Publications Mathématiques de Besançon

Michael Temkin

Wild coverings of Berkovich curves

Thong Nguyen Quang Do

Descente galoisienne et capitulation entre modules de Bertrandias-Payan

Karim Belabas

Jean-François Jaulent

The logarithmic class group package in PARI/GP

Sur la capitulation pour le module de Bertrandias-Payan

Fabien Pazuki

Publications mathématiques de Besançon
A l g è b r e  e t  t h é o r i e  d e s  n o m b r e s
Fabien Pazuki
Erratum and addendum to “Heights and regulators of number fields and elliptic
curves”
2016, p. 81-83.
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Publications mathématiques de Besançon – 2016, 81-83
ERRATUM AND ADDENDUM TO “HEIGHTS AND REGULATORS
OF NUMBER FIELDS AND ELLIPTIC CURVES”
by
Fabien Pazuki
Abstract. — We give a correct version of the Northcott property for regulators of elliptic
curves over number fields.
Résumé. — (Erratum et addendum de "Hauteurs et régulateurs de corps de nombres et de
courbes elliptiques") Nous donnons une version corrigée de la propriété de Northcott pour les
courbes elliptiques définies sur un corps de nombres.
1. A correction
A square root is missing in the application of the Minkowski inequality on page 59 of [Paz14].
We apologize and correct the statement of Theorem 4.8 page 59 of [Paz14]. We keep the
notation of the original article, the regulator is defined using the divisor L = 3(O).
Theorem 1.1. — Assume the Lang-Silverman Conjecture 4.5 page 58 of [Paz14]. Let K be
a number field. There exists a quantity c10 = c10(K) > 0 only depending on K such that for
any elliptic curve E defined over K with positive rank mK ,
Reg(E/K) ≥
( c10
mK
max{hF+(E/K), 1}
)mK
.
Thus the set of Q-isomorphism classes of elliptic curves defined over a fixed number field K
such that E(K) has positive bounded rank mK and with bounded regulator is finite.
Proof. — Let L = 3(O) and let ĥ = ĥE,L be the associated canonical height on E and
consider the euclidean space (E(K)⊗ R, ĥ1/2) ' (RmK , ĥ1/2). Apply Minkowski’s successive
minima inequality to the Mordell-Weil lattice ΛK = E(K)/E(K)tors viewed as a lattice inside
this euclidean space,
λ1(ΛK) · · ·λmK (ΛK) ≤ m
mK/2
K Reg(E/K)
1/2.
Now apply mK times the inequality of Conjecture 4.5 page 58 of [Paz14] to get
Mathematical subject classification (2010). — 11G50, 14G40.
Key words and phrases. — Heights, elliptic curves, regulators, Mordell-Weil.
82 Erratum and addendum to “Heights and regulators of number fields and elliptic curves”
(1) Reg(E/K) ≥ c
mK
4 max{hF+(E/K), 1}mK
mmKK
,
Finally, if the regulator and the rank is bounded then the height is bounded as soon as
mK 6= 0, hence the claimed finiteness. 
It implies the following new statement of Theorem 1.2 page 48 of [Paz14].
Theorem 1.2. — Assume the Lang-Silverman Conjecture 4.5 page 58 of [Paz14]. The set
of Q-isomorphism classes of elliptic curves E, defined over a fixed number field K with E(K)
of positive bounded rank mK and bounded regulator is finite.
2. The number field analogy and a new question
To recover the statement without the boundedness condition, the first idea would be to
improve Lemma 4.7 page 58 of [Paz14] (which is still valid and unconditional) to get an
inequality of the form mK ≤ c(K) hF+(E/K)1−ε for ε > 0 universal and c(K) > 0 depending
only on K. For K = Q and under the Generalized Riemann Hypothesis, Mestre has obtained
in II.1.2 pages 217-218 of [Mes86] the inequality mQ  logNE/ log logNE where NE is the
conductor of the elliptic curve E (inequality valid for NE big enough). It gives hope that a
stronger statement than the present Theorem 1.2 could be true.
In order to remove the boundedness condition on the rank, another idea would be to prove
an inequality between the regulator and the rank. In view of the number field case, it would
play a role similar to Friedman’s Theorem 3.4 page 53 of [Paz14]. So we formulate it here as
a question.
Question 2.1. — Let E be an elliptic curve defined over a number field K. Let mK be
the rank of the Mordell-Weil group E(K) and let Reg(E/K) be its regulator. Can one find a
positive quantity c0(K) depending only on K and a strictly increasing function f : N → R+
not depending on E such that the inequality
Reg(E/K) ≥ c0(K) f(mK)
holds?
Such an inequality would indeed imply that if one fixes K, a bounded regulator would force
the rank to be bounded, exactly as in the case of number fields where a bounded regulator
implies a bounded degree. This will be the subject of a future work.
References
[Mes86] Mestre, J.-F., Formules explicites et minorations de conducteurs de variétés algébriques.
Compositio Math. 58.2 (1986), 209–232.
[Paz14] Pazuki, F., Heights and regulators of number fields and elliptic curves. Publ. Math. Besançon
2014/2 (2014), 47–62.
Publications mathématiques de Besançon – 2016
F. Pazuki 83
September 19, 2016
Fabien Pazuki, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5,
2100 Copenhagen, Denmark. • E-mail : fpazuki@math.ku.dk
Publications mathématiques de Besançon – 2016


