Let
f =
∑
an(
f)
e2πinz
be a weight-two newform on Γ
0(
N), where
N =
DM is the product of two relatively
prime integers
D and
M and where
D is
the discriminant of an indefinite quaternion division algebra over
Q. Assume that the Fourier coefficients of
f are
rational integers, so that
f is associated with an isogeny
class
![[mathematical script A]](corehtml/pmc/pmcents/x1D49C.gif)
of elliptic curves over
Q. Among the curves in
![[mathematical script A]](corehtml/pmc/pmcents/x1D49C.gif)
is a distinguished element
A, the strong modular curve
attached to
f. Shimura (
1) has constructed
A as
an optimal quotient of
J0(
N). Thus
A is the quotient of
J0(
N)
by an abelian subvariety of this Jacobian. Composing the standard map
X0(
N)
J0(
N) with the quotient ξ:
J0(
N) →
A, we obtain a
covering π:
X0(
N) →
A
whose degree δ is an integer which depends only on
f.
The integer δ has been regarded with intense interest for the last
decade. For one thing, primes dividing δ are “congruence primes for
f”: if p divides δ, then there is a
mod p congruence between f and a weight-two cusp
form on Γ0(N) which has integral coefficients
and is orthogonal to f under the Petersson inner product.
(See, e.g., Section 5 of ref. 2 for a precise statement.) For another,
it is known that a sufficiently good upper bound for δ will imply the
ABC Conjecture (3, 4). More precisely, as R. Murty explains
in ref. 24, the ABC Conjecture follows from the conjectural
bound
(For a partial converse, see ref.
5.) While δ is easy to
calculate in practice (
6), it seems more difficult to manage
theoretically. Murty (
24), has summarized what bounds are known at
present.
This note concerns relations between δ and analogues of δ in which
J0(N) is replaced by the Jacobian of
a Shimura curve.
To define these analogues, it is helpful to give a characterization of
δ in which π does not appear explicitly. For this, note that the
map ξ
: A
J0(N)
which is dual to
ξ may be viewed as a homomorphism A →
J0(N), since Jacobians of curves (and
elliptic curves in particular) are canonically self-dual. The image of
ξ
is a copy of A which is embedded in
J0(N). The composite
ξ
ξ
End A is necessarily
multiplication by some integer; a moment’s reflection shows that this
integer is δ. Let Γ0D(M) be
the analogue of Γ0(M) in which
SL(2, Z) is replaced by the group of norm-1 units
in a maximal order of the rational quaternion algebra of discriminant
D. Let
X0D(M) be the Shimura
curve associated with Γ0D(M)
and let J′ =
J0D(M) be the Jacobian
of X0D(M). The
correspondence of Shimizu and Jacquet–Langlands (7) relates
f to a weight-two newform f′ for the group
Γ0D(M); the form
f′ is well defined only up to multiplication by a
nonzero constant. Associated to f′ is an elliptic
curve A′ which appears as an optimal quotient ξ′ :
J′ → A′ of J′. Using the techniques
of Ribet (8) or the general theorem of Faltings (9), one proves that
A and A′ are isogenous—i.e., that
A′ belongs to
. We define
δD(M)
Z as the
composite ξ′
(ξ′)
.
To include the case D = 1 in formulas below, we set
δ1(N) = δ.
Roberts (10) and Bertolini and Darmon (section 5 of ref. 11) have
pointed out that the Gross–Zagier formula and the conjecture of Birch
and Swinnerton-Dyer imply relations between δ and
δD(M) in
Q*/(Q*)2. Bertolini and Darmon
allude to the possibility that there may be a simple, precise formula
for the ratio δ/δD(M). The
relation which they envisage involves local factors for the elliptic
curves A and A′ at the primes
p|D.
