A study on the pro-p outer Galois representations associated to once-punctured CM elliptic curves for ordinary primes

概要

Summary : A study on the pro-p outer Galois
representations associated to once-punctured CM
elliptic curves for ordinary primes
Shun Ishii
In this thesis [3], we study the pro-p outer Galois representation associated
to once-punctured CM elliptic curves defined over imaginary quadratic fields of
class number one for ordinary primes.
Chapter I includes a brief overview of previous studies on the pro-p outer
Galois representations associated to the thrice-punctured projective line, which
motivates this thesis. In Chapter I, we also review previous results on the
pro-p outer Galois representations associated to once-punctured elliptic curves,
especially Nakamura’s construction of a certain power series explained below.
Let k be an imaginary quadratic field of class number one and p ≥ 5 a
rational prime which splits in k as (p) = pp¯ for distinct maximal ideals p, p¯ of
k. We denote the number of roots of unity in k by w and the absolute Galois
group of k by Gk . We denote the mod-p (resp. mod-p, mod-p¯) ray class field of
k by k(p) (resp. k(p), k(p¯)).
Let (E, O) be an elliptic curve over k of the Weierstrass form y 2 = 4x3 −
g2 x − g3 with g2 , g3 ∈ k which has complex multiplication by the integer ring
Ok of k and has good reduction at p. We set X := E \ O, which we refer to as
a once-punctured CM elliptic curve in this thesis.
We denote the pro-p geometric fundamental group of X by Π1,1 . We can
identify Π1,1 with a free pro-p group of rank two of the form ⟨x1 , x2 , z | [x1 , x2 ] =
z⟩. Moreover, we can choose a basis {x1 , x2 } of Π1,1 so that the image of x1 (resp.
∼
x2 ) in Πab
¯ (E) generates the p-adic Tate module Tp (E)
1,1 = Tp (E) = Tp (E) ⊕ Tp
(resp. the p¯-adic Tate module Tp¯ (E)) of E and z = [x1 , x2 ] generates the inertia
⃗ the tangential basepoint with parameter
subgroup at O corresponding to O,
¯ by
t = − 2x
.
We
denote
the
image
of x1 and x2 in Tp (E) = limn E[pn ](Q)
y
←−
(ω1,n )n and (ω2,n )n , respectively.
The p-adic Tate module Tp (E) and the p¯-adic Tate module Tp¯ (E) induce
Galois representations:
χ1 : Gk → Aut(Tp (E)) ∼
= Z×
p

and

χ2 : Gk → Aut(Tp¯ (E)) ∼
= Z×
p,

×
1 m2
respectively. For m = (m1 , m2 ) ∈ Z2 , we define χm := χm
1 χ2 : Gk → Zp and
denote the χm -twist of Zp by Zp (m). In the following, we denote m1 + m2 by
|m| and (1, 1) by 1.

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We have the pro-p outer Galois representation associated to X:
ρX,p : Gk → Out(Π1,1 ).
Nakamura [5] constructed a Gal(k(E[p∞ ])/k)-equivariant homomorphism:
α1,1 : Gk(E[p∞ ]) → Zp [[T1 , T2 ]](1)
which represents the action of the Galois group Gk(E[p∞ ]) on Π1,1 (2)/[Π1,1 (2), Π1,1 (2)]
through ρX,p . Here, (1) means the Tate twist. Moreover, Nakamura found an
explicit formula of the power series α1,1 (σ) for every σ ∈ Gk(E[p∞ ]) as an element
of Zp [[T1 , T2 ]] ⊂ Qp [[U1 , U2 ]] where Ui := log(1 + Ti ) for i = 1, 2 as follows:
Theorem 1 (Nakamura [5, Theorem A]).
∞
∑

α1,1 (σ) =

m≥2 : even

1
1 − pm

∑

κm+1 (σ)

m=(m1 ,m2 )∈Z2≥0
|m|=m

U1m1 U2m2
m1 !m2 !

n
holds for every σ ∈ Gk(E[p∞ ]) . Here, κm : Gab
k(E[p∞ ]) → Zp modulo p is a
Kummer character associated to the following element for every n ≥ 1:

