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arXiv:2306.01759v1 [math.NT] 25 May 2023
UNLIKELY INTERSECTION IN HIGHER-DIMENSIONAL FORMAL
GROUPS AND p-ADIC DYNAMICAL SYSTEMS
MABUD ALI SARKAR∗AND ABSOS ALI SHAIKH†
Abstract. In this work, we study arithmetic dynamical questions for formal groups and p-adic
dynamical systems in higher dimensions. As a generalization of Berger’s result in 1-dimensional
case, in the paper, we prove that for any two higher-dimensional formal groups over the ring
of p-adic integers, if they have infinitely many torsion points in common, then they coincide.
Finally, we prove if D1and D2are d-dimensional stable p-adic dynamical systems with infinitely
many preperiodic points in common then D1and D2coincide.
1. Introduction and Motiivation
Let Zpbe the ring of integers of the p-adic field Qp. Suppose Kis a finite extension of Qpsuch
that its ring of integers is OK. We also supppose that ¯
Kis an algebraic closure of Kand Cp
is the p-adic completion of ¯
K. Let F(X, Y ) = X+Y∈ OK[[X, Y ]] be an 1-dimensional formal
group law over OK. Let Tors(F) be the set of torsion points of Fin mCp={z∈Cp| |z|p<1}.
Then Berger proved
Theorem 1.1. 3, Theorem A If Fand Gare 1-dimensional formal groups over OKsuch that
Tors(F)∩Tors(G)is infinite, then F=G.
The preceding result is restricted to 1-dimensional formal groups. We generalised the same
result to higher-dimensional formal groups in this study.
Consider the stable p-adic dynamical system Dassociated with the formal group F. The
collection of preperiodic points of Dis denoted by Preper(D). Then we get the following
equivalent outcome;
Theorem 1.2. 13, Theorem 3.8 If D1and D2are 1-dimensional p-adic stable dynamical
systems over OKsuch that Preper(D1)∩Preper(D2)is infinite, then D1=D2.
The preceding result is restricted to a p-adic dynamical system in one dimension. In this work,
we extend the aforementioned finding to stable p-adic dynamical systems in higher-dimensions.
2020 Mathematics Subject Classification. 11S31, 11S82, 14L05, 11F85, 13J05, 37P35.
Key words and phrases. Formal group, nonarchimedean dynamical system, preperiodic points, torsion points,
unlikely intersection.
1
2 M. A. SARKAR AND A. A. SHAIKH
The following is a summary of the paper:
Preliminary facts are covered in Section 2, whereas the main results are covered in Sections
3and 4. Some open questions are presented in the final part.
2. Preliminery
Definition 2.1. 5 Let Rbe a commutative ring and consider the n-tuples X= (x1,··· , xn)
and Y= (y1,··· , yn) over R. A n-dimensional formal group F(X, Y ) = (F1(X, Y ),··· , Fn(X, Y ))
is a n-tuple power series in R[[X, Y ]]nsatisfying the following properties
(i) F(X, Y ) = X+Y+mod deg. 2, where Fr(X, Y ) = xr+yr+mod deg. 2, r = 1,2,··· , n.
(ii) F(F(X, Y ), Z ) = F(X, F (Y, Z)).
(iii) F(X, 0) = F(0, X ) = X.
(iv) an unique power series ι(X)∈R[[X]]nwith F(X, ι(X)) = 0.
Example 2.1. (i) A(X, Y ) = X+Yand M(X, Y ) = X+Y+XY are 1-dimensional
additive and multiplicative formal groups, respectively.
(ii) The 2-tuple power series F(X, Y ) = (x1+y1+x2y2, x2+y2) is a 2-dimensional formal
group with X= (x1, x2), Y = (y1, y2).
(iii) The n-dimensional additive formal group is defined by
A(X, Y ) = (A1(X, Y ),··· , An(X, Y )) = X+Y,
where Ai(X, Y ) = xi+yi, i = 1,2,··· , n, are 1-dimensional additive formal groups,
X= (x1,··· , xn), Y = (y1,··· , yn).
Definition 2.2. 5, Sec. 9.4 If F(X, Y )∈ OK[[X, Y ]]mand G(X, Y )∈ OK[[X, Y ]]nare respec-
tively m-dimensional and n-dimensional formal group, then a vector-valued map f:F(X, Y )→
G(X, Y ) of the form of n-tuple of power series f(X) = (f1(X),··· , fn(X)) ∈ OK[[X]]nin m
indeterminates, is called a homomorphism if f(X)≡0 mod (deg. 1) such that
f(X+FY) = f(X) +Gf(Y) i.e., f(F(X, Y )) = G(f(X), f (Y)).
The homomorphism f(X)∈ OK[[X]]nis an isomorphism if there exists g(X) : G(X, Y )→
F(X, Y ) such that f(g(X)) = X=g(f(X)).
