ANNIHILATION OF REAL CLASSES
by
Jean-Robert Belliard & Anthony Martin

Abstract. — Let F be a number field, abelian over Q and let p be a prime unramified
in F . In this note we prove that Solomon’s ψF element annihilates the torsion of the
Galois group of the maximal abelian p-ramified p-extension of F . This is anotherunconditional proven reinforcement of All theorem [All13] which was conjecture 4.1of [Sol92].

1. Introduction
Let F be a number field, abelian over Q, and fix a prime p 6= 2. For all n ∈ N
fix ζn a primitive nth root of unity in a coherent way, e.g. take ζn = e2iπ/n . Let f
be the conductor of F , so that F ⊂ Q(ζf ) and f is minimal with respect to thatproperty. Let XF denotes the p-part of the ideal class group of F . The Galois
group ∆F = ∆ = Gal(F/Q) acts onto XF in a natural way, turning it into a Zp [∆]module. One of the main challenge of algebraic number theory is to understand thestructure of this object. As this is a finite module, a starting point would be todescribe its annihilator or to find some canonical annihilators. If F is imaginary the
Stickelberger element is such canonical element for the minus part of XF . However
if F is real this Stickelberger element is a multiple of the absolute trace, hence theannihilation statement is trivial. Assume that F is totally real. Fix once and for
all an embedding of Q in the field of complex number C and another embeddingıv : Q −→ Cp in the Tate field Cp . This embedding uniquely defines a place v above
p in F . Let Fv ∈ Cp be the closure of F and let Ov be the local valuation ring at v.
In [Sol92] Solomon defined an element ψF ∈ Ov [∆] as follows

1 X
ψF :=
logp ıv NQ(ζf )/F (1 − ζf )δ δ −1 ,p δ∈∆Fand stated the conjecture :Conjecture 1.1 ([Sol92], conjecture 4.1). — ψF annihilates XF ⊗ Ov .
Let us state a few historical remarks, for which we need some more notations. Let
XF be the Galois group of the maximal abelian p-ramified p-extension of F and let

2

JEAN-ROBERT BELLIARD & ANTHONY MARTINtXF ⊂ XF be its Zp -torsion module. Let also DF be the sub-module of XF generated
the places dividing p.
1. If p - |∆| (so called semi-simple case) then conjecture 1.1 follows from theMain Conjecture of Iwasawa theory (here Mazur-Wiles theorem [MW84]) :see remark 4.1 (ii) of [Sol92].
2. Theorem 4.1 of [Sol92] proves that ψF annihilates DF ⊗ Ov .
3. If f is composite and if p is totally split in F , then theorem 5.4 of [BNQD05]
proves conjecture 1.1.
4. In full generality the theorem 1.1 of [All13] is a proven reinforcement of conjecture 1.1.
The point here is to construct global explicit annihilators of all XF ⊗ Ov inside
Zp [∆] without using χ-eigenspaces, so the semi-simple cases are not relevant to thisproblem. The main result of this note is
Theorem 1.2. — ψF annihilates tXF ⊗ Ov .
As tXF maps surjectively onto XF , this theorem is another reinforcement of conjecture 1.1.
The theorem 1.2 reinforces conjecture 1.1, but it also proves that the element ψF is
not a real analogue of Stickleberger element, in the sense that ψF is more naturally
associated to tXF than to XF . The interested reader should consult [NQDN11]
and the pioneering work [Sol09] where a better real analogue of Stickelberger ideal
are introduced and studied. The strategy to prove theorem 1.2 is quite simple. One
first use Iwasawa theory to consider objects X∞ and X∞ defined by replacing F
with its cyclotomic Zp -extension F∞ . Then one use Coleman theory to define a
global power series Col(ηf ) ∈ Ov [∆][[T ]] such that ψF is recovered from Col(ηf )simply by setting T = 0. Now by classical Iwasawa’s Main Conjecture, which is astatement about χ-parts, all χ components of Col(ηf ) annihilates all χ-parts of X∞ .
But the point here is that X∞ has simultaneously no finite sub-module and trivialµ-invariant, contrary to X∞ which is conjectured to be finite. Therefore the global
elements Col(ηf ) actually annihilates full X∞ and theorem 1.2 follows by taking
co-invariants.

