LINEAR FORMS IN TWO ELLIPTIC LOGARITHMS IN THE p ... - J-Stage

Kyushu J. Math. 64 (2010), 239–260 doi:10.2206/kyushujm.64.239

LINEAR FORMS IN TWO ELLIPTIC LOGARITHMS IN THE p-ADIC CASE Noriko HIRATA-KOHNO and Rina TAKADA (Received 23 December 2009 and revised 11 April 2010)

Abstract. The principal aim of the present paper is to provide a new lower bound for linear forms in two p-adic elliptic logarithms. We refine a result due to Rémond and Urfels. Our improvement lies in the dependence on the height of algebraic coefficients of the linear forms. Our bound is the best possible one concerning the height of the coefficients, where all the relevant constants are explicit. We use the argument that relies on the interpolation method of Laurent, on the variable change introduced by Chudnovsky, and on Faà di Bruno’s formula adapted to matrices whose elements are p-adic elliptic logarithmic functions. The bounds would be useful to determine the set of S-integer points on elliptic curves defined over a number field.

1. Introduction Let K be a number field of finite degree D over Q, whose ring of integers is denoted by O = OK . Let E be an elliptic curve defined over K, equipped with a Weierstraß model: Y 2 = X3 − AX − B

(A, B ∈ K, := 4A3 − 27B 2 = 0).

We may assume that A, B ∈ O; indeed, if either A or B does not lie in O, then there exists a suitable c ∈ O such that the elliptic curve E : Y 2 = X3 − A X − B with A = c4 A ∈ O and B = c6 B ∈ O. The discriminant of E is = c12 , thus E is isomorphic to E since the j -invariant remains equal under these multiplications. For a given finite set S = {p1 , . . . , pk , ∞} of places in Q, we define the set of S-integers (of Q) to be ZS = {x ∈ Q : |x|v ≤ 1 ∀v ∈ S}. We are then concerned with the set of S-integer points in the elliptic curve E: E(ZS ) = {(X, Y, 1) ∈ P2 (Q) : |X|v ≤ 1, |Y |v ≤ 1 ∀v ∈ S}. Siegel proved in 1929 that the number of sets of integer points on E, that is, E(Z) = E(ZS ) with S = {∞}, is finite. Then in 1934, Mahler generalized the finiteness assertion to E(ZS ) for any finite S including all the infinite places. Their method was based on Diophantine approximations due to Thue, Siegel and Mahler, which were later refined by Roth and generalized to higher-dimensional theory by Schmidt. However, all of the results on finiteness were not effective, namely, there was no way to determine the set E(ZS ). With the transcendence theory due to Baker, new discoveries with effectivity were made. For example, 2000 Mathematics Subject Classification: Primary 11J61, 11J86, 11J89, 11K60. Keywords: linear forms in logarithms; elliptic logarithm; S-integer points; p-adic logarithm. c 2010 Faculty of Mathematics, Kyushu University

240

N. Hirata-Kohno and R. Takada

in 1970, Coates showed that all the points in E(ZS ) are effectively computable. Baker’s theory on linear forms in ordinary logarithms endorses that the finitely many solutions to S-unit equations are all effectively (not practically) computable. Historical surveys of these developments can be found in [3, 11, 52] for instance. An alternative approach to the finiteness assertion of the points in E(Z) was proposed by Lang [29, 30] and investigated by Masser [35] and Zagier [55]. That started the applications of the theory of linear forms in elliptic logarithms. The exponential function is now replaced by the Weierstraß elliptic function ℘, since it can be viewed as the exponential map of the elliptic curve. With this, the quantitative theory of linear forms serves as a bridge between the transcendence theory and the determination of all the points in E(Z), under a natural assumption that a basis of the Mordell–Weil group E(Q) be known. It has been revealed in part by works in computational number theory that the use of linear forms in elliptic logarithms is more direct and practical than the use of linear forms in ordinary logarithms via S-unit equations. The determination of the points in E(Z) is thus reduced to the problem of finding explicit lower bounds for linear forms in elliptic logarithms. Such an effective lower bound for linear forms in arbitrary n elliptic logarithms, without assuming complex multiplications of the elliptic curves, was first obtained by Philippon and Waldschmidt [38] in 1988, which was later improved by one of the present authors in 1991 [19]. In 1995, a totally explicit lower bound was first achieved by David [10] where the constant part of the latter [19] was completely calculated. The work of David made it possible to determine all the integer points on various elliptic curves (see e.g. [15, 46–48]). The dependence on the height of algebraic coefficients of the linear forms in n elliptic logarithms is refined as the best possible one by David and Hirata-Kohno in [13]. One naturally proceeds to the investigation on E(ZS ) with S an arbitrary finite set of places including all the infinite places. It is then crucial to establish explicit lower bounds for linear forms in p-adic elliptic logarithms. One should obviously try to obtain highly refined lower bounds, as a better numerical estimate of relevant constants certainly yields a substantial reduction in the computation time in practice. One finds in the work of Bertrand [6] an effective lower bound for linear forms in p-adic elliptic logarithms for elliptic curves with complex multiplications. Rémond and Urfels [39] showed an estimate for a linear combination of two p-adic elliptic logarithms without assuming complex multiplications. We shall refine one part of the last work by closely following their argument, while employing the variable change method of Chudnovsky [9], with the p-adic elliptic logarithmic function proposed by Lang [30]. Our contribution, if any, is in giving a way that one may use the variable change in conjunction with the interpolation matrix argument due to Laurent [31]. A straightforward proof by means of Faà di Bruno’s formula on derivatives of composed functions in several variables enables us to use the fact that the Taylor coefficients at the origin of a p-adic elliptic logarithmic function have better arithmetic properties than those of a p-adic elliptic function. In our main result, it concerns only two p-adic elliptic logarithms, and thus the problem is equivalent to measuring the distance between nonzero algebraic numbers and a quotient of two non-zero elliptic logarithms. We mention that explicit lower bounds for linear forms in p-adic logarithms, not elliptic, have been obtained by Yu [53, 54]. The organization of this paper is as follows. In Section 2, we state our main result (Theorem 2.1). In Section 3, we introduce our basic tool, the p-adic elliptic logarithmic