Remark 1.It is worth noting that, according to Definition 2.2, the endomorphism power series
of a n-dimensional formal group belongs to the power series ring OK[[X]]nrather than OK[[X]].
UNILIKELY INTERSECTION IN FORMAL GROUPS AND p-ADIC DYNAMICAL SYSTEMS 3
In ( 5 , 6 ), and 12 , there are multiple distinct constructions of higher dimensional formal
groups. The authors of 1 recently constructed a 2-dimensional formal group with features
similar to a 1-dimensional Lubin-Tate formal group. It should be noted that in 11 , Lubin and
Tate introduce a 1-dimensional Lubin-Tate formal group, which is a very useful tool in local
class field theory. Lubin and Tate prove (local) Artin-Reciprocity law with the help of their
formal group’s torsion points, thus providing a parallel interpretation of local class field theory.
Example 2.2. In 1 , the authors construct a 2-dimensional Lubin-Late formal group Fover
Zpwith multiplication-by-pendomorphism
[p]F(X) = (px1, px2) (mod deg. 2) and [p]F(X)≡(xh1
2, xh2
1) (mod p),
where gcd(h1, h2) = 1, X= (x1, x2). The logarithm Lfor the 2-dimensional Lubin-Tate formal
group Fis given recursively by
L(x1, x2) = (L1(x1, x2), L2(x1, x2)) = x1+1
pL2(xph1
1, xph1
2), x2+1
pL1(xph2
1, xph2
2),
where
L1(x1, x2) = x1+1
pxph1
2+1
p2xph1+h2
1+1
p3xp2h1+h2
2+1
p4xp2h1+2h2
1+1
p5xp3h1+2h2
2+···
L2(x1, x2) = x2+1
pxph2
1+1
p2xph1+h2
2+1
p3xph1+2h2
1+1
p4xp2h1+2h2
2+1
p5xp2h1+3h2
2+···
so that the 2-dimensional Lubin-Tate formal group can be obtained by the formula
F(X, Y ) = L−1(L(X) + L(Y)).
Remark 2.It would be an interesting research subject to generalise the construction ( 1 ) of a
2-dimensional (Lubin-Tate) formal group in upper dimensions along the same paths as 1 .
The following definition gives formula for composition of two power series in the ring OK[[X]]n.
Definition 2.3. Let f(X) = (f1(X), f2(X),··· , fn(X)), g(X) = (g1(X), g2(X),··· , gn(X)) be
two arbitrary members in OK[[X]]n, X = (x1,··· , xn) with fi(X), gi(X)∈X· OK[[X]]. Then
f◦g= (f1(g1,··· , gn),··· , fn(g1,··· , gn))
g◦f= (g1(f1,··· , fn),··· , gn(f1,··· , fn)).
4 M. A. SARKAR AND A. A. SHAIKH
3. Unlikely intersection in higher dimensional formal groups
This section contains main results. We note that the higher dimensional formal group F
is not formal OK-module i.e., the multiplication-by-amap [a]F(X) is not an endomorphism
for some a∈ OK. However, as usual we have multiplication-by-nmap [n]F(X) for all n∈Z,
which, by continuity of p-adic topology, extends to [a]F(X) for all a∈Zp.
Definition 3.1. If Fis a d-dimensional formal group over Zpwith a multiplication-by-pendo-
morphism [p]F, then the set of torsion points of Fis given by
Tors(F) := [
n
{X∈¯
Qd
p|[pn]F(X) = X}, X = (x1,··· , xd).
The height of a n-dimensional formal group is defined as follows:
Definition 3.2. 5, Sec. 18.3.8 Let F(X, Y ) be an n-dimensional formal group F(X, Y ) over
the ring of p-adic integers Zpwith endomorphism [p]F(X) = (f1(X),··· , fn(X)) (mod p). If the
ring Fp[[x1,··· , xn]] is a finitely generated and free module over the subring Fp[[f1(X),··· , fn(X)]]
of rank ph, h ∈N, then his the height of F. We use the notation Fpfor finite field with p
elements.
If Fis a formal group with height h, then the p-adic Tate module TpFis a free Zp-module
with rank h, yielding a Galois representation ρF: Gal( ¯
Qp/Qp)→GLh(Qp). Then we have the
following corollary:
Lemma 3.1. 3, Corollary 1.2 The image of ρFcontains an open subgroup of Z×
p·Id.
So if σ∈Gal( ¯
Qp/Qp) is such that ρF(σ) = a·Id, then σ(z) = [a]F(z) for all z∈Tors(F).
Definition 3.3. A power series h(X)∈Qp[[X]]d, X = (x1,··· , xd) with h(0) = 0 is said to be
stable if J0(h) is neither zero matrix nor a matrix with entries roots of unity along diagonals.
The following result proves that if J0(h)6= 0, then there exists a unique power series h◦−1∈
Qp[[X]]dsuch that h◦h◦−1=h◦−1◦h=X.
