SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
MAHESH KAKDE
King’s College London
अातां कोठे धांवे मन ।
तुझे चरण दे ख लया ॥
(Sant Tukaram)
To John Coates on his 70th birthday.
Abstract. Let p be an odd prime and let G be a p-adic Lie group. The group
K1 (Λ(G)), for the Iwasawa algebra Λ(G), is well understood in terms of congruences between elements of Iwasawa algebras of abelian sub-quotients of G due to
the work of Ritter-Weiss and Kato (generalised by the author). In the former one
needs to work with all abelian subquotients of G whereas in Kato’s approach
one can work with a certain well-chosen sub-class of abelian sub-quotients of
G. For instance in [11] K1 (Λ(G)) was computed for meta-abelian pro-p groups
G but the congruences in this description could only be proved for p-adic Lfunctions of totally real fields for certain special meta-abelian pro-p groups. By
changing the class of abelian subquotients a different description of K1 (Λ(G)),
for a general G, was obtained in [12] and these congruences were proven for
p-adic L-functions of totally real fields in all cases. In this note we propose a
strategy to get an alternate description of K1 (Λ(G)) when G = GL2 (Zp ). For
this it is sufficient to compute K1 (Zp [GL2 (Z/pn )]). We demonstrate how the
strategy should work by explicitly computing K1 (Zp [GL1 (Z/p)])(p) , the pro-p
part of K1 (Zp [GL2 (Z/p)]), which is the most interesting part.
1. Introduction
We begin with a discussion of the main conjecture for non-CM elliptic curves. Let
E be an elliptic curve defined over Q. Assume that E does not admit complex
multiplication. Let p be a prime and put E[pn ] for the set of pn -torsion points
of E (over Q). Let F∞ := Q(E[p∞ ]) := ∪n≥1 Q(E[pn ]). By a famous theorem
of Serre [17] Gal(F∞ /Q) is an open subgroup of GL2 (Zp ) and is in fact equal
E-mail address: mahesh.kakde@kcl.ac.uk.
Date: version 2.
The author is supported by EPSRC First Grant EP/L021986/1.
1
2
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
to GL2 (Zp ) for almost all p. From now on fix a prime p > 3 (so that GL2 (Zp )
does not have an element of order p) such that G := Gal(F∞ /Q) ∼
= GL2 (Zp )
and such that E has good ordinary reduction at p. Let µn be the group of nth
roots of 1. Put Q∞ := Q(µp∞ ) := ∪n≥1 Q(µpn ). By the Weil pairing Q∞ is
contained in F∞ . Let Qcyc be the unique extension of Q (contained in Q∞ ) such
that Γ := Gal(Qcyc /Q) is isomorphic to the additive group of p-adic integers Zp .
We put H := Gal(F∞ /Qcyc ).
F∞
H
Q∞
G
Qcyc
Γ
Q
For a profinite group P = limi Pi , we put Λ(P ) := limi Zp [Pi ] and Ω(P ) :=
←−
←−
limi Fp [Pi ]. Following [5] we put
←−
S := {f ∈ Λ(G) : Λ(G)/Λ(G)f is a f.g. Λ(H)-module}.
Put S ∗ := ∪n≥0 pn S. By [5, theorem 2.4] S and S ∗ are multiplicatively closed subsets of Λ(G), do not contain any zero-divisors and satisfy the Ore condition. Hence
we get localisations Λ(G)S and Λ(G)S ∗ of Λ(G). Put MH (G) for the category of
finitely generated S ∗ -torsion Λ(G)-modules i.e. all finitely generated Λ(G)-modules
M such that Λ(G)S ∗ ⊗Λ(G) M = 0. Following [5, section 5], for any algebraic extension L of Q, we define the Selmer group
∏
)
(
S(E/L) := Ker H 1 (L, E[p∞ ]) →
H 1 (Lw , E(Lw )) ,
w
where w runs through all non-archimedian places of L and Lw denotes the union
of completions at w of all finite extensions of Q contained in L. Also put
X(E/L) := Hom(S(E/L), Qp /Zp )
for the Pontrjagin dual.
Conjecture 1 (Conjecture 5.1 [5]). The Λ(G)-module X(E/F∞ ) lies in the category MH (G).
Every continuous homomorphism ρ : G → GLn (O), where O is the valuation ring
in a finite extension L of Qp , induces a map (see [5, equation (22)])
(1)
K1 (Λ(G)S ∗ ) → L ∪ {∞},
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
3
x 7→ x(ρ).
∫
(Classically, this would be denoted as G ρdx).