While these factors may well be different for the two elliptic curves,
we will ignore this subtlety momentarily and introduce only those
factors which pertain to A. Suppose, then, that p
is a prime dividing D, so that A has
multiplicative reduction at p. Let cp
be the number of components in the fiber at p for the
Néron model of A; i.e., cp =
ordpΔ, where Δ is the minimal discriminant
of A. As was mentioned above, δ controls congruences
between f and newforms other than f in the space
S of weight-two forms on Γ0(N);
analogously, δD(M) controls
congruences between f and other forms in the
D-new subspace of S. At the same time,
level-lowering results such as those of Ribet (12) lead to the
expectation that the cp control congruences
between f and D-old forms in S. This
yields the heuristic formula:
Equivalently, one can consider factorizations
N =
MpqD, where
p and
q are distinct prime
numbers,
D ≥ 1 is the product of an even number of
distinct primes, and the four numbers
p, q, D, and
M are relatively prime. The formula displayed above amounts
to the heuristic relation
for each factorization
N =
MpqD.
Although simple examples show that Eq.
1 is not correct as
stated, we will prove that a suitably modified form of it is valid in
many cases.
To state our results, we need to be more precise about the numbers
cp and cq which appear
above. We set:
Let ξ :
J →
A and ξ′ :
J′
→
A′ be the optimal quotients of
J and
J′ for which
A and
A′ lie in
![[mathematical script A]](corehtml/pmc/pmcents/x1D49C.gif)
.
(This is a change of notation, since we have been taking
A
to be an optimal quotient of
J0(
N);
the new elliptic curve
A is the unique curve isogenous to
the original
A which appears as an optimal quotient of
J.) Let
cp and
cq be defined for
A as above; i.e.,
cp =
ord
pΔ(
A) and
cq =
ord
qΔ(
A). Note that
cp, for instance, may be viewed as the order of
the group of components of the fiber at
p of the Néron
model for
A. This group is cyclic. Let
c′
p and
c′
q be defined analogously,
with
A′ replacing
A. Notice that
ord
cp =
ord
c′
p and
ord
cq =
ord
c′
q for
each prime
![[ell]](corehtml/pmc/pmcents/x2113.gif)
such that
A[
![[ell]](corehtml/pmc/pmcents/x2113.gif)
] is irreducible. Indeed,
the curves
A and
A′ are isogenous over
Q. The irreducibility hypothesis on
A[
![[ell]](corehtml/pmc/pmcents/x2113.gif)
]
implies that any rational isogeny
A →
A′ of
degree divisible by
![[ell]](corehtml/pmc/pmcents/x2113.gif)
factors through the multiplication-by-
![[ell]](corehtml/pmc/pmcents/x2113.gif)
map
on
A. Hence there is an isogeny ϕ :
A →
A′ whose degree is prime to
![[ell]](corehtml/pmc/pmcents/x2113.gif)
. If
d = deg
ϕ, the map ϕ induces an isomorphism between the
prime-to-
d parts of the component groups of
A and
A′, both in characteristic
p and in
characteristic
q.
Theorem 1. One has
where the “error term” ![x2130](corehtml/pmc/pmcents/x2130.gif)
(
D,
p,
q,
M)
is a positive divisor
of c′pcq. Further, suppose that M is
square free but not a prime
number,
† and let
be a prime number which divides ![x2130](corehtml/pmc/pmcents/x2130.gif)
(
D, p, q, M).
Then the
Gal(
![equation M6 equation M6](picrender.fcgi?artid=34176&blobtype=equ&blobname=M6)
/
Q)-
module
A[
![[ell]](corehtml/pmc/pmcents/x2113.gif)
]
is reducible.
In our proof of the theorem, we shall first prove a version of the
displayed formula in which
(D, p, q, M) is expressed in
terms of maps between component groups in characteristics p
and q. (See Theorem 2 below.) We then prove the
second assertion of Theorem 1.
Before undertaking the proof, we illustrate Theorem 1 by
considering a series of examples. As the reader will observe, these
examples show in particular that the “error term”
(D, p,
q, M) is not necessarily divisible by all primes
for which
A[
] is reducible.