Proposition 3.2. The sufficient condition for h∈ OK[[X]]d, X = (x1,··· , xd)with h(0) = 0
being invertible is that J0(h)∈GLd(O×
K), where J(h) := h∂
∂xjhiid
i,j=1 is the Jacobian matrix of
h(X)and J0(h) = J(h)(0).
Proof. Because h(X) is invertible, there must exist some inverse h−1(X)∈ OK[[X]]dfor
h−1(0) = 0 so that h(h−1(X)) = X. By chain rule, J(h(h−1(X)) = J(h)(X)J(h−1)(X) = J(X)
UNILIKELY INTERSECTION IN FORMAL GROUPS AND p-ADIC DYNAMICAL SYSTEMS 5
such that evaluating at 0 yields J0(h)J0(h−1) = Id, the d-by-didentity matrix Id. As a result,
both J0(h), J0(h−1)∈GLd(O×
K).
Conversely, suppose J0(h)∈GLd(O×
K). The construction of an inverse h−1(X) is sufficient.
Induction will be used. If we use the linear algebraic expression f1(X) = J0(h−1)X∈ OK[[X]]d,
we get
h(f1(X)) ≡X(mod deg 2).
Suppose the polynomial fn(X)∈ OK[[X]]dwith fn(X)≡fn−1(X) (mod deg n) fulfilling
h(fn(X)) ≡X(mod deg n+1)
for n≥1. The following polynomial fn+1(X)∈ OK[[X]]dmust then have the formula
fn+1(X) = fn(X) + Rn+1 (remainder),
where Rn+1(X)≡0 (mod deg n+ 1) and Rn+1 is actually a homogeneous polynomial with
degree n+ 1. Since J0(h) is invertible, we can solve for Rn+1 from the condition
h(fn+1(X)) = h(fn(X) + Rn+1 )) ≡h(fn(X)) + J0(h)Rn+1(X)≡X(mod deg n+ 2)
giving us Rn−1≡J0(h)−1(X−h(fn(X))) ≡0 (mod deg n+ 1). So we derive the inverse of
h(X) via induction on fn(X). The proof is now complete.
Thus if h(X)∈Zp[[X]]dwith h(0) = 0 and h′(0) ∈GLd(Z×
p), then h◦−1(X)∈Zp[[X]]dwith
h◦−1(0) = 0.
Theorem 3.3. Let u(X)∈Qp[[X]]d, X = (x1,··· , xd)with u(0) = 0 be a stable power series.
A power series h(X)∈Qp[[X]]dwith h(0) = 0 such that h◦u=u◦his determined by J0(h),
the Jacobian of hat 0.
Proof. Let h1and h2be two power series in Qp[[X]]dsuch that J0(h1) = J0(h2) and they both
commute with u. Let gmbe the sum of the terms of hi, i = 1,··· , d of total degree ≤m.
We can decompose hi=gm+ ∆i, i = 1,··· , d, where the remainder series ∆ihas total-degree
≥m+ 1. Therefore
(gm+ ∆i)◦u=gm◦u+ ∆i◦u≡gm◦u+J0(u)m+1∆imod deg (m+ 2),(1)
u◦(gm+ ∆i)≡u◦gm+J(u)(gm)∆i≡u◦gm+J0(u)∆imod deg (m+ 2),(2)
6 M. A. SARKAR AND A. A. SHAIKH
where J(u) is the Jacobian of u,J0(u) = J(u)(0). Since u◦h=h◦u, from equations (1) and
(2), we get
gm◦u+J0(u)m+1∆i≡u◦gm+J0(u)∆imod deg (m+ 2)
⇒∆i≡(u◦gm−gm◦u)J0(u)m+1 −J0(u)−1mod deg (m+ 2)
and because J0(u)m+1 6=I2,2, ∆iis defined. If h1and h2agree in total-degrees ≤m, they must
also agree in total-degrees ≤m+ 1. Induction on myields h1=h2. To put it another way, the
mapping h(X)7→ J0(h) is a bijection.
Corollary 3.3.1. Let u(X)∈Qp[[X]]d, X = (x1,··· , xd)with u(0) = 0 be a stable power
series. A power series h(X1, X2)∈Zp[[X1, X2]]dwith h(0) = 0 such that h◦u=u◦his
determined by J0(h), the Jacobian of hat 0, for X1= (x11,··· , x1d),··· , X2= (x21,··· , x2d).
Proof. The proof is an immediate consequence of Theorem 3.3, because we can simply replace
h(X)∈Zp[[X]]dwith h(X1, X2)∈Zp[[X1, X2]]d.
Corollary 3.3.2. Assume that uis a stable endomorphism of the formal group Fand that
dis the dimension of F. If h(X)∈Zp[[X]]dwith h(0) = 0 such that h◦u=u◦h, then
h∈EndZp(F).