Conjecture 2 (Conjecture 5.7 [5]). There is a finite unramified extension A of
Zp and an element LE ∈ K1 (Λ(G)S ∗ ⊗Zp A) such that, for all Artin representation
ρ of G we have
LE (ρ) =
LΣ (E, ρ̂, 1)
Pp (ρ, u−1 ) −fρ
·
e
(ρ)
·
·u .
p
+
−
Pp (ρ̂, w−1 )
Ω+ (E)d (ρ) Ω− (E)d (ρ)
(see [5, section 5] for all unexplained notation and the paragraph before proposition
7.5 [3] for correction to loc. cit. We thank the referee for pointing this out).
Recall the following part of the localisation sequence of K-theory
∂
→ K0 (MH (G)) → 1.
K1 (Λ(G)) → K1 (Λ(G)S ∗ ) −
The surjection of ∂ is shown in [5, proposition 3.4]. Assuming conjecture 1 a
characteristic element of X(E/F∞ ) is defined as any element of K1 (Λ(G)S ∗ ) that
maps to the class of X(E/F∞ ) in K0 (MH (G)).
Conjecture 3 (Conjecture 5.8 [5]). Assume conjectures 1 and 2. Let ξE be a
characteristic element of X(E/F∞ ). Then the image of ξE in
K1 (Λ(G)S ∗ ⊗Zp A)
Image of K1 (Λ(G) ⊗Zp A)
coincides with the image of LE .
For simplicity we assume that the ring A in the above Conjecture 2 is Zp as we
discuss the strategy for attacking the above conjecture. This strategy is due to
Burns and Kato (see [1]) and is well-known by now. Put S(G) for the set of all
open subgroups of G. For every U in S(G), there is a map
θU : K1 (Λ(G)) → K1 (Λ(U ab )) ∼
= Λ(U ab )×
defined as composition of the norm map K1 (Λ(G)) → K1 (Λ(U )) and the map
induced by natural project Λ(U ) → Λ(U ab ). Hence we get a map
∏
∏
θ :=
θU : K1 (Λ(G)) →
Λ(U ab )×
U ∈S(G)
Similarly, there are maps
θS : K1 (Λ(G)S ) →
U ∈S(G)
∏
Λ(U ab )×
S,
U ∈S(G)
θS ∗ : K1 (Λ(G)S ∗ ) →
∏
U ∈S(G)
Λ(U ab )×
S∗ .
4
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
As G has no element of order p by “Weierstrass preparation theorem” [3, proposition 3.4] there is an isomorphism
K1 (Λ(G)S ∗ ) ∼
= K1 (Λ(G)S ) ⊕ K0 (Ω(G)).
We have the map
θ0 : K0 (Ω(G)) →
∏
K0 (Ω(U ab ))
U ∈S(G)
that fits into the following commutative diagram
∼
=
K1 (Λ(G)S ∗ )
θS ∗
∏
U ∈S(G)
?_
o
Λ(U ab )×
S∗
/ K1 (Λ(G)S ) ⊕ K0 (Ω(G))
∏
U ∈S(G)
(θS ,θ0 )
ab
Λ(U ab )×
S ⊕ K0 (Ω(U )).
Here, we abuse the notation by denoting Ore sets for U ab by the same symbols
S and S ∗ . The injection
is not surjective in general. Nevertheless,
∏ in lowerabrow
×
an element (xU ) ∈ U ∈S(G) Λ(U )S ∗ lies in the image of θS ∗ if and only if it
“factorises” as xU = (xU , µU ) and (xU ) and (µU ) lie in the images of θS and θ0
respectively.
Let P be a pro-p normal subgroup of G. Put G0 (Fp [G/P ]) for the Grothendieck
group of the category of finitely generated Fp [G/P ]-modules. The group is isomorphic to the group Brauer characters of G/P by [8, proposition 17.14]. The
group is also independent of the choice of the pro-p normal subgroup P (by [8,
proposition 17.16(i)]. The group K0 (Ω(G)) is a Green module over the Green ring
G0 (Fp [G/P ]) (this is [18, 17.1(*)(c)]. For the notion of Green rings and Green
modules see[15, Chapter 11]). Moreover, there is a Cartan homomorphism [18,
15.1]
c : K0 (Ω(G)) → G0 (Fp [G/P ])
which is injective by [8, lemma 18.22(ii)] (see also [3, 3.4.2]). Hence
(2)
x ∈ K0 (Ω(G)) is 0 ⇐⇒ χ(x) = 0 ∀ Brauer characters χ of G
(note that χ induces a map χ : K0 (Ω(G)) → K0 (Fp ) = Z). The following lemma
is a generalisation of the the ‘only if’ part of [7, theorem 3.8].
