we are given an R-linear morphism ':M ! : Fix an element P 2 S and de…ne the following elements in choose a section s : GL=K ! GH=K , so that ` GH=K = g2GL=K GH=L s (g)
L
and
: …rst
and then de…ne
s;P L=K
:=
P
g2GL=K
' NH=L s (g) P g 2
L;
' NH=K P 2 ; P where NH=L s (g) P = 2GH=L s (g) P . In other words, as an application, for every g 2 GL=K : P s;P 2GH=L ' ( s (g) P ) : L=K (g) = Note that the de…nition of the element ' NH=K P is independent of the choice of the section s and also of the choice of the element P 2 S, when GH=K acts transitively on the set S. The following remark is an easy consequence of the de…nitions.
29
Remark 6.3 There are the following equations s; P L=K
s;P L=K ,
=
NL=K s;P L=K
=
where
L=K '
2 GL=K is the image of
in GL=K ,
NH=K P .
Lemma 6.4 Suppose that the R-action on factors through a …nite ring R I, R n where # I = l . If there exists P 2 S such that ' NH=K P generates a free s;P R R I -module (for example if ' NH=K P 6= 0, when I is a …eld), then L=K 2 L generates a free rank one R I GL=K -module. Proof. The lemma follows from Lemma 6.2, where R=
R , e1 = I
L=K '
NH=K P
and ee1 =
s;P L=K ;
in light of the second formula of the above remark.
Let A=Q be the abelian variety associated to a p-new modular form g of level pM . Consider a real quadratic …eld K=Q such that p is inert in K=Q and the primes dividing M are split in K=Q; the prime pOK splits completely in every narrow ring class …eld H=K (say of conductor coprime with the discriminant D = DK of K=Q and pM ). The theory of Stark-Heegner points provides us with a family of local points S = S H=K A (Kp ), depending on the choice of an embedding p : K ,! Kp into the quadratic unrami…ed extension of Qp . This theory is recalled in the following subsection. These points are endowed with the action of GH=K and we may choose an l-power extension L=K contained in H=K. The Hecke algebra acts on A = Ag through the order R = Og contained in the totally real …eld Kg . For a triple (Ag ; Og ; l)Q , let ' be the natural morphism A (Kp ) !
l 1 A (Kp ) = lA (Kp ) Og
Og
A (Kp ) .
Proposition 6.5 Let l Og be a prime dividing the rational prime l 6= p and 1 assume p2 6= 1 (mod l). If NH=K P 6= 0 in lOg Og A (Kp ) (any P 2 S), for every l-power extension L=K contained in H=K, the F GL=K -module (where O F := lg ) L l 1 Og A (Kp ) 2GL=K Og is freely generated by where
s;P L=K
(any P 2 S) and
2 GL=K is the image of . 1
s; P L=K
=
s;P L=K
for every
2 GH=K ,
Proof. Recall that lOg Og A (Kp ) is the image in H 1 (Kp ; A [l]) of the local l-Kummer map and, since Kp =Qp is a quadratic extension, the torus of A=Kp is 30
split and hence Lemma 3.5 applies: it follows that the F-rank of s;P L=K
is
l 1 Og
Og
A (Kp )
1. By Lemma 6.4, the element generates a free F GL=K -module of L l 1 rank 1 and this is Og A (Kp ), since this last module has F-rank 2GL=K Og less than [L : K].
6.2 6.2.1
Local divisibility of Stark-Heegner points Stark-Heegner points and Shimura reciprocity law
Consider the ring 1=p
R = M0
(M ) = fM 2 M2 (Z [1=p]) : cM
0 mod N g .
Let K be a real quadratic …eld such that p is inert in K and the primes dividing M are split. Choose embeddings: : K ! R and For each 2 Hp (K) := Hp \ K (use say that has positive orientation at p if ordp (
)
p p
: K ! Kp . to view K as a sub…eld of Kp ), we
0 (mod 2) .