2. Setting
Let us recall the analytic and algebraic setting of Iwasawa Main Conjecture.
2.1. Galois setting. — Fix once and for all a number field F , abelian over Q, real,and of conductor f . Fix a prime number p 6= 2 such that p - f . For all integern, the field Fn will be the n-th step of the cyclotomic Zp -extension of F , so that
[Fn : F ] = pn . Also we put Kn = F (ζpn ) and ∆Kn = Gal(Kn /Q), ∆Fn = Gal(Fn /Q).
SS
As p - f , we have Kn = Fn (ζp ). Set also F∞ = n Fn and K∞ = n Kn . By disjoint
ramification the relevant extensions are split and we have direct product of various

ANNIHILATION OF REAL CLASSES

3

Galois groups as follows :∆Fn ' ∆F × Gal(Fn /F ) ' ∆F × Z/pn Z;∆Kn ' ∆F × Gal(K/F ) × Gal(Kn /K) ' ∆F × Z/(p − 1)Z × Z/pn Z.
The whole Galois setting is summarized as follows
Q(f p∞ )

F∞

Fn

||
||
|
|
||

Q

F

s
sss
s
s
ss
sss
s
ss
ss
s
s
ss
ss

K∞

Kn

pp
ppp
p
p
pp
ppp

Q(f pn )

pp
ppp
p
p
pp
ppp

p
ppp
p
p
pp
ppp

Q(f p)

F (µp ) = K

s
sss
s
s
ss
sss

.

For any finite abelian group G and any valuation ring O of a finite extension of
Qp , we will denote by ΛO [G] the Iwasawa algebraΛO [G] = (lim O[Z/pn Z])[G] ' O[G][[T ]].
←−If O = Zp we simply note Λ[G] = ΛO [G]. Moreover we will note ω the Teichmüller character defined by the embedding ıv . Precisely we have for all a in
Z either p | a and ω(a) = 0 either (a, p) = 1 and ω(a) is the unique (p − 1)throot of unity (simultaneously in C and Cp ) such that v(ω(a) − a) > 0. Identifying
Gal(Q(ζp )/Q) with (Z/pZ)× , this character describes the action of Galois on thepth -power root of unity as for all g and all such root of unity ζ we have g(ζ) = ζ ω(g) ,the right hand side being well defined because ω(g) ∈ Zp . We also will denote by∼κ : Gal(Q(f p∞ )/Q(f )) −→ Z×
p the canonical isomorphism (abusively called cyclo
∞ |Q)tomic character). For all a in Z×
wewill
denoteσtheelement
of
Gal
Q(ζ
a
f
p
p
such that κ(σa ) = a.
2.2. Analytic side. — The p-adic L function is defined by interpolating the valuesof the complex L functions at negative integers. To be more precise let us recall
a fex classical notations. Let χ be a Dirichlet character, assumed to be primitive
modulo f . Via the Artin map the character χ may be seen as a Galois Character,actually a character of Gal(Q(ζf )|Q). Let Q(χ) ⊂ Q be the field generated over Q
by the values of χ. The χ-twisted Bernoulli numbers Bn,χ ∈ Q(χ) are defined by

4

JEAN-ROBERT BELLIARD & ANTHONY MARTIN

the power series expansionfXχ(a)seas∞X

sn=
Bn,χ .
ef s − 1
n!
a=1
n=0

The complex L function L(s, χ) takes algebraic values at negative integers. Indeed
according to [Was97] theorem(1) one has :Bn,χ.
n
Now by making use of the fixed embedding ıv it is possible to interpolate these values
with a Cp valued function.
L(1 − n, χ) = −Theorem 2.1 (cf. [Was97] theorem 5.11). — Exists a unique meromorphic
function Lp (., χ) (which is holomorphic if f 6= 1) defined on the open ball{s ∈ Cp | |s| < pp−1/p−1 } and such that for all n ≥ 1 :
Lp (1 − n, χ) = −(1 − χω −n (p)pn−1 )Bn,χω−n= (1 − χω −n (p)pn−1 )L(1 − n, χω −n ).
nRemark 2.2. — If f 6= 1, then one has the Leopoldt formula (see [Was97] theorem
5.18)