Elliptic logarithms in the p-adic case

241

function, and supply precise estimates related with its arithmetic behavior. In Section 4, we collect known results from previous works in the literature, especially [39], which make the preparation for later sections. In Section 5, we carry out the variable change to replace p-adic elliptic functions by p-adic elliptic logarithmic functions. Section 6 is devoted to estimates of the height of the determinant of the interpolation matrices, using the p-adic elliptic logarithmic function. In Section 7, the proof of our main theorem is completed. 2. Notation and results Let us denote by Q the algebraic closure of Q in C. Let K be a number field of finite degree D over Q. Denote O = OK the ring of integers in K. Let p be a rational prime, p ∈ Q, and | · |∞ be an Archimedean valuation on K. For a place v of K over p, we write the valuation | · |v normalized such that |x|v = p−ordp (x) for x ∈ Q. Denote by Kv the completion of K by v, and write Qp for the completion of Q by p.

The field Kv is a finite extension of Qp of local degree nv = [Kv : Qp ] with v|p nv = D. Let Cp be the completion of the algebraic closure of Kv . We note that the algebraic closure of Kv is not complete itself. It is well known that Cp is an algebraically closed complete field of characteristic zero, in which the algebraic closure of Kv is dense, and that there are D distinct embeddings of K into Cp . Denote again by | · |v the extension of | · |v on Cp . For x ∈ PN (Q) having coordinates x = (x0 , . . . , xN ) ∈ PN (K), recall the absolute logarithmic height of x defined by 1 h(x) = nv log(max{|x0 |v , . . . , |xN |v }), [K : Q] v where the sum runs over all the normalized places of K. This definition is independent of the choice of the projective coordinates and of the choice of the field containing x0 , . . . , xN . For a ∈ Q, we set h(a) := h(1 : a), the absolute logarithmic height of the algebraic number a. Eventually, we also write h(a) = h∞ (a) + hf (a), where the sum in h∞ (a) runs over all the infinite places and the sum in hf (a) runs over all the finite places: h∞ (a) = hf (a) =

1 [K : Q]

nv log(max{1, |a|v }),

v infinite

1 nv log(max{1, |a|v }). [K : Q] v finite

From now on, we fix a place v over p and denote | · | = | · |v throughout the paper. For a formal power series f (z) =

∞

ak zk ∈ Cp [[z]],

k=0

f (z) converges at z ∈ Cp if and only if |ak zk | → 0. As in the Archimedean case, the radius of convergence is given by Hadamard’s formula: (lim supk→∞ |ak |1/ k )−1 .

242


We recall the Lutz–Weil p-adic elliptic function, which corresponds to the p-adic version of the Weierstraß elliptic function ℘. Let E be an elliptic curve, E ⊂ P2 (Cp ), defined by ZY 2 = X3 − AXZ 2 − BZ 3

(A, B ∈ O, 4A3 = 27B 2 ).