Lemma 4. The map θ0 is injective and its image consists of all tuples (µU ) such
that for any finite collection {U
∑i } ⊂ S(G), if there are mod p representation χi of
(χi ) = 0, then
Uiab and integers ni such that i ni IndG
Uiab
∑
ni χi (µUi ) = 0.
i
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
5
Proof. There is a canonical inflation map K0 (Ω(U ab )) → K0 (Ω(U )). By the Brauer
induction theorem [8, theorem 21.15] there are finitely many open subgroups Ui of
G and Brauer characters χi of Uiab such that the trivial character 1 of G is given
by
∑
1=
IndG
Ui χi ,
i
where we use the same symbol χi for the character of Ui obtained by inflating χi .
For any tuple (µU )U satisfying the condition given in the statement of the lemma
define an element µ ∈ K0 (Ω(G)) as follows - firstly, by abuse of notation, let µU
denote the inflation of µU to K0 (Ω(U ). Let iU : K0 (Ω(U )) → K0 (Ω(G)) be the
map induced by the inclusion U ,→ G. Define
∑
iUi (χi · µUi ).
µ :=
i
Here χi is considered as an element of G0 (Fp [Ui /Pi ]) for a suitable (i.e. χi is trivial
on Pi ) open normal pro-p subgroup Pi of Ui and χi · µUi ∈ K0 (Ω(Ui )) is obtained
by the action of G0 (Fp [Ui /Pi ]) on K0 (Ω(Ui )). The element µ is independent of the
choice of Ui ’s and χi ’s because the hypothesis on (µU )U and (2). It is easy to check
that this gives the inverse of θ0 for the claimed image.
□
Remark 5. The above discussion reveals two surprising consequences of S ∗ -torsion
conjecture. Firstly, as observed in [1], the p-adic L-function in Λ(U ab )×
S ∗ should
factorise canonically into an element in Λ(U ab )×
and
a
“µ-invariant”
part
(even
S
ab ×
though there is no Weierstrass preparation theorem for Λ(U )S ∗ in general) . Secondly, both these parts of the p-adic L-functions should satisfy “Artin formalism”
(one part satisfies Artin formalism if and only if the other does since the p-adic
L-function satisfies Artin formalism).
Remark 6. Coates-Sujatha [7] proved that for P ∼
= Zp ×Zp , the S ∗ -torsion conjecture is equivalent to Artin formalism for µ-invariant parts of p-adic L-functions in
Λ(Zp × pn Zp ) (in this case this is equivalent to growth of µ-invariant in a certain
way). However, in general, Artin formalism for µ-invariants seems to be weaker
than S ∗ -torsion conjecture.
In any case, to prove above conjectures at present it seems necessary to study the
maps θ, θS , θS ∗ , θ0 and compute their images explicitly. There are two ways to do
this. One is due to Ritter-Weiss ([16, 19]) and other is due to Kato, generalised by
the author ([13, 14, 12]). The first one is very elegant to state and can be easily
stated for a general p-adic Lie group (as opposed to one dimension pro-p groups
treated in [16, 19]) as will be shown in [2]. However, in this approach it is necessary
to use all open subgroups of G. The advantage of Kato’s approach is that in special
situations one can restrict to smaller classes of open subgroups of G. In this short
6
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
note we propose to begin the study of K1 (Λ(G)) using a smaller class of open
subgroups of G. By [9, proposition 1.5.1] K1 (Λ(G)) ∼
K1 (Zp [GL2 (Z/pn )]).
= lim
n
←
−
Hence it is enough to compute K1 (Zp [GL2 (Z/pn )]). The strategy to compute
K1 (Zp [P ]) for a finite group P can roughly be stated as follows: Let S(P ) be a
fixed class of subgroups of P . Then we have a map
∏
θ : K1 (Zp [P ]) →
Zp [U ab ]× .
U ∈S(P )
There is an additive analogue of this map
ψ : Zp [Conj(P )] →
∏
Zp [U ab ],
U ∈S(P )
where Conj(P ) is the set of conjugacy classes of P . For the definition of ψ see
next section. Precise relation between θ and ψ via integral logarithm is often hard
to describe. In short there are two ingredients to compute K1 (Zp [P ])
(1) Explicit knowledge of Conj(P ) so that the map ψ can be described explicitly.
