P-adic Measures Of Algebraic Independence For The Values Of Ramanujan Functions And Klein Modular Functions

Acta Mathematica Sinica, English Series Jan., 2007, Vol. 23, No. 1, pp. 83–88 Published online: Apr. 2, 2006 DOI: 10.1007/s10114-005-0708-0 Http://www.ActaMath.com

p-adic Measures of Algebraic Independence for the Values of Ramanujan Functions and Klein Modular Functions Tian Qin WANG Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, P. R. China and Institute of Data and Knowledge Engineering, Henan University, Kaifeng 475001, P. R. China E-mail: [email protected]

Guang Shan XU Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, P. R. China Abstract In this paper, we give the p-adic measures of algebraic independence for the values of Ramanujan functions and Klein modular functions at algebraic points. Keywords p-adic value, Measure of algebraic independence, Ramanujan function, Klein modular function MR(2000) Subject Classification 11J61, 11J82, 11J91

1 Introduction and Main Results Recently, an essential progress in the development of the theory of transcendental numbers has been made, and some important results have been obtained. In particular, the transcendence, algebraic independence and their measures for the values of some modular functions and forms have been given. In 1996, Barr´e et al. [1] proved the famous Mahler–Manin conjecture. Let J(z) be the Klein modular function, which can be expressed in the series ∞  1 c(n)z n , c(n) ∈ N; J(z) = + 744 + z n=1 they proved that if q is an algebraic number with 0 < |q| < 1 (or 0 < |q|p < 1), then J(q) is a transcendental number. In 1997, Barr´e [2] also obtained the measure and p-adic measure of transcendence of J(q). ∞ The  definition of Ramanujan function is as follows: P (z) = 1 − 24 n=1 σ1 (n)z n , Q(z) =  ∞ n n k 1 + 240 ∞ d|n d ; these functions n=1 σ3 (n)z , R(z) = 1 − 504 n=1 σ5 (n)z , where σk (n) = are connected to the Eisenstein series E2 (τ ), E4 (τ ), E6 (τ ) by the identities E2 (τ ) = P (e2πiτ ), E4 (τ ) = Q(e2πiτ ), E6 (τ ) = R(e2πiτ ), Imτ > 0. It is well known that the Eisenstein function E2k (τ ) (k ≥ 2) is a modular form of weight 2k. In 1996, Nesternko [3, 4, 5] proved that if q is an algebraic number with 0 < |q| < 1, then P (q), Q(q), R(q) are algebraically independent. He also gave the measure of algebraic independence of P (q), Q(q), R(q). Meanwhile he obtained the algebraic independence and their measure of J(q), J  (q), J  (q). Received March 18, 2004, Accepted December 2, 2004 Supported by the NSFC (No. 10171097; 10571180)

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In this paper, we follow the methods of Nesternko [4, 5] and use some properties of Schnirelman integrals of p-adic analytic function introduced by Adams [6]. We will respectively give the p-adic measures of algebraic independence of P (q), Q(q), R(q) and J(q), J  (q), J  (q). As usual, deg P and H(P ) denote the degree and the height of polynomial P respectively, set t(P ) = deg P + ln H(P ). Theorem Let q be an algebraic number in Cp , 0 < |q|p < 1, and let ω1 , ω2 , ω3 ∈ Cp be such that all numbers P (q), Q(q), R(q) are algebraic over the field Q(ω1 , ω2 , ω3 ). Then for any polynomial A ∈ Z[x1 , x2 , x3 ], A ≡ 0, the following inequality holds : |A(ω1 , ω2 , ω3 )|p > exp(−cSd3 ln9 S), (1.1) where S and d are arbitrary numbers satisfying the inequalities S ≥ max{ln H(A) + deg A × ln t(A), e}, d ≥ deg A, and c is a positive constant depending only on q and ωi . Corollary 1 Let q be an algebraic number, 0 < |q|p < 1. Then for any polynomial A ∈ Z[x1 , x2 , x3 ], A ≡ 0, the following inequality holds : |A(P (q), Q(q), R(q))|p > exp(−c Sd3 ln9 S), where c is a positive constant depending only on q and P (q), Q(q), R(q). 2

d DJ . From [5] we have the following identities: P = 6 DDJJ − 4 DJ Let D = z dz J − 3 J−1728 , Q =

(DJ)2 J(J−1728) ,

3

(DJ) R = − J 2 (J−1728) . The following corollary holds immediately from the Theorem.