Erratum and addendum to “Heights and regulators of number fields and elliptic curves”

Christian Wuthrich

Publications mathématiques de Besançon
A l g è b r e  e t  t h é o r i e  d e s  n o m b r e s
Christian Wuthrich
The sub-leading coefficient of the L-function of an elliptic curve
2016, p. 95-96.
<http://pmb.cedram.org/item?id=PMB_2016____95_0>
© Presses universitaires de Franche-Comté, 2016, tous droits réservés.
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Publications mathématiques de Besançon – 2016, 95-96
THE SUB-LEADING COEFFICIENT OF THE L-FUNCTION OF AN
ELLIPTIC CURVE
by
Christian Wuthrich
Abstract. — We show that there is a relation between the leading term at s = 1 of an L-
function of an elliptic curve defined over an number field and the term that follows.
Résumé. — On montre une relation entre le terme dominant de la série L en s = 1 d’une
courbe elliptique définie sur un corps de nombre et le terme suivant.
Let E be an elliptic curve defined over a number field K. We will assume that the L-function
L(E, s) admits an analytic continuation to s = 1 and that it satisfies the functional equation.
By modularity [1], we know that this holds when K = Q. The conjecture of Birch and
Swinnerton-Dyer predicts that the behaviour at s = 1 is linked to arithmetic information.
More precisely, if
L(E, s) = ar (s− 1)r + ar+1 (s− 1)r+1 + · · ·
is the Taylor expansion at s = 1 with ar 6= 0, then r should be the rank of the Mordell-
Weil group E(K) and the leading term ar is equal to a precise formula involving the Tate-
Shafarevich group of E. It seems to have passed unnoticed that the sub-leading coefficient
ar+1 is also determined by the following formula.
Theorem 1. — With the above assumption, we have the equality
(1) ar+1 =
(
[K : Q] · (γ + log(2π))− 12 log(N)− log |∆K |
)
· ar
where γ = 0.577216 . . . is Euler’s constant, N is the absolute norm of the conductor ideal of
E/K and ∆K is the absolute discriminant of K/Q.
In particular, the conjecture of Birch and Swinnerton-Dyer also predicts completely what the
sub-leading coefficient ar+1 should be. One consequence for K = Q is that for all curves with
conductor N > 125, and this is all but 404 isomorphism classes of curves, the sign of ar+1 is
the opposite of ar. Of course, it is believed that ar is positive for all E/Q.
Mathematical subject classification (2010). — 11G40.
Key words and phrases. — L-functions of elliptic curves, Birch-Swinnerton-Dyer conjecture.
96 The sub-leading coefficient of the L-function of an elliptic curve
Proof. — Set f(s) = Bs · Γ(s)n with n = [K : Q] and B =
√
N · |∆K |/(2π)n. Then Λ(s) =
f(s) · L(E, s) is the completed L-function, which satisfies the functional equation Λ(s) =
(−1)r · Λ(2 − s), see [3]. For i ≡ r + 1 (mod 2) it follows that di
dsi
Λ(s)
∣∣
s=1 = 0. Hence for
i = r + 1, we obtain that
(r + 1) · f ′(s) · d
r
dsr
L(E, s) + f(s) · d
r+1
dsr+1
L(E, s)
is zero at s = 1. Therefore (r + 1) f ′(1) r! ar + f(1) (r + 1)! ar+1 = 0. It remains to note that
f(1) = B and f ′(1) = B ·
(
log(B) + n · Γ′(1)
)
together with Γ′(1) = −γ. 
Obviously a similar formula holds for the L-function of a modular form of weight 2 for Γ0(N).
More generally, for any L-function with a functional equation there is a relation between the
leading and the sub-leading coefficient of the Taylor expansion of the L-function at the central
point.
Sub-leading coefficients of Dirichlet L-functions have been investigated; for instance Colmez [2]
makes a conjecture, which is partially known. However these concern the much harder case
when s is not at the centre but the boundary of the critical strip of the L-function.
References
[1] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor. On the modularity of elliptic
curves over Q: wild 3-adic exercises. J. Amer. Math. Soc., 14(4):843–939, 2001.
[2] Pierre Colmez. Périodes des variétés abéliennes à multiplication complexe. Ann. of Math. (2),
138(3):625–683, 1993.
[3] Dale Husemöller. Elliptic curves, volume 111 of Graduate Texts in Mathematics. Springer-Verlag,
New York, second edition, 2004. With appendices by Otto Forster, Ruth Lawrence and Stefan
Theisen.
January 18, 2016
Christian Wuthrich, Room C58 Mathematics Building, University Park, Nottingham, NG7 2RD,UK
E-mail : christian.wuthrich@nottingham.ac.uk
Publications mathématiques de Besançon – 2016