(2) Explicit relation between θ and ψ via integral logarithm. Here we again
need to know Conj(P ) explicitly.
In this short note we carry out this strategy for n = 1. For n > 1 a similar computation (with a lot more blood, sweat and tears but essentially with no new ideas)
should go through and will be carried out in future. In the end the congruences
turn out to be rather trivial for n = 1 but that is expected as p-Sylow subgroups of
GL2 (Z/pZ) are cyclic groups of order p. However, we hope the calculations in this
simple case are illuminating and indicate how the general case will proceed.
There are several interesting aspects of this construction. For example, it is already
seen from computations in meta-abelian case in [11] and in general in [12], that
the shape of the congruences can be very different if one chooses different class of
subgroups. It seems hard to pass from the congruences in [11] to the congruences
in [12] directly in cases when they both apply. As [12] shows having an alternate
description can be useful to prove the congruences. Secondly, we hope that open
subgroups of G that, modulo pm , are one of the “standard subgroups” (Borel,
Cartan, Centre etc.) will suffice. These are closer to the theory of automorphic
representations. Therefore one may hope that the theory of automorphic representations and automorphic forms weighs in significantly in understanding these
congruences. This will only be clear after the congruences are explicitly written
down in general. Lastly, in the case of elliptic curves admitting CM, the main
conjecture for symmetric power representations attached to the curve has been
deduced from the two variable main conjecture by Coates-Schmidt [6]. It was
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
7
mentioned to the author by John Coates that a similar deduction should be possible for non-CM elliptic curves using the main conjecture stated above. This seems
to be an extremely hard problem. Specially because in the main conjecture above
we allow evaluation of p-adic L-function only at Artin representations. Hence one
needs to understand reduction modulo powers of p of Artin representations and
symmetric power representations. This study is implicit in the calculations proposed here. L-functions of symmetric power representations are extremely hard
to study and the author does not claim that the calculations proposed here would
provide a way to do this.
Remark 7. Throughout the paper we restrict to Zp coefficients, however, the same
computation should work for a more general class of coefficient rings for which
integral logarithm has been constructed (for example rings considered in [4]). However, all the results we need are not stated or proven in this generality yet and a
satisfactory discussion of these will take us too far off our modest goal.
Acknowledgement: John Coates introduced me to non-commutative Iwasawa
during my PhD. He has been a constant source of encouragement and inspiration
for me and I am sure will continue to be so for many years. It is my great pleasure
and honour to dedicate this paper to him on the occasion of his seventieth birthday.
These calculations were carried out during a visit to TIFR in fall, 2014 and I thank
TIFR for its hospitality. The author would like to thank the anonymous referee
for several helpful comments and careful reading of the manuscript.
2. An additive result
From now onwards G denotes GL2 (Fp ). Let us denote the set of conjugacy classes
of G by Conj(G). Fix a non-square element ϵ in Fp . It is well-known that the
conjugacy classes of G are (for example see [10])
(
)
a 0
ia :=
for a ∈ F×
p.
0 a
(
)
a 1
ca,1 :=
for a ∈ F×
p.
0 a
(
)
a 0
ta,d :=
for a ̸= d ∈ F×
p.
0 d
(
)
a ϵb
ka,b :=
for a ∈ Fp and b ∈ F×
p.
b a
We first describe the free Zp -module Zp [Cong(G)] explicitly in terms of abelian
subgroups of G. For this we define the following subgroups of G.
{
(
)
}
a 0
×
Z := ia =
: a ∈ Fp = centre of G,
0 a
8
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
{
(
)
}
a b
×
C := ca,b =
: a ∈ Fp , b ∈ Fp ,
0 a
{
(
)
}
a 0
×
T := ta,d =
: a, d ∈ Fp = split Cartan,
0 d
and
{
K :=
(
ka,b =
a ϵb
b a
}
)
: a, b ∈ Fp
= non-split Cartan,
where ϵ is the fixed non-square element of Fp . Note that they are all abelian. Put
S(G) := {Z, C, T, K}. Define a map
∏
ψ := (ψU )U ∈S(G) : Zp [Conj(G)] →
Zp [U ],
U ∈S(G)
where (the trace map) ψU : Zp [Conj(G)] → Zp [U ] is a Zp -linear map defined
by
n
∑
−1
g 7→
{h−1
i ghi : hi ghi ∈ U }
i=1
for any g ∈ G and a fixed set {h1 , . . . , hn } of left coset representatives for U in G.