Corollary 2 Let q be an algebraic number, 0 < |q|p < 1. Then for any polynomial A ∈ Z[x1 , x2 , x3 ], A ≡ 0, the following inequality holds: |A(J(q), DJ(q), D2 J(q))|p > exp(−c Sd3 ln9 S),  where c is a positive constant depending only on q and J(q), DJ(q), D2 J(q). Remark The paper [2] only gives the p-adic measure of transcendence of J(q). The proof of the Theorem depends on the following proposition: Proposition Let q be an algebraic number, 0 < |q|p < 1. Then for each integer r, 1 ≤ r ≤ 3, there exists a constant μr > 0 such that for any homogeneous unmixed ideal I ⊂ K[x0 , x1 , x2 , x3 ], dim I = r − 1, the following inequality holds : r 3r ¯ p ≥ −μr (h + d ln(h + d))d 4−r (h + d ln(h + d)) 4−r , ln |I(ξ)| where d and h are arbitrary numbers satisfying the inequalities deg I ≤ d, h(I) ≤ h, and ξ¯ = (1, P (q), Q(q), R(q)). The definitions of deg I and h(I) are as in the following section. 2 Preliminaries Let p denote a fixed prime number, Q be rational field and Qp the completion of Q with respect to the p-adic valuation | |p on Q. We denote by Cp the completion of the algebraic closure of Qp , K  Cp an algebraic number field which is a finite extension of Q, ν = [K : Q]. Let M = M∞ ∪M0 , where M∞ and M0 be the sets of all Archimedean and non-Archimedean valuation on K respectively. From now on, we use c1 , c2 , . . . , γ1 , γ2 , . . . , κ1 , κ2 to denote positive constants depending only on α ∈ K, α = 0, there holds the  q, ωi and K. We can assume q ∈ K. For product formula v∈M |α|v = 1. For any polynomial P= ¯j a¯j xj11 · · · xjmm ∈ K[x1 , . . . , xm ], define the logarithm height h(P ) by the relation h(P ) = v∈M ln |P |v , where |P |v = max¯j |a¯j |v . It is easy to verify from the product formula that h(P ) ≥ 0 and h(λP ) = h(P ) for all nonzero λ ∈ K. Let r be an integer, 1 ≤ r ≤ m, and u ¯i = (ui0 , . . . , uim ), 1 ≤ i ≤ r be sets of variables. We m x) = j=0 uij xj , i = 1, . . . , r. Let I be an unmixed homogeneous introduce the linear forms Li (¯ x, u ¯1 , . . . , u ¯r ] = K[¯ x][¯ u]. Let ideal of the ring K[¯ x]. Let (I, L1 , . . . , Lr ) denote the ideal in K[¯ ¯ I(r) denote the ideal of K[¯ u1 , . . . , u ¯r ] consisting of all polynomials G ∈ K[¯ u] which satisfy the conditions GxTi ∈ (I, L1 , . . . , Lr ), 0 ≤ i ≤ m, for some integer T . By Proposition 4.4 ¯ ¯r ) such that I(r) = of [5], we know that there exists a symmetric polynomial F (¯ u1 , . . . , u m+1 ¯ = (ω0 , ω1 , . . . , ωm ) ∈ Cp , we set (F ), deg I = degu¯1 F, h(I) = h(F ). For each point ω (i)

|¯ ω |p = max0≤j≤m |ωj |p . Let S (i) = (sj,k )0≤j,k≤m , i = 1, . . . , r be skew-symmetric matrices.

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Set F  = F (¯ u(1) , . . . , u ¯(r) ), where u ¯(i) = (S (i) ω ¯ t )t , i = 1, . . . , r. For any unmixed homogeneous  −1 deg I ω |−r . Then |I(λ¯ ω )|p = |I(¯ ω )|p for all nonzero ideal I ⊂ K[¯ x], we define |I(¯ ω )|p = |F |p |F |p |¯ p λ ∈ Cp . Some Nesterenko’s results are now to be quoted. Lemma 2.1 (See Proposition 4.7 of [5]) Let I be an unmixed homogeneous ideal  of the ring K[¯ x], dim I ≥ 0, I = I1 ∩ · · · ∩ Is be a reduced primary decomposition, ℘j = Ij , and let kj be the exponent of the primary idealIj . Let 0 = ω ¯ ∈ Cm+1 . Then p s s s (i) j=1 kj deg ℘j = deg I, (ii) j=1 kj h(℘j ) ≤ h(I) + νm2 deg I, (iii) j=1 kj log |℘j (¯ ω )|p ≤ log |I(¯ ω )|p . deg P ω )|p |P |−1 ω |− . Let P ∈ K[¯ x] be a homogeneous polynomial, define ||P ||ω¯ = |P (¯ p |¯ p