Proof. Note that
(F◦h)◦u=F◦u◦h, since ucommute with h
=u◦(F◦h),since uis endomorphism of F ,
(h◦F)◦u=h◦u◦F, since uis endomorphism of F
=u◦(h◦F),since ucommute with h.
The power series F◦hand h◦Fcommute with uin the same way that their Jacobians at 0
are equivalent, i.e., J0(F◦h) = J0(h◦F). Theorem 3.3 now implies that F◦h=h◦F, which
leads to the conclusion that h∈EndZp(F).
UNILIKELY INTERSECTION IN FORMAL GROUPS AND p-ADIC DYNAMICAL SYSTEMS 7
Remark 3.The set of zeros of any p-adic power series h(X) =
h1(x1,··· , xd)
.
.
.
h2(x1,··· , xd)
∈Zp[[X]]d
with h(0) = 0 in ¯
Qd
pis 0-dimensional because we are solving the system
h1(x1,··· , xd) = 0
.
.
.
hd(x1,··· , xd) = 0
of d-equations in d-variables. To put it another way, the set of zeros of h(X) can be a finite or
discrete subset of ¯
Qp. It is worth noting that the failure of ¯
Qp’s local compactness establishes
a contrast between finiteness and discreteness.
Remark 4.For technical reasons, we will assume for the rest of this article that Fis a simple
formal group with no proper sub-formal group. Furthermore, if Fis a simple formal group
of finite height, say h, then any endomorphism of Fhas a finite number of zeros. In ¯
Qd
p, the
multiplication-by-pmap [p]F(X) has phzeros.
Lemma 3.4. If Fis a d-dimensional formal group over Zpand if h(X)∈Zp[[X]]dis such that
h(0) = 0 and h(Z)∈Tors(F)for all Z∈ Z, where Z ⊂ Tors(F)is infinite, then h∈EndZp(F).
Proof. By Lemma 3.1, there exists σ∈Gal( ¯
Qp/Qp) and a stable endomorphism uof Fwith
σ(Z) = u(Z) for all Z∈Tors(F).(3)
Since h(Z)∈Tors(F), for every Z∈ Z, substituting Zby h(Z) in equation (3) yields
σ(h(Z)) = u(h(Z)) for all Z∈Tors(F).(4)
Moreover, according to equation (3), σ(h(Z)) = h(σ(Z)) = h(u(Z) holds true for Z∈ Z. Thus,
we derive from equation (4)
u(h(Z)) = h(u(Z)) for all Z∈ Z
⇒u◦h−h◦u= 0 on Z.
Because Zis infinite and both u, h have finitely many zeros, we get u◦h=h◦u. Therefore
by Corollary 3.3.2, we conclude h∈EndZp(F).
8 M. A. SARKAR AND A. A. SHAIKH
Theorem 3.5. If Fand Gare d-dimensional formal groups over Zpsuch that Tors(F)∩Tors(G)
is infinite, then F=G.
Proof. According to Lemma 3.1, there exists σ∈Gal( ¯
Qp/Qp) and a stable endomorphism uof
Fwith
σ(Z) = u(Z) for all Z∈Tors(F).(5)
The set Z:= Tors(F)∩Tors(G) is stable under the action of Gal( ¯
Qp/Qp), and therefore Z∈ Z
implies σ(Z)∈ Z. Since Z∈ Z ⇒ Z∈Tors(G)⇒σ(Z)∈Tors(G), equation (5) gives
u(Z)∈Tors(G) for all Z∈ Z. Because Zis infinite, we obtain u∈EndZp(F) using Lemma
3.4. Consequently, uis an endomorphism of both d-dimensional formal groups Fand G, in
other words,
u◦F=F◦u
u◦G=G◦u.
Because Fand Ghave the same Jacobian at 0, we derive F=Gby Corollary 3.3.1.
4. Unlikely intersection in p-adic dynamical system
In this part, higher dimensional stable p-adic dynamical systems connected to higher dimen-
sional formal groups over Zpare discussed. A power series and its iterations are typically used
to analyse a dynamical system. However, in our research, we use Definition 4.2 to define a more
broad dynamical system. Such dynamical systems are central to Lubin’s research ( 10 ).
Definition 4.1. A power series u(X)∈Zp[[X]]d, X = (x1,··· , xd) with u(0) = 0 is said to be
stable if the d-by-dJacobian matrix J0(u) of u(X) at 0 is neither zero matrix nor J0(u) a roots
of unity in the ring of d-by-dmatrices over ¯
Qp.
Example 4.1. If u(X)∈Zp[[X]]2, X = (x1, x2) be defined by
u(X) = (px1, px2) (mod deg. 2) and u(X)≡(xh1
2, xh2
1) (mod p),
then the Jacobian matrix of u(X) at 0 is given by
J0(u) = p0
0p!and u(X)6≡ 0 (mod p).
This is a stable power series.