This map is explicitly given in the following table
ψZ
ψC
ψT
ψK
ia p(p2 − 1)ia (p2 − 1)ia p(p + 1)ia p(p − 1)ia
∑p−1
ca,1
0
0
0
i=1 ca,i
ta,d
0
0
ta,d + td,a
0
ka,b
0
0
0
ka,b + ka,−b
∏
Definition 8. We put Ψ for the set of all tuples (aU ) ∈ U ∈S(G) Zp [U ] satisfying
the following conditions
(A1) For every U ∈ S(G), the trace map trU : Zp [U ] → Zp [Z] maps aU to aZ
(as Zp [U ] is a free finitely generated module over Zp [Z] we have the trace
map trU ).
(A2) Let NG U be the normaliser of U in G. We require that every aU is fixed
under the conjugation action of NG U .
(A3) The element aZ lies in the ideal pZp [Z] of Zp [Z].
Theorem 9 (additive theorem). The map ψ induces an isomorphism between
Zp [Conj(G)] and Ψ.
Proof. From the above table it is clear that the image of ψ lies in Ψ. We simply
define a left inverse δ of the map ψ and then show that δ is injective on Ψ. Define
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
∑
∏
δ :=
δ
,
with
each
δ
:
U
U
U ∈S(G)
V ∈S(G) Zp [V ] → Qp [Conj(G)] =
defined by
(
)
{
1
1
a
−
a
if U ̸= Z
U
[NG U :U ]
[U :Z] Z
δU ((aV )) =
1
a
if U = Z.
[G:Z] Z
9
Q[G]
[Q[G],Q[G]]
First we show that δ ◦ ψ = idZp [Conj(G)] . As all the maps are Zp -linear it is enough
to check only on conjugacy classes Conj(G).
(1) (For classes ia ) It is clear from the above table that δZ (ψZ (ia )) = ia .
For every U ∈ S(G), we have trU (ψU (ia )) = [U : Z]ψU (ia ) (considered as
an element of Zp [Z] as it already lies in that subring of Zp [U ]). Therefore
δU (ψU (ia )) = 0 for all U ̸= Z by (A1). Hence δ(ψ(ia )) = ia .
(2) (For classes ca,1 ) From the above table it is clear that δU (ψU (ca,1 )) = 0
for all U ̸= C. Moreover, it is easy to check that NG
∑C is the set of upper
1
triangular matrices in G. Hence δC (ψC (ca,1 )) = p−1
( h∈NG C/C h−1 ca,1 h) =
ca,1 . Hence δ(ψ(ca,1 )) = ca,1 .
(3) (For classes ta,d ) Again from the above table it is clear that δU (ψU (ta,d )) =
0 for U ̸= T . The normaliser
{(
)
}
(
)
a b
0 1
NG T =
: a = d = 0 or b = c = 0 = T ∪
T.
c d
1 0
Therefore [NG T : T ] = 2. Hence δT (ψT (ta,d )) = ta,d (we abuse the notation
and denote the conjugacy class of ta,d by the same symbol).
(4) (For classes ka,b ) Again from the above table it is clear that δU (ψU (ka,b )) =
0 for U ̸= K. The normaliser
{(
)
}
a ϵb
NG K =
: a = d, b = c or a = −d, b = −c
c d
(
)
0 −ϵ
=K∪
K.
1 0
Therefore [NG K : K] = 2. Hence δK (ψK (ka,b )) = ka,b .
This show that δ ◦ ψ = idZp [Conj(G)] .
Next we show that
δ|∑
Ψ is injective. Let (aU ) ∈ Ψ be such that δ((aU )) = 0. First
∑p−1
∑p−1
xa,0 ca,0 which,
= [C : Z] a=1
consider aC = i=0 p−1
a=1 xa,i ca,i . Then trC (aC ) ∑
p−1 ∑p−1
1
by (A1), is equal to aZ . Hence aC − [C:Z] aZ = i=1 a=1 xa,i ca,i . As ca,i , for
1 ≤ i ≤ p − 1, are all conjugates of ca,1 we have, by (A2),
∑
∑
1
aC −
aZ =
xa,1
ca,i .
[C : Z]
a=1
i=1
p−1
p−1
10
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
∑
Therefore δC ((aU )) = p−1
a=1 xa,1 ca,1 . Moreover these conjugacy classes ca,1 cannot
appear in the image of δU for any U ̸= C. Therefore δ((aU )) = 0 implies that
1
xa,1 = 0 for all a. Hence aC = [C:Z]
aZ . Hence δC ((aU )) = 0.