Put λp = 1/(p − 1) if p = 2, and λ2 = 3. We set Cp := {z ∈ Cp : |z| < p−λp } and Cv := Cp ∩ Kv . After Lutz [33] and Weil [51] (see the revisit by Bertrand [6]), it is known that there exist two solutions ϕ and −ϕ to the differential equation (ϕ )2 = 1 − Aϕ 4 − Bϕ 6 with ϕ(0) = 0, defined over Cv → Kv , analytic in Cv . In fact putting ϕ 2 = 1/℘ϕ we have (℘ϕ /2)2 = ℘ϕ3 − A℘ϕ − B and ϕ (0) = 1. The function ϕ(z) is called the Lutz–Weil p-adic elliptic function. The elliptic curve can be given the structure of the p-adic Lie group E(Cp ) ⊂ P2 (Cp ), and the exponential map is expressed by the Lutz–Weil p-adic elliptic function as follows. We may enlarge the domain of the definition of the function ϕ to Cp . See the definition of the Lutz–Weil p-adic elliptic function on Cp in [4, p. 151] (see e.g. [5, 44]). Definition 2.1. For the p-adic Lie group E(Cp ) ⊂ P2 (Cp ) we have the exponential map exp = expE : Cp → E(Cp ) ⊂ P2 (Cp ), ϕ 3 (z) ϕ(z) 3 , −1, z → (ϕ(z), −ϕ (z), ϕ (z)) = . ϕ (z) ϕ (z) The expression leads our elliptic curve to be parameterized by (X, Y, Z) = (ϕ, −ϕ , ϕ 3 ). This p-adic exponential map is locally analytic only. The function ϕ is odd and injective; indeed, |ϕ(z)| = |z|, |ϕ (z)| = 1 for any z ∈ Cp , hence expE has no period [6]. There are corresponding addition formula and derivation formula, similar to those of ℘. We collect in Section 4 the properties shown in [5, 6, 30, 39]. Denote by exp(x, y) = (exp(x), exp(y)) = (expE (x), expE (y)) the exponential map on E(Cp ) × E(Cp ). Assumption 2.1. Let β ∈ K. Take u1 and u2 in Cp and assume that ϕ 2 (ui ),

ϕ (ui ) ∈ K ϕ

(i = 1, 2).

Put = βu1 − u2 . We have assumed that exp(ui ) ∈ E(K) (i = 1, 2). Here is a linear form in two p-adic elliptic logarithms u1 and u2 . We shall give a lower bound at the non-Archimedean valuation v of . We write ˆ ) := 1 lim 1 h(2n P ) h(P 2 n→∞ 4n the Néron–Tate height defined on E for a rational point P ∈ E(K). We may suppose that none of these three numbers β, u1 , u2 equals zero. (Otherwise our statement trivially follows by Liouville’s inequality: |α| ≥ e−[K:Q]h(α) for α ∈ K − {0}.)


243

Assumption 2.2. Denote non-negative real numbers h1 , h2 , h3 , ρ, E, a1 , a2 and d by ˆ hi = h(exp(u i )) (i = 1, 2),

h3 = max(1, h(β)),

ρ = p−λp ,

E = ρ/max(|u1 |, |u2 |), [K : Q] a1 = max(1, h1 ), a2 = max(1, h2 ), d = max 1, . log E We denote further by h = h4 = h(E) := max{1, h(1, A, B)} the height of the elliptic curve E. Finally put g = max(1, h4 , log(h1 ), log(h2 ), log(d)). Now we are ready to state our result. T HEOREM 2.1. Under the assumptions (2.1) and (2.2), if we have || ≤ exp(−1.16 × 1035 × a1 a2 h3 g 3 d 6 log E), then we obtain =0 and β = u2 /u1 is an algebraic number of degree at most two over Q with √ h(β) ≤ log(5.89 × 1017 × g 2 d 3 max(a1 , a1 a2 )). C OROLLARY 2.1. Whenever we have = 0, then we obtain || > exp(−1.16 × 1035 × a1 a2 h3 g 3 d 6 log E). We compare our result with that of Rémond and Urfels. Put b = max(h3 , h4 , h1 , h2 , d) and c = max(1, h4 , log b). The result in [39] shows that, if || ≤ exp(−5.7 × 1026 × a1 a2 bc3 d 6 log E), then =0 and β = u2 /u1 is an algebraic number of degree at most two over Q of height √ log(4.29 × 1014 × c2 d 3 max(a1 , a1 a2 )). Our bound does not contain log h3 and express in an explicit manner the part of h3 separately from other data. This is the best possible concerning the height h3 . However, our numerical constant is larger than that of the statement of [39]. 3. The p-adic elliptic logarithmic function We define the p-adic logarithmic function in the elliptic case as a reversed function of expE with an expression of the formal group over O, following [30, 45] (see also [11, 12]). Take a point P = (X, Y, 1) ∈ E(K). We let t = t (P ) = −X/Y,

ω(t) = −1/Y.

244


We have

P = (X, Y, 1) = (t, −1, ω(t)) =

ϕ(z) ϕ 3 (z) , −1, . ϕ (z) ϕ (z)

Putting ψ(z) = ϕ 2 (z), we have ψ(z) =

ω(t) t

ψ (z) ω(t) = 2 . 2 t

and

Let r be a positive real number. We put E(r) the set of points P in E(K) with |t (P )| ≤ p−r . We include the origin in E(r) by convention, and then E(r) is a subgroup of E(K). Denote by pr the set of elements t ∈ K with |t| ≤ p−r . The map P → t (P ) establishes a bijection between E(r) and pr (see [30, Theorem 3.2, Ch. III]). Moreover, there is a power series expansion of ω(t) in t where the coefficients are polynomials in A, B with coefficients in Z (see [30, Theorem 3.1, Ch. III]). This power series expansion is studied in [12]. Below we rewrite estimates obtained in Theorem 1.1† in [12], replacing the height of the j -invariant by the height of the elliptic curve h = h(E), for later convenience. L EMMA 3.1. Under the notation above, we have ω(t) = An t n , n≥3

where An ∈ Z[A, B] is homogeneous of degree n − 3 (of weights 4,6 on A, B) of form A3 = 1 and (n) aλ,μ Aλ B μ , An = 4λ+6μ=n−3, λ,μ≥0 (n) ∈ Z with where aλ,μ (n) |aλ,μ |∞ ≤