Lemma 2.2 (See Proposition 4.8 of [5]) Let I = (P ) be the principal ideal in the ring K[¯ x] . Then generated by the homogeneous polynomial P , and let 0 = ω ¯ ∈ Cm+1 p ω )|p ≤ log P ω¯ . (i) deg I = deg P, (ii) h(I) ≤ h(P ) + νm2 deg P, (iii) log |I(¯ ¯ = (b0 , . . . , bm ) ∈ Cm+1 , define For any nonzero vectors A¯ = (a0 , . . . , am ), B p −1 ¯ −1 ¯ ¯ ¯ max |ai bj − aj bi |p . ||A − B|| = |A|p |B|p  0≤i<j≤m  ¯ | β( ¯ = 0) ∈ Cm+1 is a zero of ℘ . For any prime ideal ℘ ⊂ K[¯ x], define ρ = min ||¯ ω − β|| p Lemma 2.3 (See Corollary 4.9 of [5]) If A is a homogeneous polynomial of the ring K[¯ x] ¯ = 0, then ||A||ω¯ ≤ ρ. , A( ξ) and 0 = ω ¯ , ξ¯ ∈ Cm+1 p Lemma 2.4 (See Corollary 4.12 of [5]) Suppose that P ∈ K[¯ x], P ≡ 0 is a homogeneous polynomial, ℘ is a homogeneous prime ideal of the ring K[¯ x], dim ℘ ≥ 0, P ∈ ℘, and |℘(¯ ω)| ≤ e−S , S > 0, ||P ||ω¯ ≤ e−2mν deg P . ¯ < 1 If the integer η > 0 satisfies −η log ||P ||ω¯ ≥ 2 min{S, log ρ1 }, where ρ = min ||¯ ω − β|| m+1 ¯ and the last minimum is taken over all the nontrivial zeros β of the ideal ℘ in Cp , then for r = dim ℘ + 1 ≥ 2, there exists a homogeneous unmixed ideal J ⊂ K[¯ x], dim J = dim ℘ − 1, whose zeros coincide with those of the ideal (℘, P ), such that (i) deg J ≤ η deg ℘ deg P, (ii) h(J) ≤ η (h(℘) deg P + h(P ) deg ℘ + νm(r + 2) deg P deg ℘), (iii) log |J(¯ ω )|p ≤ −S + η (h(℘) deg P + h(P ) deg ℘ + 12νm2 deg P deg ℘). Inequality (iii)is also valid for r = 1 if we formally assume that |J(¯ ω )|p = 1 in this case. Lemma 2.5 (See Proposition 4.13 of [5]) Let I be an unmixed homogeneous ideal of the ring K[¯ x], r = dim I + 1 ≥ 1. For each nonzero point ω ¯ ∈ Cm+1 , there exists a zero β¯ ∈ Cm+1 of p p 1 1 ¯ the ideal I such that deg I log ||¯ ω − β|| ≤ r log |I(¯ ω )| + r h(I) + νm3 deg I.

Lemma 2.6 (See [3]) Let L1 and L2 be integers, L1 ≥ 1, L2 ≥ 1. Then for any polynomial A(z, x1 , x2 , x3 ) ∈ Q[z, x1 , x2 , x3 ], A ≡ 0, degz A ≤ L1 , degxi A ≤ L2 , the following inequality holds : ordA(z, P (z), Q(z), R(z)) ≤ c1 L1 L32 , where c1 = 1047 .

Lemma 2.7 (See Theorem 1 in Appendix of [6]) Let a, r ∈ Cp and f (z) be a function such that  (a) For all z such that |z|p = |r|p , f (a + z) is defined, (b) a,r f exists.  Then | a,r f (z)dz|p ≤ max|z|p =|r|p |f (a + z)|p . Lemma 2.8 (See Theorem 8 in Appendix of [6]) Let f (x) be a power series converging for all z such that |z|p < R (R > 0). Suppose x, a, r ∈ Cp are such that |x|p , |a|p , |r|p < R. Then  f (n) (x) = n! a,r f (z)(z−a)dz (z−x)n+1 for |x − a|p < |r|p . Lemma 2.9 (See Theorem 13 in Appendix of [6]) Let f (x) be a power series converging for all z such that |z|p < R (R > 0). Let r ∈ Cp , |r|p < R. G(z) = (z − a1 )k1 · · · (z − an )kn be a polynomial with |ai |p < |r|p for 1 ≤ i ≤ n, and 0 = t ∈ Cp such that |ai − aj |p > |t|p for all    (z)z 1) n) i, j(i = j). Then 0,r fG(z) dz = a1 ,t f (z)(z−a dz + · · · + an ,t f (z)(z−a dz. G(z) G(z)