The group naturally acts on Hp (K) and the subset Hp+ (K) of positive oriented by Möbius transformations. We denote by E = Emb (K; M2 (Q)) the set of Q-algebra embeddings of K into M2 (Q). We say that 2 E has a positive orientation at p if one of the two …xed point of (K ) in Hp (K) has a positive orientation at p (here M2 (Q) acts on Hp by Möbius transformations). The group naturally acts on E and the subset E + of positive oriented via Möbius transformations. Let K+ K be the subgroup of elements having positive norm. De…ne L0 (M ) as being the set of couples ( 1 ; 2 ) of Z [1=p]-lattices in K such that 1 ' MZZ . 2 There are natural bijections: L0 (M ) ' ' ! nHp+ (K) ! nE + . K+ The …rst bijection is obtained by sending (
1;
2)
to the ratio
= Z [1=p] e1 + Z [1=p] e2 ; = Z [1=p] M e1 + Z [1=p] e2 ; 2 (det (e1 ; e2 )) > 0; ordp ( p (det (e1 ; e2 ))) 0 (mod 2) ; 1
where det (e1 ; e2 ) := det 31
e1 e1
e2 e2
.
p
e1 e2
, where
The second bijection is obtained by associating to the class represented by 2 Hp+ (K) the representation obtained expressing the multiplication in K through the base ( ; 1). De…ne the Z [1=p]-order O( 1 ; 2 ) associated to ( 1 ; 2 ) 2 L0 (M ) as being the set of those 2 K such that i i for i = 1; 2. De…ne the Z [1=p]-order 1 associated to 2 E as being O := (R). Consider the collection of M 2 R such that M for some
M
1
=
M
1
2 K.
The association : M 7!
M
identi…es the above set of matrices with a Z [1=p]-order O in K (note the is the embedding associated to ). De…ne the Z [1=p]-order associated to O O as being O . Denote by LO 0 (M ) (resp. Hp (K) and E ) the subset of those + + elements in L0 (M ) (resp. Hp (K) and E ) whose associated order is O. The above bijections restricts to give bijections LO ' 0 (M ) ' ! nHpO (K) ! nE O . K+ The theory of Stark-Heegner points provides us with a uniformization map : nHp (K) ! J (Kp ) ! J0p
new
(pM ) (Kp ) ,
where J is an abelian variety over Kp isogeneous (over Kp ) to two copies of J0p new (pM ) via + . The proof of the uniformization gives a certain control over the set of primes appearing in the degree of the isogeny + . More precisely recall the Hecke algebra TN (resp. TM ) acting faithfully on H1 (X0 (N ) ; Z) (resp. H1 (X0 (M ) ; Z) ) and the maps induced in homology by contravariant and covariant functoriality: f f
2
: H1 (X0 (M ) ; Z) ! H1 (X0 (N ) ; Z) , 2
: H1 (X0 (N ) ; Z) ! H1 (X0 (M ) ; Z) .
There is also a morphism of Hecke modules 2
fcusp : H1 (X0 (M ) ; cusp; Z) ! H1 (X0 (N ) ; cusp; Z) . De…ne H (resp. Hcusp ) to be the torsion free quotient of the cokernel of f (resp. fcusp ); one can show that they are TpN new -modules (see [9, Proposition 4.2]). The natural Hecke equivariant map from H to Hcusp is an injection H H ' Z (see [9, Proposition 3.2]). Since the Hecke action on cusp is and cusp H H 32
Eisenstein (see [4, Proposition 1.1]) there is an Hecke equivariant morphism from Hcusp to H such that the composite H ! Hcusp ! H has …nite cokernel. The isogeny equivariant morphism.
depends on the choice of such a Hecke
+
De…nition 6.6 We denote by P = PpM; the set of primes appearing in the degree of this isogeny, depending on the factorization N = pM and the choice of (see [9, Section 5] for a bound on P ). The narrow Picard group P ic+ (O) is de…ned as being the quotient of the group of invertible ideals by the group of principal ideals that are generated by an element having positive norm. The class …eld theory identi…es canonically + P ic+ (O) with the Galois group of an extension HO of K: '
rec : P ic+ (O) ! GH + =K . O
The group P ic+ (O) acts on a (
LO 0 (M ) K+ 1;
2)
by the formula
:= (a
1; a 2) ,
so that it acts on nHpO (K) too. Denote by a this action. The main conjecture on Stark-Heegner points is the following conjectural Shimura Reciprocity law: Conjecture 6.7 If
2 HpO (K) then
( ) 2 J0p
(a
1
) = rec (a)
new
+ (pM ) HO and
( )
for all a 2 P ic+ (O). We also note that the action of P ic+ (O) on
LO 0 (M ) K+
(or nHpO (K)) splits this
set into 2#fljM g sets on which the action is simply transitive. More precisely, to …x an orbit corresponds to …x an orientation on O and the involutions Wl at l change the orientation at the prime l. 6.2.2
Stark-Heegner points and special values of complex L-functions
Let D = DK be the discriminant of the …eld K, let O be the Z [1=p]-order + associated to the narrow ring class …eld HO =K, …x an orientation :O! 33
Z MZ
and suppose D, the conductor of the order and N to be pairwise coprime. There is a unique Z-order OZ , having the same conductor of O, such that O = OZ [1=p]; the orientation on O induces an orientation on OZ . Let j be a family of representative for the M0 (M ) -conjugacy classes of oriented optimal embeddings of OZ in the ring M0 (M ) of the matrices of M2 (Z) having lower left entry congruent to zero modulo M . Denote by the same symbol 1=p (M ) obtained j the -oriented optimal embeddings of O in the ring R = M0 from them: they are a set of representative for the -conjugacy classes of oriented optimal embeddings of O in the ring R. The group of units in O of norm 1 is a free abelian group of rank 1. There is only one generator u with the property: (u ) > 1 if (u ) < 1 if
( )> ( )>
( ), ( ).