For the first example, take M = 1,
D = 1, and pq = 14. Thus
N = 14, so that the curves A and
A′ lie in the unique isogeny class of elliptic curves over
Q with conductor 14. [There is a unique weight-two newform
on Γ0(14).] According to the tables of Antwerp IV, there
are six curves in this isogeny class [ref. 13, p. 82]. The curve
A is identified as [14C] in the notation of ref. 13. We
have A = J0(14); and
A′ = J014(1), so that
δ1(14) = δ14(1) = 1. Since
c2 = 6 and c7 = 3,
Theorem 1 yields the pair of equalities
there are two equalities because there are two choices for the
ordered pair (
p, q). By
Theorem 1, the integers
![x2130](corehtml/pmc/pmcents/x2130.gif)
(1, 2, 7, 1) and
![x2130](corehtml/pmc/pmcents/x2130.gif)
(1, 7, 2, 1) are divisible only by the primes 2
and 3. Indeed, we see (once again from ref.
13) that these are the only
primes
![[ell]](corehtml/pmc/pmcents/x2113.gif)
for which
A[
![[ell]](corehtml/pmc/pmcents/x2113.gif)
] is reducible. Looking further
at the tables, we see that there is a unique curve
A′ in the
isogeny class of
A for which
3
c′
2 is the square of an integer. This
curve is [14D]. Thus we have
A′ = [14D], as Kurihara
determined in ref.
14.
There are five similar examples of products pq for which
J0pq(1) has genus one, namely
2
17, 2
23, 3
5, 3
7, and 3
11. [See, e.g.,
the table of Vignéras (ref. 15, p. 122).] In each of the five
cases, we shall see that A′ =
J0pq(1) can be determined as a
specific elliptic curve of conductor pq with a small amount
of detective work. [We suspect that this detective work was done 15
years ago by J.-F. Michon (see refs. 16 and 17).]
To begin with, we note that in each case there is a single weight-two
newform on Γ0(pq) with integral coefficients,
i.e., a single isogeny class of elliptic curves of conductor
pq. The strong modular elliptic curve A of
conductor pq is identified in ref. 13. Knowing this curve,
we have at our disposal cp and
cq. Further, the integer
δ1(pq) is available from Cremona’s table
(ref. 6, pp. 1247–1250).
In the two cases pq = 3
5 and 3
7,
A coincides with the Jacobian
J0(pq). In this circumstance, an easy
argument based on Proposition 1 below shows that the local
invariants of A and A′ =
J0pq(1) are “flipped”—we
have c′p =
cq and
c′q =
cp. After glancing at p. 82 of ref. 13, one sees
that A′ = [15C] in the first of the two cases and
A′ = [21D] in the second.
Let us now discuss the remaining three cases, 2
17, 2
23, and
3
11, where Cremona’s table gives the values 2, 5, and 3
(respectively) for δ1(pq). Using Theorem
1 and the value δpq(1) = 1 in each case,
we obtain equations which express
c′p and
c′q as products of known
rational numbers and unknown square integers. These are enough to
determine A′. Indeed, when pq = 34, we have
so that 3
c′
17 is a square. We then
must have
A′ = [34C]. When
pq = 46,
2
c′
23 is a square, and we conclude
A′ = [23B]. When
pq = 33,
2
c′
11 is a square and thus
A′
= [33B].
In the six examples we have discussed so far, an alternative approach
would have been to read off the numbers
c′p and
c′q from a formula of Jordan
and Livné (section 2 of ref. 18; see Theorem 4.3 of ref. 8). As
we have seen, A′ is determined in each case by these local
invariants.
For an example with a different flavor, we take f to be the
modular form associated with the curve A = [57E] of
ref. 13. This curve is isolated in its isogeny class; i.e.,
A[
] is irreducible for all
. In particular,
A′ = A. Because A[
] is
irreducible for all
, the theorem gives
(1, 3, 19, 1) = 1. Hence
Now Cremona’s table (ref.
6, p. 1247) yields the value
δ
1(57) = 4; also, one has
c3 = 2,
c19 = 1. Thus we find δ
57(1) = 2.
