Elliptic curves and p-adic linear independence

Let E be an elliptic curve defined over a number field and L the field of endomorphisms of E. We prove a result on p-adic elliptic linear independence over L which concerns algebraic points of the elliptic curve E.

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ELLIPTIC CURVES AND p-ADIC LINEAR INDEPENDENCE

Pham Duc Hiep

Faculty of Mathematics, Hanoi National University of Education

Abstract.Let E be an elliptic curve defined over a number field and L the field

of endomorphisms of E We prove a result on p-adic elliptic linear independence

over L which concerns algebraic points of the elliptic curve E.

Keywords: Elliptic curves, linear independence, p-adic.

The problem of finding roots of a given polynomial is always a natural and big question in mathematics It is well-known that every polynomial with complex coefficients of positive degree has all roots in the complex fieldC and in particular, so does every polynomial with rational coefficients of positive degree

Dually, to study the arithemetic of complex numbers, that is given α ∈ C, one may naturally ask whether there is a non-zero polynomial P in one variable with rational coefficients such that P (α) = 0?

If there exists such a P we call α algebraic, otherwise we call α (complex) transcendental The most prominent examples of transcendental numbers are e (proved

by C Hermite in 1873) and π (proved by F Lindemann in 1882) Apart from the complex

fieldC, there is another important field, the so-called (complex) p-adic number field (first described by K Hensel in 1897) for each prime number p Namely, it is a p-adic analogue

ofC which is denoted by Cp Note that by construction,Cpis an algebraically closed field containingQ, therefore one can analogously give the definition of p-adic transcendental

numbers as follows

An element α ∈ C p is called (p-adic) transcendental if P (α) ̸= 0 for any non-zero polynomial P (T ) ∈ Q[T ].

Transcendence theory in both domainsC and Cp has been studied and developed

by many authors In order to investigate the theory more deeply, one can naturally put the

problem in the context of linear independence For instance, if α is a number (inC or Cp)

Received August 12, 2014 Accepted September 11, 2014.

Contact Pham Duc Hiep, e-mail address: phamduchiepk6@gmail.com

such that 1 and α are linearly independent over Q, then α must be transcendental Indeed,

it follows from the trivial equality: α · 1 − 1 · α = 0 One of the most celebrated results

in this direction is due to A Baker Namely, in 1967 he proved the following theorem (see [1])

Theorem 1.1(A Baker) If α1, , α n are algebraic numbers, neither 0 nor 1, such that log α1, , log α n are linearly independent over Q, then 1, log α1, , log α n are linearly independent over Q.

J Coates extended the Baker’s method to the p-adic case in 1969 (see [5]) It is

natural to think of similar problems in the language of arithmetic algebraic geometry, in

particular, for the elliptic curves over number field Such a theory is called elliptic linear independence theory Note that the theory of elliptic curves plays a very important role,

not only in pure mathematics (e.g contribution to solve Fermat’s Last Theorem), but also

in real life (e.g in cryptography)

The aim of this paper is to formulate and prove a new result on elliptic linear

independence over p-adic fields which is given by the following theorem.

Theorem 1.2 Let E be an elliptic curve defined over a number field and let ℘ p be the p-adic Weierstrass function of E Denote by A p the set of algebraic points of ℘ p Then, elements u1, , u n ∈ A p are linearly independent over Q if and only if u1, , u n are linearly independent over the field of endomorphisms of E.

2 The arithemtic of elliptic curves

In this section, we briefly recall the theory of elliptic curves (see [9] for detailed theory) Let Λ be a lattice in C, i.e Λ is a set of the form Λ = {mα + nβ; m, n ∈ Z} where α, β ∈ C such that α, β are linearly independent over R The Weierstrass ℘-function

(relative to Λ) is defined by the series

℘(z) := ℘(z; Λ) := 1

w ∈Λ\{0}

( 1

(z − w)2 − 1

w2

)

.

The function ℘ is meromorphic on C, analytic on C \ Λ and periodic with period w ∈ Λ.

We also call Λ the lattice of periods of ℘ Furthermore one has

(℘ ′ (z))2 = 4(℘(z))3− g2℘(z) − g3

with

g2 := g2(Λ) = 60G4(Λ), g3 := g3(Λ) = 140G6(Λ), where the Eisenstein series of weight 2k (relative to Λ) are defined by

G 2k(Λ) := ∑

ω ∈Λ\{0}

ω −2k , ∀k ≥ 1.