The de…nition of u is independent of the choice of . Set Mj := j (u). + De…ne the norm from HO to K of any ( j ) 2 J (Kp ) using the GH + =K -action O on the Stark-Heegner points: PK;J :=
h X j=1
( j ) 2 J (Kp ) .
Let now f be a modular form on 0 (M ) and let Af be the associated abelian variety. Popa’s formula states that, for any choice of z0 in the extended complex upper Poincarè halfplane, it holds the equality 0 12 M j z0 Z h BX C L (Af =K; 1) = @ f (z) dz A ; j=1
z0
where = means equality up to explicit non-zero factors. For a …nite set S of primes let ZS be the localization of Z at the primes contained in S, denote by [Mj ]S 2 H1 (X0 (M ) ; ZS ) the homology class corresponding to Mj and set h X [MK ]S := [Mj ]S : j=1
Finally de…ne L (Af =K; 1)S as being the image of [MK ]S in H1 (X0 (M ) ; ZS )f =:
H1 (X0 (M ) ; ZS ) : If H1 (X0 (M ) ; ZS )
Note that H1 (X0 (M ) ; ZS )f is a rank 2 module over Of and possibly enlarging S we may assume it to be a free rank 2 module over Of;S = OKf ;S . When this assumption is satis…ed, Popa’s formula implies the equivalence: L (Af =K; 1)S 6= 0 , L (Af =K; 1) 6= 0: 34
Let := (J) be the group of the connected component at p of the Néron model of J over the maximal unrami…ed extension of Qp , denote by S the localization of at a …nite set of primes S and let @S : J (Kp ) !
S
be the canonical map into the group of connected components. De…nition 6.8 Let SpN; be the set of primes dividing 6' (M ) p2 size of the cokernel of the composite
1 or the
H ! Hcusp ! H p new
f ree de…ned when we discussed the "Uniformization of J0 (pM ) ". Write SpN; to denote the set SpN; , enlarged in order to make H1 (X0 (M ) ; ZS )f a free Of;S = OKf ;S -module.
Proposition 6.9 When S = SpN; there are isomorphisms: 2
S
'
H1 (X0 (M ) ; ZS ) ' Im (f f ) Tp2
H1 (X0 (M ) ; ZS ) 2
(p + 1)
;
H1 (X0 (M ) ; ZS )
where the …rst identi…cation is of TN -modules and in the second identi…cation the action of Tn 2 TN corresponds to the action of Tn 2 TM for every n such that (n; p) = 1. Proof. See [4, Propoposition 3.3 and Corollary 3.4]. Let L be a prime of TN in the support of S : such a prime is both p-new and p-old by the above proposition. This means that it is a prime of fusion, i.e. there exist two eigenforms g 2 0 (N ) and f 2 0 (M ) that are congruent modulo a prime of Q (see Section 5.2). For any such prime it makes sense, by the above proposition, to consider the image L (Af =K; 1)S;L of tL (Af =K; 1)S 2 H1 (X0 (M ) ; ZS )f in ( S )L := ab S , an element of ZS since S contains the primes dividing L S . Here t = # 6' (M ) p2 1 and hence contains the primes dividing # ab (see [9, Proposition 3.7]); thus L (Af =K; 1)S;L 6= 0 if and only if the image of L (Af =K; 1)S in ( S )L is non-zero. f ree Theorem 6.10 When S = SpN; it holds the formula:
@S (PK;J ) = L (Af =K; 1)S;L in ( Proof. See [4, Theorem 3.9].