This relation is confirmed by results of D. Roberts (
10), who shows,
more precisely, that
A is the quotient of
X057(1) by its Atkin–Lehner involution
w57.
Next, we consider the elliptic curves of conductor N =
714 = 2
3
7
17, which are tabulated in Cremona’s
book (19). These curves fall into nine isogeny classes, A–I. Four of
these classes, B, C, E and H, contain precisely one element. In other
words, the four elliptic curves 714B1, 714C1, 714E1, and 714H1 are
isolated in their isogeny classes. For each elliptic curve,
Theorem 1 expresses δ714(1) as well as the six
degrees δpq(714/pq) for
pq|714 in terms of δ1(714) and the integers
cp for p|714. These numbers are
available from refs. 6 and 19. The most striking of the four elliptic
curves is perhaps 714H1. For this curve, c2 =
c3 = c7 =
c17 = 1 and δ1(714) = 40. Hence
δ714(1) and all degrees
δpq(714/pq) are equal to 40.
For a final example, we consult further tables of John Cremona
which are available by anonymous ftp from euclid.ex.ac.uk in
/pub/cremona/data. Let A be the curve denoted
1001C1, which has Weierstrass data [0, 0, 1, −199, 1092]. Its
minimal discriminant is −72113132.
This curve is isolated in its isogeny class, which suggests that
(1,
7, 13, 11) = 1. Since 11 is a prime, the second part of Theorem
1 yields no information. However, by Proposition 3
below,
(1, 7, 13, 11) divides both c7 and
c13. Each of these integers is 2, so that we may
conclude at least that
(1, 7, 13, 11) is 1 or 2. Cremona’s tables
give the value δ1(1001) = 1008; hence
δ7
13(11) is either 252 or
1008.‡ On the other hand, since
c7 and c11 are relatively
prime, we find that
(1, 7, 11, 13) = 1. Thus
δ7
11(13) = 1008/6 = 168. Similarly,
δ11
13(7) = 168.
The First Assertion of Theorem 1 If V is an abelian variety over Q and
is
a prime, let Φ(V,
) be the group of components of the
fiber at
of the Néron model of V. This group is a
finite étale group scheme over Spec F
,
i.e., a finite abelian group furnished with a canonical action of
Gal(![equation M10 equation M10](picrender.fcgi?artid=34176&blobtype=equ&blobname=M10)
/F
).
The association V
Φ(V,
) is functorial.
For example, as we noted above, if A is an elliptic curve
with multiplicative reduction at p, then Φ(A,
p) is a cyclic group of order cp.
The maps ξ and ξ′ induce homomorphisms
Here is a version of the first assertion of
Theorem 1
in which
![x2130](corehtml/pmc/pmcents/x2130.gif)
(
D, p, q, M) appears with a precise value.
Theorem 2. One has
where ![x2130](corehtml/pmc/pmcents/x2130.gif)
(
D,
p,
q,
M) =
#
imageξ
![[low asterisk]](corehtml/pmc/pmcents/lowast.gif)
![[center dot]](corehtml/pmc/pmcents/middot.gif)
#
coker
ξ′
![[low asterisk]](corehtml/pmc/pmcents/lowast.gif)
.
To prove the theorem, we compare the character groups of algebraic tori
which are associated functorially to the mod p reduction of
J′ and the mod q reduction of J.
Recall that the former reduction is described by the well known theory
of Cerednik and Drinfeld (20–22), while the latter falls into the
general area studied by Deligne and Rapoport (23). [Although Deligne
and Rapoport provide only the briefest discussion of the case
D > 1, what we need will follow from recent results of
K. Buzzard (31).] Our comparison is based on the oft-exploited
circumstance that the two reductions involve the arithmetic of the same
definite rational quaternion algebra: that algebra whose discriminant
is Dq.
