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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

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(1983) 349-366.

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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.

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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