33 × 8n−3 n3 (λ + 1)3 (μ + 1)3

(n ≥ 3, λ ≥ 0, μ ≥ 0).

Moreover, we have h(An ) ≤ 3n + (n − 3)h. Proof. We estimate h(An ) in terms of h thanks to the differential equation and ϕ 2 (z) = ω(t)/t, being back to the proof of [12, Theorem 1.1]. Consider any place v satisfying |An |v > 1. The cardinality of such places is finite. By using the fact that #{λ, μ ∈ Z, λ, μ ≥ 0 : 4λ + 6μ = n − 3} ≤ (n + 3)/6, if v is an infinite place, we have (n) λ μ aλ,μ A B |An |v = v

4λ+6μ=n−3, λ,μ∈Z, λ,μ≥0

≤

(n)

|aλ,μ |v |A|λv|B|μ v

4λ+6μ=n−3, λ,μ∈Z, λ,μ≥0

†In the statement of Theorem 1.1 [12], the definition of β = −g3 /6 should be replaced by β = −g3 /4.


≤ 8n × ≤ 8n

245

|A|λv |B|μ v

4λ+6μ=n−3, λ,μ∈Z, λ,μ≥0

n+3 6

max{1, |A|v , |B|v }n−3

≤ e3n max{1, |A|v , |B|v }n−3 . (n) ∈ Z, we have If v is a finite place, by the fact that aλ,μ

|An |v ≤

(n)

max

4λ+6μ=n−3

|aλ,μ Aλ B μ |v ≤

max

4λ+6μ=n−3

|Aλ B μ |v ≤ max{1, |A|v , |B|v }n−3 .

Then we obtain the estimate of h(An ) by definition of the height.

2

Lemma 3.1 yields the estimate of the height of Taylor coefficients of the functions ψ(z) = ϕ 2 (z) = and

ω(t) = An t n−1 t n≥3

ψ (z) ω(t) = 2 = An t n−2 . 2 t n≥3

Since expE (z) =

(1)

(2)

−ϕ (z) 1 , , 1 = (t, −1, ω(t)), ϕ 2 (z) ϕ 3 (z)

the function z = z(t) corresponds to the logarithmic function that is introduced in [30] (see [12, 45]). By writing X, Y in terms of t and ω(t), the differential form (t) = dX/2Y is viewed as a formal power series in t, and we define as in [30, 45] the formal integral logE (t) = (t). With this formal integral we have d(1/ϕ 2 )z(t) dX (−2ϕ /ϕ 3 )z(t) = (t) = dt = z (t) dt = z(t), 2Y (−2ϕ /ϕ 3 )z(t) (−2ϕ /ϕ 3 )z(t) which is indeed the local reversed function around the origin, of the function t = t (P ) = −X/Y = ϕ(z)/ϕ (z). Definition 3.1. Put logE (t) = z(t). We call the function an elliptic p-adic logarithmic function associated to E. We rewrite the statement in [12] for convenience in the lemma below, by using h = h(E). L EMMA 3.2. The Taylor expansion of logE (t) is given by logE (t) = z(t) = Bn t n , n≥1

where B1 = 1, Bn = Cn /2n, Cn =

4λ+6μ=n−1, λ,μ≥0

(n) λ μ bλ,μ A B

(n) (n ≥ 1) with bλ,μ ∈Z

246


and (n) |bλ,μ |∞ ≤

(25 × 3 × 52 )n (n + 2)3 (λ + 1)3 (μ + 1)3

(n ≥ 1, λ ≥ 0, μ ≥ 0).

Concerning the height, we have h(Cn ) ≤ 9n + (n − 1)h. Moreover, the domain of convergence of logE (t) is {z ∈ Cp : |z| < 1}. Proof. We estimate h(Cn ) in terms of h, tracing the proof of [12, Theorem 1.2]. We notice that the upper bound in that theorem, (n) |∞ ≤ |bλ,μ

(25 × 3 × 52 )n (n + 2)2 (λ + 1)3 (μ + 1)3

(n ≥ 1, λ ≥ 0, μ ≥ 0),

should be replaced by the better estimate (n)

|bλ,μ |∞ ≤

(25 × 3 × 52 )n (n + 2)3 (λ + 1)3 (μ + 1)3

(n ≥ 1, λ ≥ 0, μ ≥ 0).