To state the result which is needed, we introduce some notation:
if V is an abelian variety over Q and
is a
prime number, let T be the toric part of the fiber over
F
of the Néron model for V
and write
(V,
) for the character group
Hom![equation M13 equation M13](picrender.fcgi?artid=34176&blobtype=equ&blobname=M13)
(T,
Gm). Thus
(V,
) is a free
abelian group which is furnished with compatible actions of
Gal(![equation M14 equation M14](picrender.fcgi?artid=34176&blobtype=equ&blobname=M14)
/F
)
and EndQ V. At least in the case when
V has semistable reduction at
, there is a canonical
bilinear pairing
which was introduced by Grothendieck (Theorem 10.4 of ref.
25).
If, moreover,
V is canonically self-dual (e.g., if
V is the Jacobian of a curve or a product of Jacobians),
then the monodromy pairing
uV is a pairing on
![X](corehtml/pmc/pmcents/1D4B3.gif)
(
V,
![[ell]](corehtml/pmc/pmcents/x2113.gif)
) (in the sense that it is defined on the product
of two copies of this group).
The relation between Φ(V,
) and the character groups
is as follows (Theorem 11.5 of ref. 25): there is a natural exact
sequence
in which α is obtained from
uV by the
standard formula (α(
x))(
y) =
uV(
x, y).
Proposition 1. There is a canonical exact sequence
where J" =
J0D(
qM).
The sequence is
compatible with the action of Hecke operators Tn
for n prime to N, which operate in the usual way on J, J′, and J".
Moreover, the map ι
is compatible with the monodromy
pairings on ![X](corehtml/pmc/pmcents/1D4B3.gif)
(
J′, p)
and ![X](corehtml/pmc/pmcents/1D4B3.gif)
(
J,
q)
in the sense that uJ′(
x, y) =
uJ(
ιx, ιy)
for all x, y
(J′, p).
When D = 1, the proposition was proved in ref.
12. (See especially Theorem 4.1 of ref. 12.) The case D
> 1 can be handled in an analogous way, thanks to K. Buzzard’s
analogue (31) of the Deligne–Rapoport theorem (23). This theme is
explored in the work of Jordan and Livné (26) and L. Yang (27).
Let
be the “f-part” of
(J, q),
defined for example as the group of characters x
(J, q) such that Tnx =
an(f)x for
all n prime to N. [Recall that
an(f) is the
nth coefficient of f.] It is not hard to check
that
is isomorphic to Z and that in fact it is contained
in
(J′, p), viewed as a subgroup of
(J, q) via ι. Indeed, consider the decomposition of
J as a product up to isogeny of simple abelian varieties
over Q. One of the factors is A, which occurs
with multiplicity 1, and the other factors are non-f: they
correspond to newforms of level dividing N whose
nth coefficients cannot coincide with the
an(f) for all
n prime to N. Hence
Q is the
tensor product with Q of the character group of the toric
part of AFq;
this shows that
has rank 1. A similar computation shows that
∩
(J′, p) has rank 1, since A
occurs up to isogeny exactly once in J′ and since
A has multiplicative reduction at p. The image of
in
(J", q) ×
(J",
q) is thus finite; it is zero since
(J",
q) is torsion free.
Fix a generator g of
and set τ =
uJ(g, g). An arbitrary nonzero
element t of
may be written ng, where
n is a nonzero integer. We then have
uJ(t, t) =
n2τ.