1
Similarly, we show that aT = [T1:Z] aZ and aK = [K:Z]
aZ and so δT ((aU )) = 0 =
δK ((aU )). Hence δZ ((aU )) = 0. Therefore aZ = 0 and so aC = 0, aT = 0 and
aK = 0. This show that δ|Ψ is injective.
□
Remark 10. The above proof goes through with any coefficient ring which is a
Z(p) -algebra.
3. The main result
We have a map
θ : K1 (Zp [G]) →
∏
Zp [U ]× .
U ∈S(G)
(F) Let χU be representations of U and nU be integers such that
∑
nU IndG
U χU = 0.
U ∈S(G)
This sum takes∏
place in the group of virtual characters of G. We say that
a tuple (xU ) ∈ U ∈S(G) Zp [U ]× satisfies (F) if for any χU and nU as above
∏
χU (xU )nU = 1.
U
It is clear that the image of θ satisfies (F).
∏
Proposition 11. Let (xU ) ∈ U Zp [U ]× satisfy (F). Then
(M1) (xU ) satisfies analogue (A1) i.e. the norm map nr : Zp [U ] → Zp [Z] maps
xU to xZ for any U ∈ S(G).
(M2) (xU ) satisfies analogue of (A2) i.e. xU is fixed by NG U for any U ∈ S(G).
Proof. This is an easy consequence of (F). We demonstrate (M1). Let χ be a
representation of Z and ρ := IndUZ (χ). Then
χ(nr(xU )) = ρ(xU ).
G
As IndG
Z (χ) = IndU (ρ), it is plain from (F) that ρ(xU ) = χ(xZ ). Hence χ(nr(xU )) =
χ(xZ ). Hence nr(xU ) = xZ .
□
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
11
Next we observe that by Oliver [15, proposition 12.7] SK1 (Zp [G]) = 1. Let P be
a finite group and put JP for the Jacobson radical of group ring Zp [P ]. By [15,
theorem 2.10]
K1 (Zp [P ]) ∼
= K1 (Zp [P ]/JP ) ⊕ K1 (Zp [P ], JP ).
The group K1 (Zp [P ]/JP ) is a finite group of order prime to P . The group K1 (Zp [P ], JP )
is a Zp -module. Hence K1 (Zp [P ])(p) = K1 (Zp [P ], JP ). Hence the map θ induces
∏
θ|K1 (Zp [G]/JG ) : K1 (Zp [G]/JG ) →
(Zp [U ]/JU )×
U ∈S(G)
and
θ|K1 (Zp [G])(p) : K1 (Zp [G])(p) →
∏
K1 (Zp [U ])(p) .
U ∈S(G)
In this paper we will ignore the prime to p-part K1 (Zp [G]/JG ) as interesting congruences come from the p-part K1 (Zp [G])(p) .
We first recall the integral logarithm of Oliver and Taylor ([15, chapters 6 and
12])
L : K1 (Zp [G])(p) → Zp [Cong(G)]
(
)
defined as L := 1 − φp ◦ log, where φ is the map induced by g 7→ g p for every
g ∈ Conj(G).
Proposition 12. The integral logarithm on K1 (Zp [G]) induces an isomorphism
∼
=
L : K1 (Zp [G])(p) −
→ Zp [Conj(G)].
In particular, K1 (Zp [G])(p) is torsion-free.
Proof. For a finite group P let P ′ denote the set representative of p-regular conjugacy classes of P . Then kernel and cokernel of integral logarithm L on K1 (Zp [P ])
are equal to H1 (P, Zp [P ′ ])φ and H1 (P, Zp [P ′ ])φ respectively. Here H1 is the Hochschild
homology and P acts on∑
the coefficients
by conjugation. The operator φ is the
∑
p
one induced by the map
ag g 7→
ag g on the coefficients (this is [15, theorem
12.9]). We apply this to the group G. Firstly note that φ is identity on p-regular
elements of G. To compute the homology group H1 (G, Zp [G′ ]) notice that by the
sentence after equation (1) on page 286 in [15]
H1 (G, Zp [G′ ]) = ker(L) = tor(K1 (Zp [G])(p) ),
which by [15, theorem 12.5(ii)] is trivial in our case (this can be computed using
the explicit conjugacy classes of G given in the previous section).
□
12
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
Next we find a relation between ψ and θ. Let η = IndC
Z 1 be a representation of C.
It induces a Zp -linear map η : Zp [C] → Zp [C] given by g 7→ tr(η(g))g. The image of
this map lies in Zp [Z] ⊂ Zp [C]. The representation
η also induces∑a map Zp [C]× →
∑
Zp [C]× , that we again denote by η, given by g∈C ag g 7→ det( g ag η(g)g). It is
easy to verify that this is just the norm map Zp [C]× → Zp [Z]× .