The sub-leading coefficient of the $L$-function of an elliptic curve

Georges Gras

Sur le module de Bertrandias–Payan dans une $p$-extension – Noyau de capitulation

Laurent Berger

Publications mathématiques de Besançon
A l g è b r e  e t  t h é o r i e  d e s  n o m b r e s
Laurent Berger
Lubin’s conjecture for full p-adic dynamical systems
2016, p. 19-24.
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© Presses universitaires de Franche-Comté, 2016, tous droits réservés.
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Publications mathématiques de Besançon – 2016, 19-24
LUBIN’S CONJECTURE FOR FULL p-ADIC DYNAMICAL SYSTEMS
by
Laurent Berger
Abstract. — We give a short proof of a conjecture of Lubin concerning certain families of
p-adic power series that commute under composition. We prove that if the family is full (large
enough), there exists a Lubin-Tate formal group such that all the power series in the family are
endomorphisms of this group. The proof uses ramification theory and some p-adic Hodge theory.
Résumé. — (La conjecture de Lubin pour les systèmes dynamiques p-adiques pleins) Nous
donnons une démonstration courte d’une conjecture de Lubin concernant certaines familles de
séries formelles p-adiques qui commutent pour la composition. Nous montrons que si la famille
est pleine (assez grosse), il existe un groupe formel de Lubin-Tate tel que toutes les séries de
la famille sont des endomorphismes de ce groupe. La démonstration utilise la théorie de la
ramification et un peu de théorie de Hodge p-adique.
Introduction
Let K be a finite extension of Qp, and let OK be its ring of integers. In [5], Lubin studied
p-adic dynamical systems, namely families of elements of T · OK [[T ]] that commute under
composition, and remarked that “experimental evidence seems to suggest that for an invertible
series to commute with a noninvertible series, there must be a formal group somehow in the
background”. This observation has motivated the work of a number of people (Hsia, Laubie,
Li, Movaheddi, Salinier, Sarkis, Specter, ...) who proved various results in that direction. The
purpose of this note is to give a proof of a special case of the above observation, which is
referred to as “Lubin’s conjecture” in §3.1 of [7]. Let us consider a family F of commuting
power series F (T ) ∈ T · OK [[T ]]. We say that such a family is full if for all α ∈ OK there
exists Fα(T ) ∈ F such that F ′α(0) = α and if wideg(Fπ(T )) = q, where wideg(F (T )) denotes
the Weierstrass degree of F (T ), π is any uniformizer of OK and q is the cardinality of the
residue field of OK .
Mathematical subject classification (2010). — 11S82, 11S15, 11S20, 11S31, 13F25, 13F35, 14F30.
Key words and phrases. — p-adic dynamical system, Lubin-Tate formal group, p-adic Hodge theory.
20 Lubin’s conjecture for full p-adic dynamical systems
Theorem. — If F is a full family of commuting power series, there exists a Lubin-Tate
formal group G such that Fα(T ) ∈ End(G) for all α ∈ OK .