Consider any place v satisfying |Cn |v > 1. If v is an infinite place, we have (n) λ μ bλ,μ A B |Cn |v = v

4λ+6μ=n−1, λ,μ∈Z, λ,μ≥0

≤

(n)

|bλ,μ |v |A|λv |B|μ v

4λ+6μ=n−1, λ,μ∈Z, λ,μ≥0

≤ (25 × 3 × 52 )n

|A|λv |B|μ v

4λ+6μ=n−1, λ,μ∈Z, λ,μ≥0

≤ (25 × 3 × 52 )n n max{1, |A|v , |B|v }n−1 ≤ e9n max{1, |A|v , |B|v }n−1 . (n) For a finite place v, since bλ,μ ∈ Z, we have

|Cn |v ≤

max

4λ+6μ=n−1

(n) λ μ |bλ,μ A B |v ≤

max

4λ+6μ=n−1

|Aλ B μ |v ≤ max{1, |A|v , |B|v }n−1 .

Then we obtain the estimate of h(Cn ) by definition of the height. The radius of convergence is obtained by Hadamard’s formula. 2 4.

Preliminaries

Consider a point u = (0, u1 , u2 ) ∈ Cp × Cp2 and the hyperplane W defined by z0 = βz1 − z2 . To prove our theorem, with respect to the fixed non-Archimedean valuation | · | = | · |v , we note that there is no restriction to suppose |β| ≤ 1, otherwise we may consider (1/β)u2 − u1 instead of . We are going to look at (, u1 , u2 ). We choose as in [19] a basis of W : (β, 1, 0) and (−1, 0, 1). Put σ = (σ1 , σ2 ) ∈ Z2 , σ1 , σ2 ≥ 0, and a differential operator over C3p along W


(see [38]):

247

∂ ∂ σ1 ∂ ∂ σ2 Dzσ = β + ◦ − + . ∂z0 ∂z1 ∂z0 ∂z2

Introduce also a ‘divided differential operator’ along W as in [13]: Dzσ Dσ 1 ∂ σ1 ∂ ∂ σ2 ∂ σz := z = = + ◦ − + . β σ! σ1 !σ2 ! σ1 !σ2 ! ∂z0 ∂z1 ∂z0 ∂z2 Put τ = (τ0 , τ1 , τ2 ) ∈ Z3 , τ0 , τ1 , τ2 ≥ 0 and define, with ψ = ϕ 2 : fτ : Cp × Cp2 → Cp , (z0 , z1 , z2 ) → z0 τ0 ψ(z1 )τ1 ψ(z2 )τ2 . For T0 , T1 , T2 , S0 , S1 ≥ 0 which are parameters in Z with S0 ≥ 5, define a matrix M = (σz fτ (su))τ ;(σ,s) = (mτ,σ,s ),

(3)

where the lines are indexed by T = {τ ∈ Z3 | 0 ≤ τi ≤ Ti } and the columns by S = {(σ, s) = (σ1 , σ2 , s) ∈ Z3 | σ1 ≥ 0, σ2 ≥ 0, |σ | := σ1 + σ2 < S0 , 0 ≤ s ≤ S1 }. The number of lines is L := (T0 + 1)(T1 + 1)(T2 + 1). The elements of the matrix are ‘divided derivatives’ instead of the ordinary derivatives in [39]. Denote by H (F ; D1 , D2 , D3 ) (respectively H (F ; D1 , D2 )) the Hilbert–Samuel polynomial [22, 38] which is multi-homogeneous on a connected algebraic group F of Ga × E 2 (respectively a connected algebraic group F of E 2 ). We recall here useful estimates in [39]. By definition of ϕ, we have (see [6]) the following lemmas. L EMMA 4.1. The function ϕ is an injective odd function such that for any z ∈ Cp we have |ϕ(z)| = |z|,

|ϕ (z)| = 1.

L EMMA 4.2. For any (z, u) ∈ Cp2 we have T (z, u)ψ(z + u) = (ψ(z) − ψ(u))2 with T (z, u) = ψ(z) + ψ(u) − 12 ψ (z)ψ (u) − ψ(u)ψ(z)[Aψ(z) + Aψ(u) + 2Bψ(z)ψ(u)]. 2

Proof. This is the addition formula [39, Lemma 3.5]. The next lemma is useful to calculate the height of the rational points. L EMMA 4.3. Let P ∈ E(K) and z ∈ Cp . The following hold: ˆ are related by (1) the absolute logarithmic height h(·) and Néron–Tate height h(·)

(2) (3)

ˆ ) ≤ h(P ) + 3 h4 + h(P ) − 32 h4 − 6 log 2 ≤ h(P 4 ψ (z) h ψ(z), ,1 = h(exp(z)); 2ψ(z) ψ (z) 1 h ψ(z), , , 1 ≤ 2h(exp(z)); and 2ψ(z) ψ(z)

7 2

log 2;

248

(4)


ψ (suj ) 1 , , 1 ≤ 2s 2 hj + 3h4 + 12 log 2. h ψ(suj ), 2ψ(suj ) ψ(suj )