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3

Construction of Auxiliary Functions

The first step of the proof is the construction of the auxiliary functions with certain bounds. Lemma 3.1 For all numbers L1 ≥ 1, L2 ≥ 1 such that L1 + L2 is sufficiently large, there exists a polynomial A ∈ Z[z, x1 , x2 , x3 ], A ≡ 0 such that degz A ≤ L1 , degxi A ≤ L2 , (i = 1, 2, 3) ln H(A) ≤ 85L2 ln(L1 + L2 ), and the function F (z) = A(z, P (z), Q(z), R(z)) satisfies the relations F (k) (0) = 0, k = 0, 1, . . . , [ 21 L1 L32 ] − 1. Proof For the proof, see Lemma 2.1 of [4]. ∞ n Now we introduce the notation M = ordF (z), F (z) = n=M bn z . From the proof of Lemma 3.1, we know bn ∈ Z, (n ≥ M ). Note that Lemma 2.6 and Lemma 3.1 imply the inequalities

1 3 L1 L2 ≤ M ≤ c1 L1 L32 , (3.1) 2 and |bM | ≤ exp(γ1 L2 ln(L1 + L2 )). From the latter, we can deduce (see [7]) that |bM |p ≥ exp(−c2 L2 ln(L1 + L2 )).

(3.2)

Lemma 3.2 There exists an integer T, 0 ≤ T ≤ γL2 ln(L1 + L2 ), for which the following inequality holds: |F (T ) (q)|p > ( 12 |q|p )2M , where γ is a constant which will be chosen later Proof Suppose that the following inequalities hold:  2M 1 |F (k) (q)|p ≤ |q|p , 0 ≤ k ≤ L = [γL2 ln(L1 + L2 )]. 2 Let r ∈ Cp , ρ ∈ Cp such that

(3.3)

1

(3.4) 0 < p p−1 |r|p < |ρ|p < |q|p < 1. M Let G(z) = F (z)/z . Then G(z) is an analytic function in the discrete circle |z|p ≤ 1. From Schnirelman integral and Lemmas 2.8 and 2.9, we have the following interpolation formula; fix z such that |z|p = |r|p , then

L+1

L+1 k+1



L  z(z − q) z(z − q) G(ξ)ξ G(ξ) G(k) (0) ξ dξ = dξ − dξ G(z) = ξ − z ξ(ξ − q) ξ − z k! ξ(ξ − q) ξ −z 0,1 0,1 0,ρ k=0

L+1

L  z(z − q) G(k) (q) (ξ − q)k+1 dξ = I1 + I2 + I3 . − (3.5) k! ξ−z q,ρ ξ(ξ − q) k=0

By Lemma 2.7, we have  

 z(z − q) L+1 G(ξ)   ≤ |r|L+1  |I1 |p ≤ max  |q|L+1 ≤ exp(−γ1 γL2 ln(L1 + L2 )), p p ξ(ξ − q) ξ − z |ξ|p =1 where γ1 = | ln |r|p |q|p |.



k

|I2 |p ≤ max

0≤k≤L

(3.6)

p

p p−1 |r|L+1 |q|L+1 |ρ|k+1 p p p |ρ|L+1 |q|L+1 |ρ|p p p

≤

1

p p−1 |r|p |ρ|p

L+1 ≤ exp(−γ2 γL2 ln(L1 + L2 )),

(3.7)

1

p p−1 |r|

where γ2 = | ln A|, A = |ρ|p p < 1. In the same way as above, and under the assumption (3.3),   

L+1

 1 z(z − q) ξ k+1 l (l) −(M +k−l)  |I3 |p ≤ max  dξ  Bl Ck F (q)q 0≤k≤L k! ξ+q−z p 0,ρ ξ(ξ + q) 0≤l≤k

≤p

L p−1

+L) (|q|p /2)2M |q|−(M p

|r|L+1 |q|L+1 |ρ|p p p

L+1 ≤ |q|M ≤ exp(−γ3 γL2 ln(L1+L2 )). (3.8) p (A) |ρ|L+1 |q|L+1 |ρ|p p p On the other hand, | Fz(z) M |p = |bM |p ≥ exp(−c2 L2 ln(L1 + L2 )). This together with (3.5), (3.6), (3.7) and (3.8) yields exp(−c2 L2 ln(L1 + L2 )) ≤ max(|I1 |p , |I2 |p , |I3 |p ), which is impossible if γ is large enough. Lemma 3.2 is proved.