35
S )L
:
De…nition 6.11 Let SpM; be the set of primes SpM; := PpM; [ SpM; and ;f ree ;f ree f ree let SpM; be the set SpM; := PpM; [ SpM; . We remark that, in [4], it is conjectured that PpM; SpM; , i.e. SpM; = SpM; . Since the primes appearing in the kernel of the isogeny giving the p new uniformization of J0 (N ) are contained in PpM; , the uniformization of p new J0 (N ) induces an isomorphism: S
=
p new
J0 (N )
2 S
,
;f ree where S = SpM; (and in fact we can take S = PpM; ). Hence, any prime L p new
;f ree TpM appearing in the support of J0 (N ) coprime with S = SpM; is a prime of fusion, since it appears in the support of S (as we remark this is a consequence of Proposition 6.9). Moreover, since L is coprime with SpM; , we have ( S )L = L and L
=
p new
J0 (N )
2
L
.
Note that, the fact that certain primes appearing in the support of S are primes of fusion, is also a consequence of the …rst statement of Theorem 6.10. p new Let PK; be the image of PK;J in J0 (N ) corresponding to the choice of the sign . In other words: PK; :=
h X j=1
p new
( j ) 2 J0 (N )
The image of L (Af =K; 1)L in the factor
(Kp ) .
p new
J0 (N )
L
corresponding to
the choice of the sign is denoted by L (Af =K; 1)L; . Note that L (Af =K; 1)L is non-zero if and only if L (Af =K; 1)L; is non-zero. Then, Theorem 6.10, implies: p new @ (PK; ) = L (Af =K; 1)L; in J0 (N ) . L
If g is a p-new modular form of level N = pM , the abelian variety Ag is a p new quotient of J0 (pM ) and we can de…ne the Stark-Heegner point PK;g as the image of PK; in Ag (Kp ). Corollary 6.12 Let : J0 (pM ) Ag = A be an optimal quotient, g being a p-new modular form of level pM . If l 2 SuppOg ( (A)) is coprime with 2pM , ;f ree SpM; and it is such that the residual representation
l
is irreducible, then:
L := g 1 (l) is a prime of fusion, i.e. there exists a modular form f of level M which is congruent to g (modulo some prime of Q); the image of the Stark-Heegner point PK;g in only if L (Af =K; 1)L is non-zero. 36
Ag (Kp ) lAg (Kp )
is non-trivial if and
p new
Proof. Write A = Ag , J p new = J0 (pM ) and T := TpM . Since SpM; 2 contains the primes dividing p 1 and l is coprime with cg (because Of;S = ;f ree OKf ;S for S = SpM; ) Theorem 5.7 applies. The …rst point of the claim follows. For the second point note that the second and the third points of Theorem 5.7 give us the following commutative diagram in which every arrow is an isomorphism: @ J p new (Kp ) ! (J)L LJ p new (Kp ) # # @
Ag (Kp ) lAg (Kp )
!
(A)l .
As we remarked, Theorem 6.10 implies 1
p new L
(@ (PK;g )) = @ (PK; ) = L (Af =K; 1)L;
in
p new
J0 (pM )
L
and the claim follows from the fact that L (Af =K; 1)L is non-zero if and only if L (Af =K; 1)L; is non-zero.