Lemma 13. We have the following commutative diagram
Qp [Conj(G)]
ψ
∏
U ∈S(G)
Qp [U ]
φ
/ Qp [Conj(G)]
/
φ̃
∏
U ∈S(G)
ψ
Qp [U ],
where the map φ̃ = (φ̃U ) is given by
η
φ̃Z (aZ , aC , aT , aK ) := φ(aZ ) + p(p + 1)φ(aC − (aC ))
p
η
φ̃C (aZ , aC , aT , aK ) := φ(aC ) − p(aC − (aC ))
p
p(p + 1)
η
φ̃T (aZ , aC , aT , aK ) := φ(aT ) +
φ(aC − (aC ))
p−1
p
η
φ̃K (aZ , aC , aT , aK ) := φ(aK ) + pφ(aC − (aC )).
p
η
In the above we use the fact that φ(aC − p (aC )) belongs to Zp [Z] and hence can be
considered as an element of Zp [U ] for any U ∈ S(G).
Proof. This is a simple and straightforward, though somewhat tedious, calculation
using the explicit description of conjugacy classes of G and the map ψ.
□
Proposition 14. The relation between the maps θ and ψ is given by
K1 (Zp [Conj(G)])(p)
θ
∏
U ∈S(G)
Zp [U ]×
/ Zp [Conj(G)]
L
/
L̃
where the map L̃ := (L̃U ) is given by
ψ
U ∈S(G)
Qp [U ]
)
xpZ
φ(η(xC ))p+1
·
φ(xZ ) φ(xC )p(p+1)
( p
)
1
xC
φ(η(xC ))
L̃C (xZ , xC , xT , xK ) := log
·
p
φ(xC ) φ(xC )p
1
L̃Z (xZ , xC , xT , xK ) := log
p
(
∏
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
(
p(p−1)
φ(η(xC ))p+1
xT
·
φ(xT )p−1 φ(xC )p(p+1)
( p
)
1
xK
φ(η(xC ))
L̃K (xZ , xC , xT , xK ) := log
·
p
φ(xK ) φ(xC )p
1
L̃T (xZ , xC , xT , xK ) :=
log
p(p − 1)
13
)
Proof. This is again a simple explicit calculation using lemma 13,
(
)
φ
L= 1−
◦ log
p
and the fact that ψ ◦ log = log ◦θ (by the commutative diagram (1a) in the proof
of theorem 6.8 in [15]).
□
Remark 15. We refer the reader to the discussion on page 286 after the proof of
theorem 12.9 in [15]. It may explain why the definition of L̃ is complicated.
∏
Definition 16. Let Θ be the set of all tuples (xU ) ∈ U ∈S(G) Zp [U ]×
(p) which are
not torsion and such that
(1) (xU ) satisfies (F).
(2) xZ ≡ φ(xC )(mod pZp [Z]).
Lemma 17. If (xZ , xC , xT , xK ) ∈ Θ, then L̃Z (xZ , xC , xT , xK ) becomes
(
)
xZ φ(xZ )
L̃Z (xZ , xC , xT , xK ) = log
φ(xC )p+1
Proof. As shown in proposition 11, condition (F) implies that η(xC ) = nr(xC ) =
xZ .
□
We can now state our main theorem.
Theorem 18. The map θ induces an isomorphism between K1 (Zp [G])(p) and Θ.
Proof. We prove this in two steps.
(1) First note that, as p is odd, xZ ≡ φ(xC )(mod p) is implied by
) (
(
) (
)2
xZ φ(xZ )
xZ ϕ(xZ )
xZ
≡
≡ 1(mod p).
≡
φ(xC )φ(xC )p
φ(xC )φ2 (xC )
φ(xC )
Note that −1 does not belong to K1 (Zp [G])(p) . As log induces an isomorphism between 1 + pZp [Z] and pZp [Z], lemma 17 implies that the image
of θ satisfies (C). Hence by propositions 11 and proposition 14 we get that
the image of θ is contained in Θ.
14
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
(2) We first claim that the ker(L̃|Θ ) is trivial. Let (xU )U be in the kernel of
L̃|Θ . As log induces an isomorphism between 1 + pZp [C] and pZp [C] it
follows that
φ(η(xC ))
xpC
·
= 1.