Proof. This is [39, Lemma 3.7] joint with a theorem of Zimmer [39, Lemma 3.1]). Now we write, for a connected algebraic subgroup 0 = F

2

⊂ E 2,

B(F ) = {y ∈ K, |y| ≤ 1 : ∃z ∈ Cp , z = 0 s.t. exp(z, yz) ∈ F }, U(F ) = {(x1 , x2 ) ∈ Cp2 : ∃s ∈ N s.t. exp(sx1 , sx2 ) ∈ F }. L EMMA 4.4. Suppose that there exists F , a connected algebraic subgroup of E 2 , of dimension one with β ∈ B(F ). Denote H (F ; T1 , T2 ) = 3T1T2 with a parameter . If we have 53 T1 h1 13 || < exp −[K : Q] 22 T2 + T2 h2 + h4 + log 2 , 4 2 2 then we obtain = 0. Proof. This is [39, Proposition 4.1].

2

L EMMA 4.5. Suppose that the rank of the matrix M defined in (3) is strictly less than L. Assume that the five parameters verify all of the following conditions: (S0 + 2)(S0 + 5)(S1 + 3) > 2916T0T1 T2 , (S0 + 2)(S0 + 5)(T1 + T2 ) > 324T0 T1 T2 , (S0 + 2)(S1 + 3) > 81 max{T1 , T2 }, (S0 + 2)(T1 + T2 ) > 27T1 T2 , S0 + 2 > 9T0 . Assume further that F is a connected algebraic subgroup of E 2 of dimension one, where β ∈ B(F ) and H (F ; T1 , T2 ) > 30 T1 T2 with 0 = 162T0/[(S0 + 2)(S1 + 3)]. Then we have = 0. Moreover, there exists a connected algebraic subgroup F ⊂ E 2 of dimension one such that 162T0T1 T2 and u ∈ U(F ). H (F ; T1 , T2 ) ≤ S0 + 2 Proof. This is [39, Proposition 5.1] (we note that the proof is unchanged if we employ the divided derivatives instead of the ordinary derivatives). 2 5.

Variable change

Now let us state arithmetic properties of the p-adic elliptic logarithmic function. We use the following lemma coming from prime number theory to estimate the finite part of the height of the denominator n. L EMMA 5.1. Let , σ be positive rational integers. Put E,σ = {j1 × · · · × j | j1 , . . . , j ∈ Z, j1 , . . . , j > 0, j1 + · · · + j = σ }. Let L0 , S be positive rational integers with S ≥ 319. Put also A = AL0 ,S = lcm{x | x ∈ E,σ , ≤ L0 , σ ≤ S}.


249

Then we have log A ≤ 2.3S + S log L0 ≤ 3.4S log(L0 + 1). 2

Proof. This is [13, Lemma 2.8].

The following two lemmas are the most important in this paper. First recall the fact essentially used in [19], which relies on Fel’dman’s idea: ⎧ ⎪ (li = 1 and li = 0), ⎪ ⎨β ∂ li ∂ li β (4) − [z0 ](0) = −1 (li = 0 and li = 1), ⎪ ∂z0 ∂z0 ⎪ ⎩0 (otherwise). We set σ = (σ1 , σ2 ), |σ | = σ1 + σ2 and write σ1 σ2 1 ∂ ∂ σt = ◦ . σ1 !σ2 ! ∂t1 ∂t2 L EMMA 5.2. Let L0 , L1 , S ≥ 1 be rational integers with S ≥ 319. Let β1 = β ∈ K and put β2 = −1. Let P (X0 , X1 , X2 ) ∈ K[X0 , X1 , X2 ] be a polynomial of degree at most L0 in X0 and of homogeneous degree at most Li in Xi = (X0,i , X1,i , X2,i ) for each 1 ≤ i ≤ 2. Put G(t) = P (β1 z(t1 ) + β2 z(t2 ), (t1 , −1, ω(t1 )), (t2 , −1, ω(t2 ))). Write γ = σt G(0) with |σ | = S. Then we have h(γ ) ≤ h(P ) + 6L0 + 3L1 + 3L2 + 18S + 2hS + 3.4S log(L0 + 1) + L0 max{1, h(β)}. Proof. Write P =

λ

λ

λ

λ

aλ X0 0 X1 1 X2 2 where λ stands for (λ0 , λ1 , λ2 ) with

λi = (λ0,i , λ1,i , λ2,i ) ∈ Z3 (1 ≤ i ≤ 2), λ0,i , λ1,i , λ2,i ≥ 0, We have γ=

λ0 ∈ Z, 0 ≤ λ0 ≤ L0 ,

0 ≤ λ0,i + λ1,i + λ2,i ≤ Li

(1 ≤ i ≤ 2).