f
χ(p) τ (χ) X
Lp (1, χ) = − 1 −χ(a)−1 logp ıv (1 − ζfa ),
p
f a=1
P
a
where τ = τ (χ) = a=f
a=1 χ(a)ζf is the Gauß sum associated to χ. Therefore theSolomon’s element ψF is obviously related to these values at 1.
These p-adic L functions are uniquely determined by this interpolation properties.
Their original definition using Mahler development of partial zeta functions does
not fully reveal their real nature. Later on Iwasawa discovered a far more concrete
definition by using a sequence of Stickelberger elements to obtain a power series
development of these functions. These power series are elements of Zp (χ)[[T ]].
Definition 2.3. — The Stickelberger element ξQζf pn related to Q(ζf pn ) is definedby the formula
a=f pn −1 X
1f pn
ξQ(ζf pn ) = n− a σa−1 ∈ Q[Gal(Q(ζf pn )/Q)].
fp
2(a,f p)=1;a=1

By definition the Stickelberger element related to Kn is the element deduced fromξQ(ζf pn ) by restriction from Gal(Q(ζf pn )/Q) to ∆Kn .

a=f pn −1 X1f pn
ξKn = n− a (σa−1 )|Kn ∈ Q[∆Kn ].
fp
2(a,f p)=1;a=1

One can then show that(1)completer la refANNIHILATION OF REAL CLASSES

5
1. Let ϑQ(ζf pn ) = 1 − (1 + f p)(κ−1 (1 + f p))−1 ξQ(ζf pn ) . Then one has
ϑQ(ζf pn ) ∈ Zp [Gal(Q(ζf pn )/Q)],so that its restriction ϑKn to Q[∆Kn ] is also a p-adic integral element actually
an element of Zp [∆Kn ]2. The sequences (ξKn )n∈N and therefore (ϑKn )n∈N are coherent with respect torestrictions maps along the towers Q(f p∞ )/Q(f ) and K∞ /K.
As the ϑKn are coherent along K∞ /K we may consider the limit ϑK∞ ∈ Λ(∆K ).
Let χ∗ = χ−1 ω be the mirror character of χ. Then χ∗ is a Dirichlet characterof ∆K . As all other character of ∆K our χ∗ extends linearly to a ring homomorphism χ∗ : Λ(∆K ) −→ Λ(χ), where Λ(χ) is just the ring obtained by adjoining
to Λ the values of χ. Let us fix the topological generator γ = κ−1 (1 + f p) ofΓ = Gal(K∞ /K). Then sending γ to 1 + T uniquely define the Serre’s isomorphism
S : Λ(χ) ' Zp (χ)[[T ]].
Definition 2.4. — We may now define the three relevant Iwasawa’s power series
attached to our Dirichlet character χ :1. g(T, χ) = S(χ∗ (ϑK∞ )) ∈ Zp (χ)[[T ]];1 + pf
2. h(T, χ) = 1 −∈ Zp [[T ]];1+T
g(T, χ)3. f (T, χ) =∈ Zp (χ)[[T ]].
h(T, χ)Iwasawa in [Iwa69] discovered that these series would provide another definitionof the functions Lp (s, χ). This definition turned out to be far more enlightening
than the original one.
Theorem 2.5 ([Iwa69]). — Let χ be an even character of ∆K . Then for all s ∈ Cp
such that |s| < pp−1/(p−1) (and s 6= 1 if χ is trivial) we have
Lp (s, χ) = f ((1 + f p)s − 1, χ).
2.3. Algebraic side. — For all number field F recall that
– XF is the p-part of the class group of F .
– Xk is the Galois group of the maximal abelian p-ramified p-extension of F .
We X∞ = lim XFn , X∞ = lim XFn (with respect to norms and restriction maps).
←−←−
These are finitely generated Λ(∆F )-modules and are linked together by the exact
sequence
0 → E ∞ → U∞ → X∞ → X∞ → 0
where– E ∞ is the direct limit of the E n = En ⊗ Zp where the En are the global unitsof the Fn .
– U∞ is the direct limit of the semi-local units of the Fn .