UNILIKELY INTERSECTION IN FORMAL GROUPS AND p-ADIC DYNAMICAL SYSTEMS 9
Definition 4.2. Ad-dimensional stable p-adic dynamical system Dover Zpconsists of a non-
invertible stable power series f(X)∈Zp[[X]]dwith f(0) = 0 such that J0(f)∈pZpand
f(X)6≡ 0 (mod pZp) as well as an invertible power series u(X)∈Zp[[X]]dsatisfying u◦f=f◦u.
Any stable power series which commutes with either of for ubelongs to D. This is how D
grows.
Example 4.2. If Fis a d-dimensional commutative formal group, then its ring of endomor-
phisms End(F) consists of a d-dimensional stable p-adic dynamical system.
If we want to construct a dynamical system from a d-dimensional commutative formal group
F, then its full ring of endomorphisms End(F) may or may not be commutative. However,
when the formal group is commutative, Zwill be contained in the endomorphisms End(F), and
Zpas well. Then just by looking at the commutative subgroup Z×
p, the p-adic units within the
endomorphism ring End(F), become a commutative family of transformations. In other words,
End(F) contains p-adic dynamical system of our interest.
Definition 4.3. Given a power series u(X)∈Zp[[X]]d, X = (x1,··· , xd), a point X∈¯
Qd
pis
said to be preperiodic point of u(X) defined by u◦n(X) = u◦m(X) for natural numbers m, n
with n≥m. For m= 1, the point Xis called periodic point of period n. For m=1=n, the
point Xis called fixed point of u.
A nice description of preperiodic points of a dynamical system is given in 14 .
Proposition 4.1. If u(X)∈Zp[[X]]d, X = (x1,··· , xd)is an invertible power series, then the
preperiodic points coincide with the periodic points.
Proof. For any preperiodic point α∈¯
Qpof u(X), ∃m, n ∈N,(m > n) such that u◦n(α) =
u◦m(α). Note that u◦(−n)(X) exists because u(X) is invertible, and hence u◦(m−n)(α) = α.
It should be noted that if m∈Zand m=prnwith p∤n, then the fixed points of u◦mand
u◦prare same. We denote Preper(D) = {Preper(u)|u∈ D}, where
Preper(u) = [
n
{X∈¯
Qd
p|u◦n(X) = X}, X = (x1,··· , xd).
The p-adic additive valuation vp(.) on Qpcan be generalised to affine d-space Qd
pby
vp(X) = vp((x1,··· , xd)) = min{vp(x1),··· , vp(xd)},(6)
where X∈Qd
p. This follows from sup-norm and the fact that |x|p=p−vp(x).
10 M. A. SARKAR AND A. A. SHAIKH
Let f, g ∈Zp[[X]]d, X = (x1,··· , xd), then denote
Λ(f) = [
n
{X∈¯
Qd
p|f◦n(X) = X}.
In the work 1 , the authors have given a definition of Newton coploygon for p-adic power series
in 2-variables.
Definition 4.4. 1, Definition 3.1 If f(x1, x2) = Pi,j≥0aij xi
1xj
2∈Zp[[x1, x2]] is a p-adic power
series, then its Newton copolygon in Euclidean 3-space coordinatized by (ξ1, ξ2, η) is given by
the boundary of the valuation function Vf(ξ1, ξ2), where
Vf(ξ1, ξ2) = min{iξ1+jξ2+v(aij )}.
v(.) is the additive p-adic valuation.
Proposition 4.2. For any θ= (θ1, θ2)∈¯
Q2
p, we have v(f(θ)) ≥Vf(v(θ1), v(θ2)).
Proof. The proof is an immediate consequence of 1, Proposition 4.3 . The inequality is strict
only when v(θ1) and v(θ2) are simultaneously the ξ1-coordinate and ξ2-coordinate, respectively.
In the case of 1-dimensional p-adic dynamical systems defined by Lubin 10 , we know two
useful propositions 10, Prop. 2.1 and 10, Prop. 3.2 . The following two statements are gener-
alisations in 2-dimensional p-adic dynamical systems.
Proposition 4.3. Let f, g ∈Zp[[X]]2, X = (x1, x2)be two noninvertible power series without
constant coefficient such that f◦g=g◦f, then Λ(f) = Λ(g).