σ1 σ2 ∂ ∂ 1 ◦ aλ (β1 z(t1 ) + β2 z(t2 ))λ0 σ1 !σ2 ! ∂t1 ∂t2 λ × ((ω(t1 ))λ0,1 (t1 )λ1,1 (−1)λ2,1 · (ω(t2 ))λ0,2 (t2 )λ1,2 (−1)λ2,k )(0, 0),

where z(ti )(1 ≤ i ≤ 2) is the p-adic elliptic logarithmic function defined above and |σ | = S. Write as before ω(ti ) =

∞ σi ≥3

σ

Aσi ti i ,

z(ti ) =

∞ σi ≥1

σ

Bσi ti i

for i = 1, 2.

Lemmas 3.1 and 3.2 show that Aσi is a homogeneous polynomial in A, B of degree at most σi − 3 with coefficients in Z of Archimedean absolute value at most 8σi . By writing Bσi = Cσi /2σi , we have that Cσi is a homogeneous polynomial in A, B of degree at most σi − 1 with coefficients in Z of bounded height.

250


Hence γ is indeed the coefficient of t1σ1 · t2σ2 of the following Taylor series corresponding to |σ | = σ1 + σ2 = S. By means of the fact (4), we obtain

aλ

λ

(j1 ,j2 )∈J

×

2

j1 j2 λ0 ! j1 j2 σ1 σ2 β · β2 Bσ1 t1 · Bσ2 t2 j1 !j2 ! 1 σ ≥1 σ ≥1 1

σ

i=1 σi ≥3

λ0,i

2

(ti )λ1,i (−1)λ2,i

Aσi ti i

with J = Jλ0 = {(j1 , j2 ) ∈ Z2 : j1 , j2 ≥ 0, j1 + j2 = λ0 }, λi = (λ0,i , λ1,i , λ2,i ),

0 ≤ λ0 ≤ L0 ,

0 ≤ λ0,i + λ1,i + λ2,i ≤ Li , 1 ≤ i ≤ 2.

For such (j1 , j2 ) ∈ J , we have #J ≤ 2L0 +1 , λ0 !/j1 !j2 ! ≤ 2L0 and j

j

h(β11 β22 ) ≤ L0 max{1, h(β)}. Now, by the proof of Lemma 3.1, we get the estimate of the height for the σi th Taylor coefficient of the λ0,i th power of ω(t) with λ0,i ≤ Li as follows: As1 · · · Asλ0,i ≤ 3(λ0,i + σi ) + 3σi + σi h (5) h s1 +···+sλ0,i =σi

for λ0,i ≤ Li , 1 ≤ i ≤ 2. Now consider the set E,σi = {s1 × · · · × s | s1 , . . . , s ∈ Z, s1 , . . . , s > 0, s1 + · · · + s = σi }. By Lemma 5.1, we know that the least common multiple AL0 ,S of all elements in {x : x ∈ E,σi , ≤ L0 , σi ≤ S} is less than exp[3.4(S log(L0 + 1)]. Then, we use this fact to estimate the finite place part of the height of the denominators in the p-adic elliptic logarithmic function. By using the estimate of Cσi in the proof of Lemma 3.2, concerning the height of the σi th Taylor coefficient of the ji th power of z(t) with ji ≤ L0 , we have h Bs1 · · · Bsji s1 +···+sji =σi

= hf

Cs1 · · · Csji

s1 +···+sji =σi

As we saw, we have hf s1 +···+sji =σi

s1 · · · sji

Cs1 · · · Csji s1 · · · sji

+ h∞

s1 +···+sji =σi

Cs1 · · · Csji s1 · · · sji

.

≤ hf (AL0 ,σi ) + max hf (Cs1 · · · Csji ) ≤ 3.4σi log(L0 + 1) + σi hf (1, A, B).


251

Moreover, for any infinite place v, by the proof of Lemma 3.2 we have Cs1 · · · Csji s ···s s1 +···+sji =σi

=

1

ji

p

A B

4p+6q=σi −ji

v

q

s1 +···+sji =σi

p1 +···+pji =p,q1 +···+qji =q

(sj ) ) bp(s11,q · · · bpjii,qji 1 s ···s 1

ji

v

≤ 23(ji +σi −1) (25 × 3 × 52 )σi (σi − 1) max{1, |A|v , |B|v }σi . Therefore, we get an upper bound of h∞

Cs1 · · · Csji

s1 +···+sji =σi

which yields h s1 +···+sji =σi

s1 · · · sji

Bs1 · · · Bsji

≤ 3(σi + ji ) + 9σi + σi h + 3.4σi log(L0 + 1),

(6)