6

JEAN-ROBERT BELLIARD & ANTHONY MARTIN

By the weak Leopoldt conjecture (here a theorem) the Λ-module X∞ is torsion. In
particular, for all character χ of ∆F , the χ-part Xχ∞ is a finitely generated torsion
Zp (χ)[[T ]]-module. We therefore may consider its characteristic series that we will
denote c.s.(Xχ∞ ).
Remark 2.6. — It is a well known theorem of Iwasawa (see [Iwa59]) that Xχ∞ has
no non-trivial finite Λ-submodule. In particular it is annihilated by its characteristic
series.
2.4. Main conjecture. — The main conjecture of Iwasawa relevant to this context
may be stated as follows :Theorem 2.7 (Mazur-Wiles). — For all character χ of ∆F we haves.c.(X∞χ )(T ) = f ] (T, χ)
where, by definition, the Iwasawa’s involution ] is given by
1 + fp]f (T, χ) = f− 1, χ .
1+T
The mirror Stickelberger series f ] (T, χ) plays an important role here. Another(equivalent) statement of the main conjecture is the following equality of characteristic series :s.c.(X∞ψ )(T ) = s.c.(U∞ /C∞ )ψ (T ),where C∞ stands by definition for the direct limit of circular units of the Fn .
3. Solomon’s element
Recall that for an abelian totally real field F such that p is unramified in F the
Solomon’s element is defined by the formula
1 X
logp ıv (NQ(ζf )/F (1 − ζf )δ ) δ −1 .
ψF :=
p δ∈∆
F

Our goal in that section is to use Coleman theory to relate this ψF to the constantterm of a power series that annihilates X∞ .
3.1. Coleman’s theory. — This theory is essentially a local one and it stands forany unramified extension of Qp . Let Ov [[X]] be the ring of formal power series in one
indeterminate X with coefficient in Ov . It is important here not to mix the Coleman
X with the Iwasawa’s T . Let Ov ((X)) be the field of fraction of Ov [[X]]. We will also
denote Γ× = Gal(Qp (ζp∞ )/Qp ); this notation is coherent with the previous notation
Γ = Gal(K∞ /K).
The rings Ov [[X]] and Ov ((X)) are endowed with :1. the action of a Frobenius ϕ such that f ϕ (X) = f ((1 + X)p − 1).
Q
2. a norm operator N uniquely determined by N (f ϕ )(X) = ξ∈µp f (ξ(1+X)−1).

ANNIHILATION OF REAL CLASSES

7

3. an action of Zp [[Γ× ]] which extends by linearity and continuity the action of
Γ× defined by γ.(1 + X) = (1 + X)κ(γ) for all γ ∈ Γ× (recall that κ is thecyclotomic character).
Theorem 3.1 (Coleman). — Let (αn ) = α ∈ lim Fv (ζpn )× be a norm coherent←−
sequence of local numbers. Then there exists a unique power series fα ∈ Ov ((X))×
such thatfα (ζpn+1 − 1) = αn and N fα = fαϕ .
Let Col : lim Fv (ζpn )× −→ Ov ((X)) be the map defined by←−pfα (X)ϕ1Col(α) = (1 − ) logp fα (X) = logp ϕ.
ppfα (X)
Then Col induces an exact sequence
Col



1
1 −→ Zp (1) −→ U∞−→ R −→ Zp (1) → 0.

where
1
– U∞
is the direct limit of the principal units of Fv (ζpn );– R is a rank one free Ov [[Γ× ]]-module generated by (1 + X);– Zp (1) is the direct limit of the (ζpn );d.
–  is the map f 7→ DfX=0 where D = (1 + X) dX