The quantities g2, g3 are said to be the invariants of ℘ The Laurent expansion of ℘ at 0 is

given by

℘(z) = 1

z2 +

∞

∑

n=1

b n z 2n

where

b1 = g2

20, b2 =

g3

28, b n=

3

(2n + 3)(n − 2)

n −2

∑

k=1

b k b n −k−1 , ∀n ≥ 3.

By induction we see that for n ≥ 1 there are polynomials of two variables P n (X, Y ) with

coefficients inQ such that b n = P n (g2, g3) In particular, if g2, g3 ∈ Q then b n ∈ Q for

n ≥ 1.

Definition 2.1 Let K be a field We call E an elliptic curve defined over K if E is

a projective algebraic group of dimension 1 defined over K, i.e an abelian variety of dimension 1 defined over K.

When K is a subfield of C, one can characterize E as a smooth projective curve in

the projective spaceP2

K , namely it is defined by an equation of the form Y2Z = 4X3 − aXZ2 − bZ3 with a, b ∈ K such that a3 ̸= 27b2 In addition, there is a unique lattice

Λ in C satisfying g2(Λ) = a and g3(Λ) = b The Lie algebra of E(C) is canonically isomorphic toC, and the exponential map expE is given by

expE :C → E(C), z 7→ (℘(z) : ℘ ′ (z) : 1).

One can show that Λ = ker expE ={ω ∈ C; ℘(z + w) = ℘(z)} = Zω1+Zω2 with

ω i :=

∫ ∞

e i

dx

√ 4x3− ax − b (i = 1, 2) (called fundamental periods), where e1, e2, e3are the roots of the polynomial 4x3−ax−b.

The map

ϕ : C/Λ → E(C), z 7→ (℘(z) : ℘ ′ (z) : 1)

induced by expE is a complex analytic isomorphism of complex Lie groups We also say that the Weierstrass elliptic function relative to the lattice Λ is the Weierstrass elliptic

function associated with E Let End(E) denote the ring of endomorphisms of E, and define the field of endomorphisms of E as the quotient field L := End(E) ⊗Z Q The map

[ ] :Z → End(E), n 7→ [n]

where [n] : E → E is the multiplication by n, is injective If End(E) is strictly larger

thanZ then we say that E has complex multiplication (this is the so-called CM case or

CM type) In this case the quotient τ := ω1/ω2 is a quadratic number, and the field L is Q(τ).

In the p-adic domain, it is known that there is a p-adic analogue of elliptic function which was constructed by E Lutz and A Weil (see [7] and [10]) Let E be an elliptic

curve defined overCp of the form

Y2T − 4X3− g2XT2− g3T3 = 0, where g2, g3 ∈ C p such that g23− 27g2

3 ̸= 0 The following differential equation

y ′ (z) =

(

1− g2

4y

4(z) − g3

4y

6(z)

)1/2

, y(0) = 0 admits the solutions φ(z) and −φ(z) which are analytic on the disk

Dp :={

z ∈ C p;|1/4| pmax{|g2| 1/4

p , |g3| 1/6

p }z ∈ B(r p)}

, here B(r p) := {x ∈ C p;|x| p < p − p−11 } The disk D p is called the p-adic domain of E Put ℘ p := φ −2 , we get ℘ ′2 p = 4℘3p − g2℘ p − g3 This leads to the following definition.

Definition 2.2. We call ℘ p the (Lutz-Weil) p-adic elliptic function associated with the elliptic curve E.

The function ℘ p (z) is analytic on Dp \ {0}, and can be represented by the p-adic

power series

℘ p (z) = 1

z2 +

∞

∑

n=1

P n (g2, g3)z 2n with the polynomials P ngiven as in the complex case above

We are now interested in the linear independence of elliptic functions The elliptic analogue of Baker’s theorem on the linear independence of logarithms was first proved by Masser in 1974 (see [8]) in the CM case Masser and Bertrand in 1980 completed this for the non-CM case (see [4]) To describe the theorems below, we put

A := Λ ∪ {u ∈ C \ Λ; ℘(u) ∈ Q};

recall that L denotes the field End(E) ⊗ZQ

Theorem 3.1 (Bertrand-Masser) If elements u1, , u n ∈ A are linearly independent over L, then 1, u1, , u n are linearly independent over Q.

In the p-adic domain, Bertrand in 1976 formulated and proved a p-adic analogue for

elliptic curves with complex multiplication (see [3]) which deals with the homogeneous

case Similar to the complex case, we denote the set of algebraic points of the p-adic Weierstrass function ℘ pby

A p :={0} ∪ {u ∈ D p \ {0}; ℘ p (u) ∈ Q}.