By the theorem of Grothendieck (25) that was cited above,
cq may be interpreted as
uA(x, x), where x is a
generator of
(A, q) and where uA
is the monodromy pairing arising from the mod q reduction of
A. Meanwhile, the map ξ : J → A
induces by pullback a homomorphism ξ* :
(A, q) →
(J, q) and the dual of ξ induces similarly a
homomorphism ξ* :
(J, q)
→
(A, q). The two homomorphisms are adjoint with
respect to the monodromy pairings:
Notice, however, that ξ
*![[open circle]](corehtml/pmc/pmcents/cir.gif)
ξ* is
multiplication by δ := δ
D(
pqM) on
![X](corehtml/pmc/pmcents/1D4B3.gif)
(
A, q), since it is induced by the endomorphism
“multiplication by δ” of
A. Thus
for all
x
![X](corehtml/pmc/pmcents/1D4B3.gif)
(
A, q). On taking
x to be a generator of
![X](corehtml/pmc/pmcents/1D4B3.gif)
(
A, q), we find
where we view
![X](corehtml/pmc/pmcents/1D4B3.gif)
(
A, q) as embedded in
![L](corehtml/pmc/pmcents/x2112.gif)
by ξ*. A
similar argument applied to
A′ mod
p yields
δ′
c′
p = (
![L](corehtml/pmc/pmcents/x2112.gif)
:
![X](corehtml/pmc/pmcents/1D4B3.gif)
(
A′,
p))
2![[center dot]](corehtml/pmc/pmcents/middot.gif)
τ, where δ′ =
δ
Dpq(
M). [To prove this relation,
one must view
![L](corehtml/pmc/pmcents/x2112.gif)
as a subgroup of
![X](corehtml/pmc/pmcents/1D4B3.gif)
(
J′,
p) and
interpret τ as
uJ′(
t, t), where
g = ι
t. The legitimacy of this
interpretation stems from the compatibility among ι,
uJ, and
uJ′.]
We emerge with the preliminary formula
After isolating δ′ on one side of the equation, we see that
Theorem 2 is implied by the following result:
Proposition 2. Let
ξ* and
ξ′* be the
homomorphisms Φ(J, q) →
Φ(A, q) and Φ(J′,
p) → Φ(A′, p) which are
induced by ξ and ξ′ on component groups.
Then (
:
(A, q)) = #coker
ξ* and (
:
(A′, p)) = #coker
ξ′*.
Proof: The two formulas are analogous; we shall prove only
the assertion relative to ξ*.
Because of the assumption that ξ : J →
A is an optimal quotient, the map ξ
:
A → J is injective. One deduces from this the
surjectivity of the map on character groups
ξ* :
(J, q) →
(A, q). Consider the commutative diagram with exact rows
in which the three vertical maps are induced by ξ. [For
instance, the central vertical map is Hom(ξ*,
Z), where
ξ* :
![X](corehtml/pmc/pmcents/1D4B3.gif)
(
A, q) →
![X](corehtml/pmc/pmcents/1D4B3.gif)
(
J, q) is an injective
map between free abelian groups of finite rank.] The exactness of the
rows is guaranteed by Theorem 11.5 of ref.
25. Because the left-hand
vertical map is surjective, the cokernels of Hom(ξ*,
Z)
and the right-hand ξ
* may be identified.
It is clear that the order of coker(Hom(ξ*, Z)) coincides
with the order of the torsion subgroup of coker(ξ*). Since
(J, q)/
is torsion free by the definition of
, we
obtain first the formula
and then the desired equality.
![[filled square]](corehtml/pmc/pmcents/x25AA.gif)
The Second Assertion of Theorem 1 We assume from now on that N is square free and that
is a prime for which A[
] is irreducible. We should
mention in passing that the irreducibility hypothesis holds for one
A
if and only if it holds for all
A
. Indeed, the semisimplification of the mod
Galois representation A[
] depends only on
. At
the same time, A[
] is irreducible if and only if its
semisimplification is irreducible.
Lemma 2. There is a prime r|N for which
does not divide cr.
Proof: Suppose to the contrary that
divides
cr for all r|N. Then the
mod
Gal(
/Q)-representation
A[
] is finite at all primes (section 4.1 of
ref. 28). If
= 2, this contradicts a theorem of Tate (29). If
> 2, a theorem of the first author (Theorem 1.1 of ref. 12) implies
that A[
] is modular of level 1 in the sense that it
arises from the space of weight-two cusp forms on SL(2,
Z). Since this space is zero, we obtain a contradiction in
this case as well.![[filled square]](corehtml/pmc/pmcents/x25AA.gif)
In order to prove the second assertion of Theorem 1, which
concerns the “
-part” of
(D, p, q, M), we will
consider varying decompositions N =
D
p
q
M. In
these decompositions, the isogeny class
, and the integer
N in particular, are understood to be invariant. We view the
prime
as fixed, and recall the hypothesis that A[
]
is irreducible. (If this irreducibility hypothesis holds for one
A
, then it holds for all A.) Set
so that
e(
D, p, q, M) is the
“
![[ell]](corehtml/pmc/pmcents/x2113.gif)
-part” to be studied.