Proof. So suppose, f◦g=g◦f. We need to show if α= (α1, α2)∈Λ(g) be a root of g, then
it is a root an “iterate” of f:
If αis a root of g, then f(α) is a also root of g. We have to show the sequence {v(f◦i)(α)}
becomes eventually infinite. Here we have f(x1, x2) = (f1(x1, x2), f2(x1, x2)) ∈Zp[[x1, x2]]2
with each fi(x1, x2)∈Zp[[x1, x2]], i = 1,2. Let f1(x1, x2) = Pi,j cij xi
1xj
2∈Zp[[x1, x2]] and
f2(x1, x2) = Pi,j dij xi
1xj
2∈Zp[[x1, x2]] with valuation functions respectively Vf1and Vf2then
by Proposition 4.2, we have
v(f1(α1, α2)) ≥Vf1(v(α1), v(α2)) (7)
v(f2(α1, α2)) ≥Vf2(v(α1), v(α2)),(8)
UNILIKELY INTERSECTION IN FORMAL GROUPS AND p-ADIC DYNAMICAL SYSTEMS 11
where v(.) is the p-adic additive valuation. By definition of Newton copolygon and by law (6),
Vf1(v(α1), v(α2)) > v(α) = min{v(α1), v(α2)}(9)
Vf2(v(α1), v(α2)) > v(α) = min{v(α1), v(α2)}.(10)
Finally, f(α) = (f1(α), f2(α)) tells us
v(f(α)) = min{v(f1(α)), v(f2(α))}(by law (6))
≥min{Vf1(v(α1), v(α2)), Vf2(v(α1), v(α2))}(by relations (7) and (8))
>min{v(α), v(α)}(by relations (9) and (10))
In the case, we have v(f(α)) > v(α), which by applying fboth sides gives v(f◦2(α)) >
v(f(α)) > v(α) = min{v(α1), v(α2)}. Continuing in this way, we get an increasing sequence of
p-adic numbers
···> v◦n(f(α)) >··· > v◦2(f(α)) > v(f(α)) > v(α).
For sufficiently large n,f◦i(α) = 0.This completes the proof.
Proposition 4.4. Let u, f ∈Zp[[X]]2, X = (x1, x2)be, respectively, invertible and noninvert-
ible power series without constant term such that u◦f=f◦u. Assume that uand its iterates
has finitely many fixed points. Then Preper(u) = Λ(f).
Proof. Take any zero αof fi.e., α∈Λ(f). Then f(u(α)) = u(f(α)) implies upermutes the
zeros of f. Thus Λ(f)⊂Preper(u).
Next let βbe any fixed point of ui.e., β∈Preper(u). Since u(f(β)) = f(u(β)) = f(β), f(β)
is also a fixed point of u. In other words, finduces a mapping from the set of fixed points
of uinto itself. By using the same argument of Newton copolygon in Proposition 4.3, we get
v(f(β)) > v(β). Applying the p-adic valuation von v(f(β)) > v(β) repeatedly we get the
following sequence of p-adic numbers
···> v◦n(f(β)) >··· > v◦2(f(β)) > v(f(β)) > v(β).
For nsufficiently large, f(β) = 0 i.e., β∈Λ(f). Therefore Preper(u) = Λ(f).
We have a corollary.
Corollary 4.4.1. Let u, g ∈Zp[[X]]2, X = (x1, x2)be two 2-dimensional stable series with u
invertible and fnoninvertible endomorphism of F, then Preper(u) = Λ(f)
12 M. A. SARKAR AND A. A. SHAIKH
Proof. Since uis an endomorphism of the 2-dimensional formal group F, it has only finitely
many fixed points. Then the corollary follows from Proposition 4.3.
Remark 5.The main purpose of proving Proposition 4.3 and Proposition 4.4 is just to observe
that the preperiodic points Preper(D) is indepenedent of the choice of power series in D. So
when we talk about Preper(D), we need not worry about whether we mean preperiodic points
of an invertible series or the zeros of a noninvertible series.
Based on Lubin’s results 10, Prop. 2.1 and 10, Prop. 3.2 for 1-variable p-adic power series,
as well as the above two Propositions 4.3 and 4.4 for 2-tuple power series in 2-variables, we
believe that the same is true for all d-tuple power series in d-variables. However, we lack
the necessary tools to prove the assertion. The tool Newton copolygon, which we utilise in
Propositions 4.3 and 4.4, becomes significantly difficult to control for p-adic power series with
more than two variables. So, we make the two following conjectures.
Conjecture 4.1. Let f, g ∈Zp[[X]]d, X = (x1,··· , xd), d ≥1 be two noninvertible power
series without constant coefficient such that f◦g=g◦f, then Λ(f) = Λ(g).
Conjecture 4.2. Let u, f ∈Zp[[X]]d, X = (x1,··· , xd), d ≥1 be, respectively, invertible and
noninvertible power series without constant term such that u◦f=f◦u. Assume that uand
its iterates has finitely many fixed points. Then Preper(u) = Λ(f).
Definition 4.5. Let Fbe a d-dimensional Honda formal group such that F∼
=F11 ⊕ · · · ⊕ F1d,
where F11,··· , F1dare 1-dimensional Lubin-Tate formal groups. We call Fto be unique if
Hom(F1i, F1j) = 0 = Hom(F1j, F1i), i 6=j, i, j = 1,2,3,··· , d.
Theorem 4.5. 6 If Fis a d-dimensional Honda formal group, then over some maximal
unramified extension of Qp,Fis isomorphic to the direct sum of d1-dimensional Lubin-Tate
formal groups F1, F2,··· , Fdi.e.,
F∼
=F1⊕F2⊕ · · · ⊕ Fd.