7
Lifting Stark-Heegner points to Selmer classes
In the following discussion we consider a triple (Ag ; Og ; l)Q obtained from a p-new modular form g of level pM and a (non-zero) prime ideal l Og of residue characteristic l 6= 2, coprime with cg and ram (Kg =Q), i.e. we assume (Ag ; Og ; l)Q to be regular and unrami…ed. We recall the following de…nition that was given at the beginning of Section 4. De…nition 7.1 We say that Ag [lg ] satis…es the inertia assumption if, for every q k M , the dimension of the inertia invariants of Ag [lg ]Qq is 1 and, for any other prime dividing M , the dimension of the inertia invariants is 0. We consider a real quadratic …eld K such that p is inert in K=Q and the primes dividing M are split in K=Q and an abelian l-power extension L=K, dihedral over Q and contained in the …xed narrow ring class …eld H=K. We assume the conductor of the narrow ring class …eld H=K to be coprime with the discriminant DK of K=Q and pM , so that the theory of Stark-Heegner point is available. Let PK;g = NH=K P be the Stark-Heegner point de…ned before Corollary 6.12 (here P is any Stark-Heegner point associated to the narrow ring class …eld H=K). Let F be the residue …eld of Og at l. Assume the following conditions to be satis…ed: (i) ram (L=K=Q) is coprime with lN (so that L=K splits completely over p); (ii) l - BS (Ag =K)
(Ag ; L=K) (where S = split (L=K));
37
(iii) l is a prime appearing in the support SuppOg ( (Ag )) of the Og -module ;f ree (Ag ) coprime with the set SpM; ;
(iv) The representation
l;g
attached to Ag [l]Q is surjective;
(v) There are isomorphisms FPK;g =
l 1 Og
Og
Ag (K) = Sell (Ag =K) ,
where PK;g (mod l) is assumed to be rational over K. Theorem 7.2 Let (Ag ; Og ; l)Q and L=K be as above and assume (i)-(v). Then: L := g 1 (l) is a prime of fusion, i.e. there exists a modular form f of level M which is congruent to g (modulo some prime of Q); s;P L=K
if L (Af =K; 1)L 6= 0 the local point cs;P L=K
(any P 2 S) lifts to a global class
2 Sell (Ag =L) and Sell (Ag =L) = F GL=K cs;P L=K is a free rank one
P F GL=K -module. Moreover cs; L=K =
cs;P L=K for every
2 GL=K is the image of .
2 GH=K , where
Proof. Thanks to (iii) we can apply Corollary 6.12, which assures us that L 1 is a prime of fusion and PK;g 6= 0 in lOg Og Ag (Kp ), when L (Af =K; 1)l 6= L l 1 0. By Proposition 6.5, Og Ag (Kp ) is a free rank 1 F GL=K 2GL=K Og s;P 2 L=K (here we use that p 6= 1 (mod l), ;f ree SpM; contains the primes dividing p2
module generated by with
;f ree SpM;
and
(iv), it follows Sell (Ag =L) = Sell (Ag =K), where extension L=K (Proposition 3.9). Thus resp : Sell (Ag =L) !
L
2GL=K
l 1 Og
since l is coprime 1). By (ii) and
is the Galois group of the
Og
Ag (Kp ) 1
is injective on the -invariants (thanks to (v), because PK 6= 0 in lOg Og Ag (Kp )). Hence, by Lemma 6.1, resp is injective and in fact it is an isomorphism for F-rank reasons by the Theorem [19, Second Corollary of the Introduction], which applies thanks to (i) and (iv) (which implies Ag [l] (K) = 0, by Proposition 3.9). The claim follows from the fact that resp is an isomorphism of F GL=K modules. Assume the following conditions to be satis…ed: (i) ram (L=K) is coprime with lN (so that L=K splits completely over p); (ii) l - BS (Ag =K)
(Ag ; L=K) (where S = split (L=K)); 38
(iii) l is a prime appearing in the support SuppOg ( (Ag )) of the Og -module ;f ree (Ag ) coprime with the set SpM; ;
(iv) The representation inertia assumption.
g
attached to Ag [l]Q is surjective and satis…es the
Theorem 7.3 Let (Ag ; Og ; l)Q and L=K be as above and assume (i)-(iv). Then: L := g 1 (l) is a prime of fusion, i.e. there exists a modular form f of level M which is congruent to g (modulo some prime of Q); If L (Af =K; 1)L 6= 0 and Sellf (Af =K) = 0 (where lf is the maximal ideal of Of appearing in the de…nition of congruence between modular forms) s;P the local point s;P L=K (any P 2 S) lifts to a global class cL=K 2 Sell (Ag =L)
and Sell (Ag =L) = F GL=K cs;P L=K is a free rank one F GL=K -module.
P Moreover cs; L=K = image of .
cs;P L=K for every
2 GH=K , where
2 GL=K is the
Proof. As in the proof of the previous theorem, (iii) implies that L
2GL=K
l 1 Og
Og
Ag (Kp )
2 is a free rank 1 F GL=K -module generated by s;P L=K . Thanks to (i), (ii), p 6= 1 (mod l) (which is a consequence of (iii), as explained in the proof of the previous theorem) and (iv), we can apply Theorem 4.2, that gives the claim.
Role of the funding source The paper has been written at the Federigo Enriques department of Mathematics in Milan, Italy. There is no funding involvement.
Acknowledgements Many thanks to Professor Massimo Bertolini, for suggesting me the problem and indicating me the right way to approach it. The author is also indebted to Professor Henri Darmon, for his careful reading of his PhD thesis, and to Matteo Longo, for the precious discussions.
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