∏

(

∞
∏

θp (aω1,n+1+k + bω2,n+1+k )−p )a
2k

m1 −1 m2 −1
b

0≤a,b p∤gcd(a,b)

where the function θp is a rational function on E defined in [3, Chapter II, §2].
We call the character κm : Gk(E[p∞ ]) → Zp the m-th elliptic Soul´e character.
One can observe that κm is trivial unless m1 ≡ m2 mod w.
In his previous work, Nakamura proved that certain linear combinations
of the elliptic Soul´e characters are nonzero for general once-punctured elliptic
curves defined over number fields. In Chapter II, we prove the non-vanishing or
even the surjectivity of the elliptic Soul´e characters for the once-punctured CM
elliptic curve X over k under various situations:
Theorem 2 ([3, Theorem II.4.4]). Assume that H 2 (Ok,Sp , Zp (m)) is finite.
Then the m-th elliptic Soul´e character κm : Gk(E[p∞ ]) → Zp (m) is nontrivial.
Here, H 2 (Ok,Sp , Zp (m)) is the second ´etale cohomology group of the spectrum of the ring of p-integers Ok,Sp of k, whose finiteness is a special case of
the so-called Jannsen conjecture. The key ingredients to prove Theorem 2 are
Iwasawa-theoretic machineries developed by Kings [4] to prove (a weaker version
of) the Tamagawa number conjecture for CM elliptic curves.
Moreover, we prove the surjectivity and non-surjectivity of the elliptic Soul´e
characters by using a relationship between the group of elliptic units defined by
Rubin and the class numbers of the ray class fields of k as follows:

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Theorem 3 ([3, Proposition II.5.1-5.5]). The elliptic Soul´e characters have the
following properties:
1. κm is not surjective for any m ∈ Z2≥2 \ {1} with m ≡ 1 mod p − 1.
2. κ(m,1) and κ(1,m) are surjective for every m > 1 with m ≡ 1 mod p − 1.
3. If the class number of k(p) (resp. k(p¯)) is not divisible by p, then κ(m,1)
(resp. κ(1,m) ) is surjective for every m > 1 with m ≡ 1 mod w.
4. If p is regular for k(p), then κm is surjective for every m ∈ Z2≥1 \ {1}
such that m ̸≡ 1 mod p − 1 and m1 ≡ m2 mod w.
Here, we say that p is regular for k(p) if the maximal pro-p extensions of
k(p) unramified outside p and p¯ are both Zp -extensions.
In the next Chapter III, we study the field corresponding to the kernel of
the outer pro-p Galois representation associated to X.
In the case of the pro-p outer Galois representation associated to the thricepunctured projective line P1Q \ {0, 1, ∞}, Ihara asked whether the field corresponding to the kernel of the outer pro-p Galois representation associated to
P1Q \ {0, 1, ∞} is equal to the maximal pro-p extension of the p-th cyclotomic
field Q(µp ) unramified outside p. Then Sharifi [6] showed that Ihara’s question has an affirmative answer for every regular prime p > 2, assuming that
the Deligne-Ihara conjecture for p. Later, Hain-Matsumoto [2] and Brown [1]
proved the Deligne-Ihara conjecture for every prime, so Ihara’s question is found
to have an affirmative answer for every odd regular prime.
In the case of the once-punctured CM elliptic curve X, first we prove that
ρX,p (Gk(p) ) naturally splits into the direct product of Gal(k(E[p])/k(p)) and
Gal(Ω∗ /k(p)), where Ω∗ is a pro-p extension of k(p) unramified outside p.
We denote the maximal pro-p extension of k(p) unramified outside p by Ω
(hence Ω∗ ⊂ Ω holds). The problem we treat in Chapter III, which can be
regarded as an analogue of Ihara’s question, is the following:
Question. Is the field Ω∗ equal to the field Ω ?
In Chapter III, we prove, assuming an analogue of the Deligne-Ihara conjecture (see below) and regularity of p for k(p), that the question has an affirmative
answer.
Let us explain the situation furthermore. Let
˜ 1,1 := {f ∈ Aut(Π1,1 ) | f preserves the conjugacy class of an inertia subgroup at O}
Γ
and
˜ 1,1 /Inn(Π1,1 ),
Γ1,1 := Γ
the pro-p mapping class group of type (1, 1). The image of ρX,p is known
˜ 1,1 is equipped with a descending central filtration
to be contained in Γ1,1 . Γ
˜ 1,1 }m≥1 which is defined by
{F m Γ
˜ 1,1 := ker(Γ
˜ 1,1 → Aut(Π1,1 /Π1,1 (m + 1))).
F mΓ
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This filtration induces a descending central filtration {F m Γ1,1 }m≥1 on Γ1,1
˜ 1,1 → Γ1,1 .
through the natural projection Γ
Since G∗ := Gal(Ω∗ /k) is embedded in Γ1,1 through the pro-p outer Galois
representation ρX,p , the Galois group G∗ inherits a descending central filtration
{F m G∗ }m≥1 from that of Γ1,1 .
We define g∗m := F m G∗ /F m+1 G∗ for every m ≥ 1 and g∗ := ⊕m≥1 g∗m .
Each g∗m is naturally embedded into F m Γ1,1 /F m+1 Γ1,1 , which is known to be
a free Zp -module of finite rank for every m ≥ 1. One can show that g∗m ⊗ Qp
is isomorphic to a finite direct sum of (a finite direct sum of) Qp (m) where
m ∈ Z2≥1 satisfies |m| = m.
The restricted elliptic Soul´e character κm |F |m| G∗ is nontrivial for every
m ∈ I such that κm : F 1 G∗ → Zp (m) is nontrivial. Moreover, κm |F |m| G∗
factors through g∗|m| for every m ∈ I.
Now we propose an analogue of the Deligne-Ihara conjecture:
Conjecture 1 ([3, Conjecture III.1.2]). Choose an arbitrary element σm ∈
F |m| G∗ for every m ∈ I so that: (1) κm (σm ) generates κm (F |m| G∗ ), and (2)
the image of σm in g∗|m| , denoted by the same letter, is contained in the χm isotypic component of g∗|m| . Then the Lie algebra g∗ ⊗ Qp = ⊕m≥1 g∗m ⊗ Qp is
freely generated by {σm }m∈I .
Theorem 4 ([3, Theorem III.1.5]). Assume that p is regular for k(p) and Conjecture 1 holds for p. Then Ω∗ is equal to Ω.
The strategy to prove Theorem 4 is to generalize Sharifi’s proof performed in
[6] to a certain two-variable situation. To accomplish this strategy, we introduce
a certain two-variable version of the descending central series on Π1,1 , a certain
two-variable filtration on the pro-p mapping class group Γ1,1 of type (1, 1) and
on the Galois group. We also prove certain fundamental properties of these
two-variable filtrations.
Although the Galois group appearing in Sharifi’s previous work is a free
pro-p group if p > 2 is regular, the Galois group Gal(Ω/k(p)) is known to have
a nontrivial relation for any p ≥ 5. Hence we need to take the existence of
such a nontrivial relation into consideration when we follow Sharifi’s approach.
Fortunately, since Gal(Ω/k(p)) is isomorphic to the decomposition group over
a prime above p if p is regular for k(p) by a theorem of Wingberg, we can
understand a “shape” of such a nontrivial relation in the Galois group to such
an extent that we can prove Theorem 4 with the help of local class field theoretic
arguments and Conjecture 1.
In the last Chapter IV, we point out possible directions for future studies in
terms of (1) Relations between the elliptic Soul´e characters and the cyclotomic
Soul´e characters, (2) Elliptic Soul´e characters and p-adic L-functions, (3) On
the elliptic Deligne-Ihara conjecture, and (4) Generalizations to supersingular
primes.

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References
[1] Francis Brown. Mixed Tate motives over Z. Ann. of Math. (2), 175(2):949–
976, 2012.
[2] Richard Hain and Makoto Matsumoto. ...