It is useful to consider another related exact sequence that is obtained by using
the Mellin transform L of Col with values in Ov [[Γ× ]] instead of R. Let us briefly
restate this definition : We saw that R is a rank one free Ov [[Γ× ]] generated by(1 + X). Hence all element g ∈ R may uniquely be written as g(X) = ĝ.(1 + X)
with ĝ ∈ Zp [[Γ× ]]. The element ĝ (also denoted by M el(g)) is called the Mellintransform of g and we define L = M el ◦ Col.
3.1.1. Formulas. — We need to give a dictionary of formulas between R andP
Ov [[Γ× ]]. For all ĝ =an γ n ∈ Ov [[Γ× ]] and all characters ψ of Γ× we notePψ(ĝ) = an ψ(γ)n ∈ Zp (ψ). Recall thatXXnan γ n . (1 + X) =an (1 + X)κ (γ)
where κ is the cyclotomic character. It follows that for all g ∈ R, i ∈ Z– g(0) = 1 ĝ (where
1 stands for the trivial character).

P
P
i
i
– D g = T w ĝ , where if ĝ = an γ n then T wi ĝ) = an (κi (γ)γ)nOne has in particular Di g X=0 = 1 T wi (ĝ) = κi (ĝ). With all these formulas andnotation the twin sequence of the exact sequence of Coleman is now
(1)

L

κ

1
1 −→ Zp (1) −→ U∞−→ Ov [[Γ× ]] −→ Zp (1) → 0.

8

JEAN-ROBERT BELLIARD & ANTHONY MARTIN

3.1.2. Induction. — Let L be the splitting field at p of F , so that p is totally split
in L/Q and that primes above p remains inert in F/L. In order to recover global
information for our field F from the local information collected in Fv one first needto induce along Gal(L/Q) and consider first the “semi-local” situation. We haveGal(Fv /Qp ) ' Gal(F/L). Let us abbreviate ∆L = Gal(L/Q). We apply the (exact)
functor − ⊗Zp [Gal(F/L)] Zp [∆F ] to the sequence (1). We get (see also[Tsu99]) :(2)bF [[Γ× ]] −→ Zp [∆L ](1) −→ 0,
0 −→ Zp [∆L ](1) −→ U∞ −→ O

bF := OF ⊗ Zp ' Zp [∆F ] ' Ov ⊗Zp [Gal(F/L)] Zp [∆F ] ' Ov [∆L ] because Ov iswhere O
Galois free (recall that p is unramified in F ).
3.2. cyclotomic units and Solomon’s element. — Consider the cyclotomic elements ηFn := NQ(ζf pn )/Fn (1 − ζf pn ). These numbers are cyclotomic units of Fn(recall that f 6= 1) and they form a norm coherent sequence (this follows from thewell known distribution relation satisfied by the numbers 1 − ζn ). This allows us todefine :Definition 3.2. — SoitηF,∞ = NQ(ζf pn )/Fn (1 − ζf pn ).

n∈NThis is an element of lim OF×n that may be seen as an element of U∞ using the←−
diagonal embedding.
We can now state and prove the key result of this note :Theorem 3.3. — The Solomon’s element ψF is nothing more than
1ψF =
1 (L(ηF,∞ )) .
p−1
Proof. — By the formulas given in 3.1.1 we have 1 L(ηF ) = Col ηF,∞ (0), so thatthe theorem is equivalent to the identity(p − 1)ψF = Col ηF,∞ (0).
We shall check this equality. First remark that the set of p-places of F is in one to
one correspondence with ∆L via the map∆L ↔ {v, | v is above p}
δ 7→ δ −1 v
Hence for all δ ∈ ∆L , the δ-component of ηF,∞ (seen as an element of U∞ ) is−1ıv (NQ(ζf pn )/Fn (1 − ζfδ ζpn )n .
Therefore we locally have (at the corresponding place δ −1 v) that the associatedColeman series is