This result already appears as Theorem 2 of [4]. Our proof is however considerably shorter
than that of ibid., and does not use the theory of the field of norms. It is very similar to that
of the main result of [8], which treats the case K = Qp. The main ingredients are ramification
theory and some p-adic Hodge theory. In order to simplify the use of p-adic Hodge theory,
we assume that K is a Galois extension of Qp.
1. p-adic dynamical systems
In this note, we consider a set F = {Fα(T )}α∈OK of power series Fα(T ) ∈ T ·OK [[T ]] such that
F ′α(0) = α and Fα ◦Fβ(T ) = Fβ ◦Fα(T ) whenever α, β ∈ OK . Recall that π is a uniformizer
of OK , and that q is the cardinality of the residue field k of OK . If F (T ) is a power series
and n > 0, we denote by F ◦n(T ) the n-th fold iteration F ◦ · · · ◦F (T ). If F (T ) has an inverse
for the composition, this definition extends to n ∈ Z. Recall that the Weierstrass degree
wideg(F (T )) of F (T ) =
∑+∞
i=1 fiT
i ∈ T · OK [[T ]] is the smallest integer i such that fi ∈ O×K .
By the Weierstrass preparation theorem, if wideg(F ) 6= +∞, then F has wideg(F ) zeroes in
mCp .
Proposition 1.1. — There exists a power series G(T ) ∈ T ·k[[T ]] and an integer d > 1 such
that G′(0) ∈ k× and Fπ(T ) ≡ G(T p
d).
Proof. — See (the proof of) theorem 6.3 and corollary 6.2.1 of [5]. 
Proposition 1.2. — There exists a power series LF (T ) ∈ K[[T ]] such that
1. LF (T ) = T + O(T 2);
2. LF (T ) converges on the open unit disk;
3. LF ◦ Fα(T ) = α · LF (T ) for all α ∈ OK .
Proof. — See propositions 1.2 and 2.2 of [5] for the construction of a unique power series
LF (T ) that satisfies (1), (2) and (3) for α a uniformizer of OK . If β ∈ OK \ {0}, then
β−1 ·LF ◦Fβ also satisfies (1), (2) and (3) for α as above, so that LF ◦Fβ(T ) = β ·LF (T ) for
all β ∈ OK . 
The hypothesis that F is full implies that pd = q, so that wideg(Fπ(T )) = q. For n > 1, let
Λn denote the set of u ∈ mCp such that F ◦nπ (u) = 0 and F ◦n−1π (u) 6= 0 and let Λ∞ = ∪n>1Λn.
Proposition 1.1 implies that F ′π(T )/π is a unit of OK [[T ]], so that the roots of F ◦nπ (T ) are
simple for all n > 1. The set Λn therefore has qn−1(q − 1) elements.
The series Fα(T ) is invertible if α ∈ O×K so that in this case, Fα(z) = 0 if and only if z = 0. If
u ∈ Λn and α ∈ O×K , then F ◦nπ ◦Fα(u) = Fα◦F ◦nπ (u) = 0 and F ◦n−1π ◦Fα(u) = Fα◦F ◦n−1π (u) 6=
0 so that the action of Fα(T ) permutes the elements of Λn.
Let Kn = K(Λn), so that Λi ⊂ Kn if i 6 n, and let K∞ = ∪n>1Kn. If α ∈ O×K , let n(α) be
the largest integer n > 0 such that α ∈ 1 + πnOK .
Proposition 1.3. — If n > 1 and u ∈ Λn, then
Publications mathématiques de Besançon – 2016
L. Berger 21
1. Fα(u) = u if and only if n(α) > n;
2. If n(α) = n, then wideg(Fα(T )− T ) = qn;
3. Λn = {Fα(u)}α∈O×K .
Proof. — If n = 1 and Fα(u) = u, then u is a root of Fα(T ) − T = (α − 1)T + O(T 2),
so that α − 1 ∈ πOK . This implies that {Fα(u)}α∈O×K has at least q − 1 distinct elements.
These elements are all roots of Fπ(T )/T , whose wideg is q− 1, so {Fα(u)}α∈O×K has precisely