for (j1 , j2 ) ∈ J, 1 ≤ i ≤ 2 (which is indeed a bound of the height of the σi th Taylor coefficient of the ji th power of z(t) with ji ≤ L0 ). Putting all this together we obtain the lemma. 2 L EMMA 5.3.† Let β1 , β2 ∈ Cp . Let P (X0 , X1 , X2 ) ∈ Cp [X0 , X1 , X2 ] be a polynomial in X0 = (X0 ), homogeneous in Xi = (X0,i , X1,i , X2,i ) for each 1 ≤ i ≤ 2. Let z = (z0 , z1 , z2 ) ∈ C3p . Let f1 , f2 , f3 , g1 , g2 , g3 be functions on Cp analytic around the origin, and write f (z1 ) = (f1 (z1 ), f2 (z1 ), f3 (z1 )) and g(z2 ) = (g1 (z2 ), g2 (z2 ), g3 (z2 )). Put 2 βj zj , (f (z1 )), (g(z2 )) , F (z) = F (z0 , z1 , z2 ) = P j =1

which is analytic at z = (z1 , z2 ) = 0. Put also 2 G(t) = P βj z(tj ), (f (z(t1 ))), (g(z(t2 ))) , j =1

which is also analytic at t = (t1 , t2 ) = 0. For σ1 , σ2 ∈ Z, σ1 , σ2 ≥ 0, put σ = (σ1 , σ2 ), |σ | = σ1 + σ2 . We define 1 ∂ σ2 ∂ σ1 ∂zσ F (z) = ◦ F (z) σ1 !σ2 ! ∂z1 ∂z2 and similarly

σ1 σ2 ∂ 1 ∂ ◦ G(t). σ1 !σ2 ! ∂t1 ∂t2 Write μ1 , μ2 ∈ Z, 0 ≤ μ1 ≤ σ1 , 0 ≤ μ2 ≤ σ2 , μ = (μ1 , μ2 ) and |μ| = μ1 + μ2 . Then there exist aμ1 μ2 ∈ Cp such that aμ μ ∂ μ1 ∂ μ2 1 2 σ σ ◦ G(0). ∂z F (0) = ∂t G(0) + μ !μ ∂t2 1 2 ! ∂t1 |μ| 27T1 T2 ,

(18)

S0 + 2 > 9T0 .

(19)

Assume further that D L × (2Q) + 2S0 − 1, > S0 log E L D × R, > S0 log E

(20) (21)

with Q=

log(L!) + T0 (h3 + 6) + 3.4S0 log(T0 + 1) + (S0 + 1)(18 + h4 ) L + 8S12 (T1 h1 + T2 h2 ) + (T1 + T2 )(16h4 + 60 log 2 + 12)

and with R = 220 T2 Now suppose that

T1 h1 13 53 + T2 h2 + h4 + log 2. 4 2 2

(22)

(23)

L || ≤ exp − log E . S0

Then we have = 0. Proof. Recall that L = (T0 + 1)(T1 + 1)(T2 + 1). We follow the proof of [39, Theorem 8.1] except for the choice of Q. We distinguish two cases. First, suppose that there exists F , a connected algebraic subgroup of E 2 of dimension one, with β ∈ B(F ) such that H (F ; T1 , T2 ) ≤ 30 T1 T2 . Then the condition (21) shows that we can apply Lemma 4.4 to obtain = 0. Moreover, since we have β ∈ B(F ), then we obtain u ∈ U(F ) and F satisfies the announced property because 30 T1 T2 =

162T0T1 T2 162T0T1 T2 3 × ≤ . S1 + 3 S0 + 2 S0 + 2

Next suppose that there is no such F . Then Lemma 7.1 assures that, for any L × L minor determinant of M, we have the estimate (13). Under the condition (20), the assumption of

258


Lemma 6.1 is satisfied. Therefore the rank of M is strictly inferior to L. Then Lemma 4.5 establishes the statement. 2 We now choose parameters. We have L ≤ (T0 + 1)(T1 + 1)(T2 + 1). Indeed the required assumptions are (14)–(23). Put T0 = [c0 a1 a2 g 3 d 5 ], T1 = [c1 a2 bgd 3 ], T2 = [c2 a1 bgd 3 ], S0 = 1 + 3[c3a1 a2 bg 2 d 5 ], S1 = 3[c4gd], with absolute constants c0 , c1 , c2 , c3 and c4 . Since only the quantity Q differs from the assumptions in [39] due to the calculations in [39], it is sufficient to choose c0 = 2.13 × 1028, c1 = c2 = 9.85 × 1016, c3 = 6.09 × 1028, c4 = 5.50 × 106. Thus we complete the proof of our main theorem since (1 + c0 )(1 + c1 )(1 + c2 ) ≤ 1.16 × 1035 3c3 − 2 and

& 18c0 c1 , c2 ≤ 5.89 × 1017 . c2 max 3c3 4

Acknowledgements. The first author was supported by a Research Grant for 2008, Nihon University. The second author was supported by a Research Grant of Graduate School of Science and Technology, Nihon University.

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Noriko Hirata-Kohno Department of Mathematics College of Science and Technology Nihon University Suruga-Dai Kanda Chiyoda Tokyo 101-8308 Japan (E-mail: hirata@math.cst.nihon-u.ac.jp) Rina Takada Graduate School of Science and Technology Mathematics Major Nihon University Suruga-Dai Kanda Chiyoda Tokyo 101-8308 Japan