We now demonstrate the desired outcomes of this section.
Theorem 4.6. If D1and D2be d-dimensional stable p-adic dynamical systems coming from two
(unique) d-dimensional Honda formal groups F1, F2, respectively, over some maximal unramified
extension Uof Qpsuch that Preper(D1)∩Preper(D2)is infinite, then D1and D2coincide.
UNILIKELY INTERSECTION IN FORMAL GROUPS AND p-ADIC DYNAMICAL SYSTEMS 13
Proof. Since Fi, i = 1,2, is d-dimensional Honda formal group over some maximal unramified
extension Uof Qp, by Theorem 4.5 it is isomorphic to direct sum of d1-dimensional Lubin-Tate
formal groups Fi1,··· , Fi2i.e., Fi∼
=Fi1⊕ · · · ⊕Fid . Since F1is unique Honda formal group, we
can express EndU(F1) in terms of matrices
α11 0··· 0
0α22 ··· 0
0 0 ··· αdd
,
where αdd ∈EndU(F1d), d = 1,2,3,···, acting naturally
α11 0··· 0
0α22 ··· 0
0 0 ··· αdd
β1
β2
.
.
.
βd
=
α11(β1)
α22(β2)
.
.
.
αdd(βd)
,
for βj∈EndU(F1j), j = 1,2,··· , d. In other words, we can write parallelly
EndU(F1) = EndU(F11)⊕ · · · + EndU(F1d)
⇒ D1=D11 ⊕ · · · ⊕ D1d,(11)
where D11,··· ,D1dare 1-dimensional stable p-adic dynamical systems coming from 1-dimensional
Lubin-Tate formal groups F11,··· , F1d, respectively. Similarly, from the decomposition F2∼
=
F21 ⊕ · · · ⊕ F2d, we obtain
D2=D21 ⊕ · · · ⊕ D2d.(12)
Since Preper(D1)∩Preper(D2) =infinite, we conclude
Preper(D11)∩Preper(D21 ) = infinite
··· ···
Preper(D1d)∩Preper(D2d) = infinite,
which, taking into account Theorem 1.2, implies D11 =D21,··· ,D1d=D2d. Thus by equations
(11) and (12), we have D1=D2.
The following result is more general than the Theorem 4.6. It holds true for any d-dimensional
stable p-adic dynamical system derived from a d-dimensional formal group.
14 M. A. SARKAR AND A. A. SHAIKH
Theorem 4.7. If D1and D2are two d-dimensional stable p-adic dynamical systems com-
ing from two d-dimensional formal groups F1and F2, respectively, such that Preper(D1)∩
Preper(D2)=infinite, then D1and D2coincide.
Proof. Since D1and D2are p-adic dynamical systems coming from the formal groups F1and
F2, respectively, by Example 4.2, we conclude
D1⊂EndZp(F1) and D2⊂EndZp(F2).
The Proposition 4.2 and Corollary 4.4.1 says
Preper(D1)⊂Tors(F1) and Preper(D2)⊂Tors(F2).
Thus Preper(D1)∩Preper(D2)=infinite implies Tors(F1)∩Tors(F2)=infinite. Then by Theorem
3.5, we have F1=F2which in turns implies D1=D2.
5. Questions
In this part, we propose some questions that remains to answer.
Question 1
Assume D1and D2are arbitrary d-dimensional stable p-adic dynamical systems such that
Preper(D)∩Preper(D2) is infinite. Is D1=D2?
Proposition 5.1. If Zis an infinite subset of ¯
Qd
psuch that Z=Z1× · · · × Zd, where Zi⊂¯
Qp
is infinite, i= 1,2,··· , d, then Zis Zariski dense in ¯
Qd
p.
Proof. Let ¯
Qpbe equipped with Zarski topology. Given that Zis of the form Z1× · · · × Zd,
where Z1,··· ,Zdare infinite subset of ¯
Qp. All Z1,··· ,Zdare also Zariski dense in ¯
Qpbecause
they are infinite. However, we can not say if Zis Zariski dense because the Zariski topology
on a product is not the product topology.
We use the induction principle to prove Zis Zariski dense.
(1) For d= 1, it is straight forward because Z1is infinite subset of ¯
Qp, and hence Zariski
dense.
(2) For d= 2, we need to show that every nonzero polynomial α(x1, x2)∈¯
Qp[x1, x2]
doesn’t vanish on Z. Since α(x1, x2) is non-zero polynomial, we can write α(x1, x2) =
Pαn(x1)xn
2∈¯
Qp[x1][x2] such that at least one non-zero αn(x1)∈¯
Qp. Now αn(x1) has
only finitely many zeros in Z1but since Z1is infinite, there is some z1∈ Z1such that
αn(z1)6= 0. Then α(z1, x2)∈¯
Qp[x2] is non-zero polynomial, which has finitely many
UNILIKELY INTERSECTION IN FORMAL GROUPS AND p-ADIC DYNAMICAL SYSTEMS 15
zeros in Z2. Since Z2is infinite, there is some z2∈ Z2such that α(z1, z2)6= 0. Thus
Z=Z1× Z2is Zariski dense.