φ(xC ) φ(xC )p
×
This shows that xpC lies in Zp [Z]× (since φ(Zp [C]× ) ⊂
( Zp [Z]
) ). Hence
2
η(xpC ) = xpC . Hence 0 = pL̃C (xZ , xC , xT , xK ) = log
Zp [C]×
(p) ).
xpC
φ(xC )
= pL(xC )
(here L is the integral logarithm map on
Whence L(xC ) = 0 and
xC = 1 by [15, theorem 12.9] (note that xC is not torsion
by the definition
( p )
x
φ(η(xC ))
of Θ). Hence φ(xC )p = 1 and consequently p1 log φ(xUU ) = 1 for all U .
Hence xU = 1 for all U ̸= C. Therefore ker(L̃) is trivial. This proves the
claim. (Compare this with proof of injectivity of δ above).
Now consider the commutative diagram
K1 (Zp [G])(p)
θ
L
∼
=
/ Zp [Conj(G)]
∼
= ψ
/ Ψ.
Θ
L̃
This diagram shows that L̃ : Θ → Ψ is surjective and hence an isomorphism. Therefore θ is an isomorphism.
□
References
[1] Burns, D. On main conjectures in non-commutative Iwasawa theory and related conjectures.
J. Reine Angew. Math., 698:105–160, 2015.
[2] Burns, D. and Kakde, M. Congruences in non-commutative Iwasawa theory. In preparation.
[3] Burns, D. and Venjakob, O. On descent theory and main conjectures in non-commutative
Iwasawa theory. J. Inst. Math. Jussieu, 10:59–118, 2-11.
[4] Chinburg, T. and Pappas, G. and Taylor, M. J. The group logarithm past and present. In
Coates, J. and Schneider, P. and Sujatha, R. and Venjakob, O., editor, Noncommutative
Iwasawa Main Conjectures over Totally Real Fields, volume 29 of Springer Proceedings in
Mathematics and Statistics, pages 51–78. Springer-Verlag, 2012.
[5] Coates, J. and Fukaya, T. and Kato, K. and R. Sujatha and Venjakob, O. The GL2 main
conjecture for elliptic curves without complex multiplication. Publ. Math. IHES, (1):163–
208, 2005.
[6] Coates, J. and Schmidt, C.-G. Iwasawa theory for the symmetric square of an elliptic curve.
J. Reine Angew. Math., 375/376:104–156, 1987.
[7] Coates, J. and Sujatha, R. On the MH (G)-conjecture. In Non-abelian fundamental groups
and Iwasawa theory, pages 132–161, 2012.
[8] Curtis, Charles W. and Reiner, Irving. Methods of Representation Theory with applications
to finite groups and orders, volume 1. John Wiley and Sons, 1981.
SOME CONGRUENCES FOR NON-CM ELLIPTIC CURVES
15
[9] Fukaya, T and Kato, K. A formulation of conjectures on p-adic zeta functions in noncommutative Iwasawa theory. In N. N. Uraltseva, editor, Proceedings of the St. Petersburg
Mathematical Society, volume 12, pages 1–85, March 2006.
[10] Fulton, W. and Harris, J. Representation theory. A first course, volume 129 of Graduate
Text in Mathematics. Springer-Verlag, New York, 1991.
[11] Kakde, M. Proof of the main conjecture of noncommutatve Iwasawa theory for totally real
number fields in certain cases. J. Algebraic Geom., 20:631–683, 2011.
[12] Kakde, M. The main conjecture of Iwasawa theory for totally real fields. Invent. Math.,
193(3):539–626, 2013.
[13] Kato, K. K1 of some non-commutative completed group rings. K-Theory, 34:99–140, 2005.
[14] Kato, K. Iwasawa theory of totally real fields for Galois extensions of Heisenberg type. Very
preliminary version, 2006.
[15] Oliver, R. Whitehead Groups of Finite Groups. Number 132 in London Mathematical Society
Lecture Note Series. Cambridge University Press, 1988.
[16] Ritter, J. and Weiss, A. On the ‘main conjecture’ of equivariant Iwasawa theory. Journal of
the AMS, 24:1015–1050, 2011.
[17] Serre, J-P. Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent.
Math., 15:259–331, 1972.
[18] Serre, J-P. Linear Representation of Finite Groups. Number 42 in Graduate Text in Mathematics. Springer-Verlag, 1977.
[19] Venjakob, O. On the work of Ritter and Weiss in comparison with Kakde’s approach. In
Noncommutative Iwasawa Main Conjectures over Totally Real Fields, volume 29 of Springer
Proceedings in Mathematics and Statistics, pages 159–182, 2013.