q − 1 elements. These elements all have valuation 1/(q − 1), and if n(α) = 1, the Newton
polygon of Fα(T ) − T starts at the point (1, 1), so that it can have only one segment, and
wideg(Fα(T )− T ) = q. This implies the proposition for n = 1.
Assume now that the proposition holds up to some n > 1 and take u ∈ Λn+1. If n(α) 6 n,
then Fα(T ) − T has at most qn roots by (2), contained in Λ0 ∪ . . . ∪ Λn by (1). Therefore
Fα(u) = u implies n(α) > n+1. The set {Fα(u)}α∈O×K therefore has at least q
n(q−1) distinct
elements, all of them roots of F ◦n+1π (T )/F ◦nπ (T ).
This implies that {Fα(u)}α∈O×K has exactly q
n(q − 1) elements. If n(α) = n+ 1, the Newton
polygon of Fα(T ) − T starts at the point (1, n + 1), with n + 1 segments of height one
and slopes −1/qk(q − 1) with 0 6 k 6 n, so that it reaches the point (qn+1, 0) and hence
wideg(Fα(T )− T ) = qn+1. This implies the proposition for n+ 1. 
Corollary 1.4. — The field K∞ is an abelian totally ramified extension of K, and if g ∈
Gal(K∞/K), there is a unique η(g) ∈ O×K such that g(u) = Fη(g)(u) for all u ∈ Λ∞.
The map η : Gal(K∞/K)→ O×K is an isomorphism.
Proof. — Take u ∈ Λn and α ∈ O×K . As we have seen above, Fα(u) ∈ Λn, so that the map
u 7→ Fα(u) induces a field automorphism of K(u). By (3) of Proposition 1.3, this implies that
Kn = K(u) and that every element of Gal(Kn/K) comes from u 7→ Fα(u) for some α ∈ O×K .
The extension Kn/K is therefore abelian, and so is K∞/K. Since Kn = K(u), the extension
Kn/K is totally ramified, and so is K∞/K.
The map η is surjective because every Fα(T ) gives rise to an automorphism of K∞, and it is
injective because if η(g) = 1, then g(u) = u for all u ∈ Λ∞ so that g = 1. 
In order to prove our main theorem, we study the p-adic periods of η. Corollary 1.4 and local
class field theory imply that the extension K∞/K is attached to a uniformizer $ of OK . Let
χ$ : GK → O×K denote the corresponding Lubin-Tate character.
2. p-adic Hodge theory
Let R be the p-adic completion of lim−→n>1OK [[Xn]] where OK [[Xn]] is seen as a subring of
OK [[Xn+1]] via the identification Xn = Fπ(Xn+1) (this ring is defined in [8], where it is
denoted by A∞). We define an action of GK on R by g(H(Xn)) = H(Fη(g)(Xn)). This is
well-defined since Fπ ◦ Fη(g)(T ) = Fη(g) ◦ Fπ(T ). We have R/πR = lim−→n>1 k[[Xn]].
Lemma 2.1. — The ring R/πR is perfect.
Publications mathématiques de Besançon – 2016
22 Lubin’s conjecture for full p-adic dynamical systems
Proof. — LetG(T ) be as in Lemma 1.1. The fact thatXn = Fπ(Xn+1) implies thatG◦n(Xn) =
G◦n+1(Xn+1)q. Since G′(0) ∈ k×, we have k[[T ]] = k[[G(T )]] and therefore
R/πR = lim−→
G◦n(Xn)=G◦n+1(Xn+1)q
k[[G◦n(Xn)]]
is perfect. 
Let Ẽ+ = lim←−x 7→xq OCp/π. Choose a sequence {un}n>1 with un ∈ Λn and Fπ(un+1) = un.
This sequence gives rise to a map i : R/πR → Ẽ+, determined by the requirement that
i(Xn) = (G◦1(un), G◦2(un+1), . . .). The definition of the action of GK on R and Corollary 1.4
imply that i is GK-equivariant.
Lemma 2.2. — The map i : R/πR→ Ẽ+ is injective.
Proof. — It is enough to show that i : k[[Xn]] → Ẽ+ is injective. If it was not, there