Y
σδ −1
ιv
1 − ζf (1 + X) .
σ∈Gal(Q(ζf )/F )ANNIHILATION OF REAL CLASSES

9Therefore, and still at the place δ −1 v, we obtainϕfηf,∞ (X)
Col(ηf,∞ ) =
1−
p
p
Q
−1
1 − ζfσδ (1 + X)
1=
logp Q−1p
1 − ζfσδ (1 + X)p
It follows that specializing at X = 0 we find that the δ-component of Col(ηf,∞ )(0)
is
p−1
−1
logp ιv NQ(ζf )/F (1 − ζf δ )pwhich is exactly (p − 1) times the δ-component of ψF . 
We now prove that the power series related to ηf,∞ annihilates X∞ .
Lemma 3.4. — The element L(ηF,∞ ) annihilates X∞ ⊗ Ov .
Proof. — By the main conjecture, for all character χ of ∆F , the characteristic
seriesχχχ
of X∞ (which annihilates X∞ ) is the same as the one of U∞ /C∞ . But for all
character χ of ∆F the series χ(L(ηF ) is a multiple of the characteristic series ofχ
U∞ /C∞ . This proves that χ(L(ηF )) annihilates Xχ∞ for all χ. But the whole
X∞ has no non-trivial p-torsion (Ferrero-Washington and Iwasawa for the finite Λmodule, [FW79, Iwa59]). The result follows. 
Now our main result is
Theorem 3.5. — The Solomon elements ψF annihilates the Zp -torsion tXF of XF .
Proof. — Putting together the theorem 3.3 and the lemma 3.4 and Iwasawa coinvariant we see that (p − 1)ψF annihilates (X∞ )Γ . But these Iwsawa co-invariant
are known to be canonically isomorphic to tXF . 
Remark 3.6. — It is an interesting but technically difficult question to precisely
relate the global integral Coleman power series L(ηF,∞ ) with the global integral
power series ϑK∞ ∈ Λ(∆K ) and also to relate both to the Deligne-Ribet pseudomeasure. All these objects seems to be various avatar of the p-adic L function but
there is some discrepancy between them. We hope to be able to say more in a future
work.
References[All13]

Timothy All, On p-adic annihilators of real ideal classes, J. Number Theory
133 (2013), no. 7, 2324–2338. MR 3035966[BNQD05] Jean-Robert Belliard and Thong Nguyen-Quang-Do, On modified circularunits and annihilation of real classes, Nagoya Math. J. 177 (2005), 77–115.
[FW79]
Bruce Ferrero and Lawrence C. Washington, The Iwasawa invariant µp vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no. 2, 377–395.
MR 528968 (81a:12005)[Iwa59]Kenkichi Iwasawa, On the theory of cyclotomic fields, Ann. of Math. (2) 70(1959), 530–561. MR 0124320 (23 #A1634)

10

JEAN-ROBERT BELLIARD & ANTHONY MARTIN[Iwa69], On p-adic L-functions, Ann. of Math. (2) 89 (1969), 198–205. MR
0269627 (42 #4522)[MW84]
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[NQDN11] Thong Nguyen Quang Do and Vésale Nicolas, Nombres de Weil, sommes deGauss et annulateurs galoisiens, Amer. J. Math. 133 (2011), no. 6, 1533–1571.
MR 2863370[Sol92]David Solomon, On a construction of p-units in abelian fields, Invent. Math.
109 (1992), no. 2, 329–350.
, Some New Maps and Ideals in Classical Iwasawa Theory with Appli[Sol09]
cations, arXiv:0905.4336 (2009), 36.
[Tsu99]
Takae Tsuji, Semi-local units modulo cyclotomic units, J. Number Theory 78(1999), no. 1, 1–26. MR 1706941 (2000f:11148)[Was97]
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1421575 (97h:11130)

March 23, 2014
Jean-Robert Belliard, Laboratoire de mathématiques de Besançon UMR 6623, Université
de Franche-Comté, 16 route de Gray, 25030 Besançon cedex, FRANCE
E-mail : jean-robert.belliard@univ-fcomte.fr
Anthony Martin, Laboratoire de mathématiques de Besançon UMR 6623, Université deFranche-Comté, 16 route de Gray, 25030 Besançon cedex, FRANCE
E-mail : anthony.martin@ens-lyon.org