(3) Assume the claim is true for d=ki.e., Z=Z1× · · · × Zkis Zariski dense. We want to
show Z=Z1× · · · × Zk× Zk+1 is Zariski dense.
Since Z=Z1× · · · × Zkis Zariski dense, for any nonzero polynomial f(x)∈
¯
Qp[x1,··· , xk] there is some point z∈Qk
i=1 Zisuch that f(z)6= 0. So by the previous
argument, any polynomial f(x)∈¯
Qp[x1,··· , xk, xk+1] that vanishes on the product
Z=Z1× · · · × Zd× Zk+1 must be zero polynomial, and hence Zis Zariski dense for
d=k+ 1.
The conclusion follows from induction.
Following the breakthrough work of Faltings 4 , several conjectures have been made regarding
the problems of when rational points of a variety over a number field are (potentially) Zariski
dense.
Let’s give Zariski topology to the affine d-space ¯
Qd
p. It follows that Tors(F) is an infinite
subset of ¯
Qd
pif Fis a formal group of ddimensions and finite height. Assuming that Tors(F)
is a product of dinfinite sets in this instance, Proposition 5.1 states that it is Zariski dense.
The set Preper(D) exhibits the same behavior. However, this is a strong assumption. Thus,
the following query need to be addressed:
Question 2
When is Tors(F) Zariski dense in ¯
Qd
p? When is Preper(D) Zariski dense in ¯
Qd
p?
Question 3
Assume u(X), f(X) are respectively invertible and noninvertible series in Zp[[X]]d, X =
(x1,··· , xd) with u(0) = 0 = f(0) such that u◦f=f◦u. Is there a d-dimensional formal
group somehow in the background making this possible ?
Readers should be aware that the 1-dimensional form of the above question was origi-
nally conjectured by Lubin in 10 and later by Berger in 2 . The same is stated again in
13, Conjecture 3.1 , which is proved by Berger in 2, Theorem. B under some conditions. Li
demonstrated several results supporting the conjecture in ( 7 , 8 ) and 9 .
Acknowledgement: The authors are deeply grateful to Professor Jonathan Lubin, who
taught us this subject and offered guidance in preparing this article. We thank Steven J. Miller
for inspiring us to generalise the prior version of this study. The first author is grateful to
CSIR, Govt. of India, for the grant with File no.-09/025(0249)/2018-EMR-I.
16 M. A. SARKAR AND A. A. SHAIKH
References
1 R. Abdellatif, M. A. Sarkar, 2-dimensional Lubin-Tate formal group and torsion points, preprint (2023)
available upon request.
2 L. Berger, Nonarchimedian dynamical systems and formal groups, Proceedings of the AMS, 147 (4) (2019)
1413-1419.
3 L. Berger, Rigidity and unlikely intersection for formal groups, Acta Arithmatica, 195 (3) (2020) 305-319.
4 G. Faltings, Finiteness theorems for abelian varieties over number fields, Inventiones Mathematicae, 73 (3)
(1983) 349-366.
5 M. Hazewinkel, Formal Groups and Applications, Academic Press. Inc. New York, 1978.
6 T. Honda. On the theory of commutative formal groups. Journal of the Mathematical Society of Japan,
22(2) (1970) 213-246.
7 H.-C. Li, Counting periodic points of p-adic power series, Compositio Math. 100 (1996), 351-364.
8 H.-C. Li, p-adic Dynamical systems and formal groups, Compositio Math. 104 (1996), 41-54.
9 H.-C. Li, p-typical Dynamical Systems and Formal Groups, Compositio Mathematica 130 (2002), 75-88.
10 J. Lubin, Nonarchimedean dynamical systems, Composita Math., 94 (3) (1994) 321-346.
11 J. Lubin and J. Tate, Formal complex multiplication in local fields, Ann. of Math. 81(1965), 380-387.
12 C. L. Matson, Multi-Dimensional Formal Group Laws with Complex Multiplication, PhD Thesis, Univer-
sity of Colorado, 2020.
13 M. A. Sarkar, A. A. Shaikh, Rigidity and unlikely intersections in stable p-adic dynamical system, p-adic
Numbers, Ultrametric Ananlysis and Applications, 14 (2) (2022), 157-163.
14 J. H. Silverman, The arithmetic of dynamical systems, Springer, New York (2007).
Department of Mathematics, The University of Burdwan, Burdwan-713101, India.
Email address:∗mabudji@gmail.com
Department of Mathematics,, The University of Burdwan,, Burdwan-713101, India.
Email address:†aashaikh@math.buruniv.ac.in