would be a nonzero polynomial P (T ) ∈ k[T ] such that P (i(Xn)) = 0, and then i(Xn) =
(G◦1(un), G◦2(un+1), . . .) would belong to Fp, which is clearly not the case. 
Let K0 = Qunrp ∩K and let Ã+ = OK ⊗OK0 W (Ẽ
+) (see [2]; note that Ã+ usually denotes
W (Ẽ+), and is denoted by Ainf in ibid.). We have R = OK ⊗OK0 W (R/πR) since R is a
strict π-ring, and by the functoriality of Witt vectors, the map i extends to an injective and
GK-equivariant map i : R → Ã+. We write x instead of i(X1) ∈ Ã+. The GK-equivariance
of i implies that g(x) = Fη(g)(x).
Let B+cris and BdR be some of Fontaine’s rings of periods. Recall that BdR is a field, that
there is a Frobenius map ϕ on B+cris, a filtration {FiliBdR}i∈Z on BdR, and an injective
map K ⊗K0 B+cris → BdR. There is also an action of GK on B
+
cris and BdR compatible
with the above structure, and BGKdR = K. Let ϕq = ϕf on B
+
cris, where q = pf , extended
by K-linearity to K ⊗K0 B+cris. We refer to [2] and [3] for the properties of these objects.
Let Σ = Gal(K/Qp). If τ ∈ Σ, choose a n(τ) ∈ Z>0 such that τ |K0 = ϕn(τ). The map
τ ⊗ ϕn(τ) : K ⊗K0 B+cris → K ⊗K0 B
+
cris is then well-defined and commutes with ϕq and the
action of GK .
We say that a character λ : GK → O×K is crystalline positive if there exists a nonzero
z ∈ K⊗K0 B+cris such that g(z) = λ(g)·z for all g ∈ GK . The following proposition summarizes
the input that we need from the p-adic Hodge theory of characters.
Proposition 2.3. — A character λ : GK → O×K that factors through Gal(K∞/K) is crys-
talline positive if and only if λ =
∏
τ∈Σ τ ◦ χhτ$ with hτ ∈ Z>0.
If t$ ∈ K ⊗K0 B+cris is such that g(t$) = χ$(g) · t$ for all g ∈ GK , then t$ ∈ Fil1BdR and
ϕq(t$) = $ · t$.
Sketch of proof. — If λ : GK → O×K is a crystalline positive character and hτ ∈ Z>0 denotes
the Hodge-Tate weight of λ at τ ∈ Σ, then λ ·
∏
τ∈Σ τ ◦χ−hτ$ is crystalline and has Hodge-Tate
weight zero at all τ ∈ Σ so that it is unramified, and therefore trivial if λ factors through
Gal(K∞/K), since K∞/K is totally ramified.
Let ωE and tE be the elements constructed in §9.2 and §9.3 of [1] (with E = K and πE = $).
We have tE ∈ K ⊗K0 B+cris and ϕq(tE) = $ · tE and tE ∈ Fil1BdR (proposition 9.10 of
ibid). If g ∈ GK , then (in the notation of ibid and where [ · ]LT denotes the endomorphisms
Publications mathématiques de Besançon – 2016
L. Berger 23
of the Lubin-Tate group attached to $) we have g(ωE) = [χ$(g)]LT(ωE) and therefore
g(tE) = g(FE(ωE)) = FE(g(ωE)) = FE ◦ [χ$(g)]LT(ωE) = χ$(g) · FE(ωE) = χ$(g) · tE
since FE is the logarithm of the Lubin-Tate group attached to $. If t$ ∈ K⊗K0 B+cris is such
that g(t$) = χ$(g) · t$ for all g ∈ GK , then t$/tE ∈ BGKdR = K, and this implies the rest of
the proposition. 
Recall that LF (T ) ∈ K[[T ]] is the logarithm attached to F . Since LF (T ) converges on the
open unit disk, we can view LF (x) as an element of K ⊗K0 B+cris.
Proposition 2.4. — The character η : GK → O×K is crystalline positive.
Proof. — If g ∈ GK , then g(LF (x)) = LF (g(x)) = LF (Fη(g)(x)) = η(g) · LF (x). 