Theorem 3.2 (Bertrand) Assume that E has complex multiplication If elements

u1, , u n ∈ A p are linearly independent over L, then u1, , u n are linearly independent over Q.

We generalize Theorem 3.2 to arbitrary elliptic curves which is given by the following theorem

Theorem 3.3 Elements u1, , u n ∈ A p are linearly independent over Q if and only if they are linearly independent over L.

Proof The implication ” ⇒” is trivially true because L is a subfield of Q We prove the

implication “⇐” Suppose that u1, , u nare linearly dependent overQ This means that

there is a non-zero linear form in n variables

l(X1, , X n ) = a1X1+· · · + a n X n

with a1, , a n ∈ Q such that

l(u1, , u n ) = 0.

Let G = E n be the direct product of n-copies of the elliptic curve E Then G is

commutative and defined overQ The Lie algebra Lie(G) is identified with Q n

, hence

Lie(G(Cp )) = Lie(G) ⊗Q Cp =Cn

p

We have G(Cp)f = E(Cp)n , and the p-adic logarithm map of G is given by

logG(Cp) :Cn

p → Lie(G(C p )), (z1, , z n)7→ (log E(Cp)(z1), , log E(Cp)(z n )) Since ℘ p (u1), , ℘ p (u n) are inQ, the point

γ := (exp E(Cp)(u1), , exp E(Cp)(u n))

is an algebraic point of G(Cp)f The value of logG(Cp)(γ) is

(

logE(Cp)(

expE(Cp)(u1))

, , log E(Cp)(

expE(Cp)(u n)))

= (u1, , u n)̸= 0 since u1, , u n are linearly independent over L Let

V := {v ∈ Q n ; l(v) = 0 }

be theQ-vector space defined by l We find that log G(Cp)(γ) is a non-zero point in V ⊗QCp

The p-adic analytic subgroup theorem (see [6]) says that there is an algebraic subgroup

H of G of positive dimension defined over Q such that γ ∈ H(Q) and Lie(H) ⊆ V It

is known that H is isogeneous to E m for some positive integer m ≤ n Using the same argument as in the proof of Theorem 6.2 in [2], there is an element π of the algebra of

endomorphisms

EndQ(G) := End(G) ⊗ZQ

of G given by the projection from G to H One can identify elements of EndQ(G) to be square matrices of order n × n with coefficients in

End(E) ⊗QZ = L.

This means that the endomorphism φ := id G − π can be written as an n × n matrix A with entries in L On the other hand H is a proper subgroup of G since

dimQLie(H) ≤ dimQV = n − 1 < n = dimQLie(G).

Then the matrix A is non-zero The subgroup H is isogeneous to E m, and this shows that

the Lie algebra of H can be identified with that of E m The Lie algebra of H then is the kernel of the endomorphism of Lie(G) given by the matrix A Further since γ ∈ H one

gets

(u1, , u n) = logG(Cp)(γ) ∈ Lie(H) ⊗QCp

This means that

A(u1, , u n ) = 0.

Since A is non-zero this provides a non-trivial linear dependence of the elements

u1, , u n over L This contradiction proves the theorem.

REFERENCES

[1] A Baker, 1966 Linear forms in the logarithms of algebraic numbers I, II, III, IV,

Mathematika 13, 204-216; 1967, Mathematika, 14, 102-107; 1967, Mathematika 14, 220-228; 1968, Mathematika 15, 204-216.

[2] A Baker and G W¨ustholz, 2007 Logarithmic forms and Diophantine geometry,

New Mathematics Monographs, Vol 9, Cambridge University Press

[3] D Bertrand, 1976 Problèmes arithmétiques liés à l’exponentielle p-adique sur les

courbes elliptiques, C.R Acad Sci Paris Sér A-B 282, No 24, Ai, A1399-A1401.

[4] D Bertrand and D Masser, 1980 Linear forms in elliptic integrals Invent Math.

58, No 3, 283-288.

[5] J Coates, 1969 The effective solution of some diophantine equations PhD

Dissertation, University of Cambridge

[6] C Fuchs and D H Pham, 2014 The p-adic analytic subgroup theorem revisited,

preprint

[7] E Lutz, 1937 Sur l’équation Y2 = AX3 − AX − B dans les corps p-adiques J.

reine angew Math 177, 238-247.

[8] D Masser, 1975 Elliptic functions and transcendence Lecture notes in

Mathematics 437 Springer-Verlag, Berlin.

[9] J H Silverman, 1986 The arithmetic of elliptic curves Springer-Verlag, New York [10] A Weil, 1936 Sur les fonctions elliptiques p-adiques Note aux C.R Acad Sc Paris

203, 22-24.