Proposition 3. If N = DpqM, then
e(D, p, q, M) is the order of the
-primary part of the cokernel of
Further, we have e(D, p, q, M) =
e(
D, q, p, M),
and e(
D, p, q,
M)
divides both cp and cq.Proof: In view of Theorem 2, the first statement
means that the
-primary part of the image of
ξ* : Φ(J, q) →
Φ(A, q) is trivial. For each prime number r
which is prime to N, let Tr be the
rth Hecke operator on J. It is a familiar fact
that Φ(J, q) is Eisenstein in the sense that
Tr acts on Φ(J, q) as 1 +
r for all such r. This was proved by the first
author in case D = 1 (see Theorem 3.12 of ref. 12 and
ref. 30), and the result can be extended as needed in view of results
of Buzzard (31) and Jordan and Livné (26).
It follows from the Eichler–Shimura relation that the image of
ξ* is annihilated by
ar(f) −
r − 1 for all r. One deduces from this that
the
-primary part of the image is trivial: If not, then
ar(f)
r + 1 mod
for all r, and this implies
that the semisimplification of A[
] is reducible; cf.
Theorem 5.2(c) of ref. 12.
To prove the second statement, we begin by noting that
e(D, p, q, M) divides
c′p. As we pointed out
earlier, there is an isogeny A → A′ of
prime-to-
degree. Indeed, A and A′ are
isogenous over Q; on the other hand, the hypothesis on
A[
] implies that any rational isogeny A →
A′ of degree divisible by
factors through the
multiplication-by-
map on A. Hence the
-primary
components of Φ(A, p) and Φ(A′, p)
are isomorphic, so that the largest powers of
in
cp and
c′p are the same. Thus
e(D, p, q, M) divides cp.
Also, since an analogous reasoning shows that cq
and c′q have the same
valuations at
, e(D, p, q, M) depends
symmetrically on p and q, as asserted. Finally,
e(D, p, q, M) divides both
cp and cq, since it
divides cp and depends symmetrically on
p and q.![](corehtml/pmc/pmcents/x2009.gif)
![[filled square]](corehtml/pmc/pmcents/x25AA.gif)
Corollary. If N = dpqrsm, where
p, q, r, and s are primes and d
is the product of an even number of primes, then
and e(
d, r, s, pqm) =
e(
d,
q, s, prm).
Proof: Each of the two integers in the displayed equality
may be calculated as the order of the
-primary part of the cokernel
of ξ′
:
Φ(Jdpqrs(m), p) →
Φ(A′, p). This coincidence gives the first
equality. To obtain the second from the first, we note that both
e(rsd, p, q, m)2
e(d, r, s, pqm)2 and
e(qsd, p, r, m)2
e(d, q, s, prm)2 are equal to the
-part of the quantity
δdpqrs(m)cpcqcrcs/δd(pqrsm).![[filled square]](corehtml/pmc/pmcents/x25AA.gif)
To finish the proof of Theorem 1, we assume from now on that
M is not prime. To prove that e(D, p, q,
M) = 1, it suffices to show that e(D, p, q,
M) divides cr for each
r|N. If r = p or
r = q, this divisibility is included in the
statement of Proposition 3. Assume, next, that r
is a divisor of D, and write D =
rsd, where s is a prime. We have
where the second equality follows from the
Corollary.
The latter number divides
cr, as required.
Finally, suppose that
r divides
M. Since
M is not prime, we may write
M =
rsm, where
s is a prime. We have seen that
e(
D, r, s, pqm) =
e(
D, q, s,
prm). Permuting the roles of the four primes
p, q, r,
and
s, we may write instead
e(
D, p, q,
rsm) =
e(
D, r, q, psm). The latter number is
a divisor of
cr.