Corollary 2.5. — We have LF (x) = β ·
∏
τ∈Σ(τ⊗ϕn(τ))(t$)hτ where hτ ∈ Z>0 and β ∈ K×.
Proof. — This follows from the facts that η =
∏
τ∈Σ τ ◦χhτ$ , that χ$(g) = g(t$)/t$ and that
BGKdR = K. 
Proposition 2.6. — We have ϕq(LF (x)) = µ · LF (x) for some µ ∈ πOK .
Proof. — Corollary 2.5 and Proposition 2.3 imply the proposition with µ =
∏
τ τ($)hτ , and
not all hτ can be equal to 0 since η 6= 1. 
Corollary 2.7. — We have ϕq(x) = Fµ(x).
Proof. — Proposition 2.6 implies that LF (ϕq(x)) = LF (Fµ(x)). We would like to apply
L◦−1F (T ) but we have to mind the convergence and need to proceed as follows. Since η is
nontrivial, there is a τ ∈ Σ such that hτ−1 > 1. We have
(τ ⊗ ϕn(τ))(LF (ϕq(x))) = (τ ⊗ ϕn(τ))(LF (Fµ(x)))
in K ⊗K0 B+cris and hτ−1 > 1 now implies that (τ ⊗ ϕn(τ))(LF (ϕq(x))) is divisible by t$ so
that by Proposition 2.3, it belongs to Fil1BdR. We can then apply Lτ◦−1F (T ) in BdR and get
that (τ ⊗ϕn(τ))(ϕq(x)) = (τ ⊗ϕn(τ))(Fµ(x)) in BdR. This equality also holds in Ã+, so that
ϕq(x) = Fµ(x). 
Theorem 2.8. — There is a Lubin-Tate formal group G such that Fα(T ) ∈ End(G) for all
α ∈ OK .
Proof. — By Corollary 2.7, we have ϕq(x) = Fµ(x). This implies that Fµ(T ) ≡ T q mod
πOK [[T ]]. The Weierstrass degree of Fµ(T ) is qval(µ) so that val(µ) = 1 and Fµ(T ) is a Lubin-
Tate power series. By [6], there is a Lubin-Tate formal group G such that Fµ(T ) ∈ End(G).
Since Fα(T ) commutes with Fµ(T ), we also have Fα(T ) ∈ End(G) for all α ∈ OK . 
Remark 2.9. — We have µ = $ and η = χ$. Indeed, the extension K∞/K is generated
by the torsion points of G, and is therefore attached to µ by local class field theory, so that
µ = $. This in turn implies that η = χ$.
Publications mathématiques de Besançon – 2016
24 Lubin’s conjecture for full p-adic dynamical systems
References
[1] Pierre Colmez. Espaces de Banach de dimension finie. J. Inst. Math. Jussieu, 1(3):331–439, 2002.
[2] Jean-Marc Fontaine. Le corps des périodes p-adiques. Astérisque, (223):59–111, 1994. With an
appendix by Pierre Colmez, Périodes p-adiques (Bures-sur-Yvette, 1988).
[3] Jean-Marc Fontaine. Représentations p-adiques semi-stables. Astérisque, (223):113–184, 1994.
[4] Liang-Chung Hsia and Hua-Chieh Li. Ramification filtrations of certain abelian Lie extensions of
local fields. J. Number Theory, 168:135–153, 2016.
[5] Jonathan Lubin. Nonarchimedean dynamical systems. Compositio Math., 94(3):321–346, 1994.
[6] Jonathan Lubin and John Tate. Formal complex multiplication in local fields. Ann. of Math. (2),
81:380–387, 1965.
[7] Ghassan Sarkis. Height-one commuting power series over Zp. Bull. Lond. Math. Soc., 42(3):381–
387, 2010.
[8] Joel Specter. The crystalline period of a height one p-adic dynamical system. Trans. Amer. Math.
Soc., to appear, 2016.
March 30, 2016
Laurent Berger, UMPA de l’ENS de Lyon, UMR 5669 du CNRS, IUF
E-mail : laurent.berger@ens-lyon.fr • Url : perso.ens-lyon.fr/laurent.berger/
Publications mathématiques de Besançon – 2016


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