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Lemma 3.3 Let q ∈ K, 0 < |q|p < 1. For all numbers L1 ≥ 1, L2 ≥ 1 such that L1 + L2 is sufficiently large, there exists a polynomial B ∈ Z[z, x1 , x2 , x3 ] such that degz B ≤ L1 , degxi B ≤ 2γL2 ln(L1 + L2 ), (i = 1, 2, 3), (3.9) ln H(B) ≤ 2γL2 ln2 (L1 + L2 ), exp(−κ2 L1 L32 ) ≤ |B(q, P (q), Q(q), R(q))|p ≤ exp(−κ1 L1 L32 ).

(3.10) (3.11)

Proof As Proposition 2.1 of [4], we have B(z, P (z), Q(z), R(z)) = (12z)T F (T ) (z). Lemma 3.2 and (3.1) yield the lower bound  T 2M  3M 1 1 1 T (T ) |B(q, P (q), Q(q), R(q))|p = |12q|p |F (q)|p ≥ ≥ |q|p ≥ exp(−κ2 L1 L32 ). |q|p |q|p 12 2 2 To obtain the upper bound, we have    ∞   CnT bn q n−T  |B(q, P (q), Q(q), R(q))|p ≤ |F (T ) (q)|p = T ! M −T

p

n=M bM +1 q M +1−T

−T ≤ |bM q + + · · · |p ≤ |q|M ≤ exp(−κ1 L1 L32 ). p This proves (3.11). In the same way as Proposition 2.1 of [4], we can prove (3.9) and (3.10). Lemma 3.3 is thus proved.

4 Proof of Proposition We prove it by induction on r. Let r, 1 ≤ r ≤ 3 be the least number for which the Proposition is no longer true. We choose and fix a sufficiently large number λ. The set of real numbers S for which there exist a pair of numbers d, h and a prime homogeneous ideal ℘ ⊂ K[¯ x] of dimension r − 1 satisfying the conditions r 3r ¯ p < −2λ12 Sd 4−r (ln S) 4−r , (4.0) deg ℘ ≤ d, h(℘) ≤ h, h + d ln(h + d) = S, ln |℘(ξ)| is unbounded (for reasons, see [4]). Now let S be sufficiently large, and let the numbers d, h, the prime homogeneous ideal ℘ of dimension r − 1 satisfy conditions (4.0). We define the numbers L1 and L2 as follows: r−3 3 L1 = λ3 Sd 4−r (ln S) 4−r , (4.1)   r 3r 1 2κ2 L1 L32 = min 2λ12 Sd 4−r (ln S) 4−r , ln , (4.2) ρ where ρ is the distance from the point ξ¯ to the variety of zeros of the ideal ℘ and κ2 is the con2r−4 3r stant from Lemma 3.3. It follows from (4.2) and Lemma 2.5 that κ2 L1 L32 ≥ λ11 Sd 4−r (ln S) 4−r , and therefore, for a sufficiently large λ, we have L1 ≥ 1 and L2 ≥ 1. (4.1) and (4.2) give the r−1 1 estimate L2 ≤ λ3 d 4−r (ln S) 4−r . Define polynomial B ∈ Z[z, x1 , x2 , x3 ] with the aid of Lemma 3.3. The degree n of B in the totality of variables x1 , x2 , x3 satisfies 1 3 (4.3) n ≤ 6γL2 ln(L1 + L2 ) ≤ 12γL2 ln S ≤ 12γλ3 d 4−r (ln S) 4−r . Let us define the homogeneous polynomial E ∈ K[¯ x] by the relation E = xn0 B(q, xx10 , xx20 , xx30 ). r−3

3

Then h(E) ≤ h(B) + γ1 L1 ≤ ν ln H(B) + γ1 L1 ≤ λ4 Sd 4−r (ln S) 4−r . We assert that E ∈ ℘, since otherwise, by Lemma 2.3 and the definition of ||E||ξ¯, there holds ¯ p ≤ |E|p |ξ| ¯ n ρ. |E(ξ)| (4.4) p r−3

3

Meanwhile we have ln |E|p ≤ γ2 L1 ≤ λ4 Sd 4−r (ln S) 4−r . By virtue of (4.2) and (4.3), (4.4) ¯ p = ln |E(ξ)| ¯ p ≤ ln ρ + ln |E|p + n ln |ξ| ¯ p < −κ2 L1 L3 . This contradicts the left implies ln |B(ξ)| 2 side of (3.11). By the Liouville theorem (see Lemma 3.8 of [7]), there holds ln |E|p ≥ −c(ln H(B) + r−3 3 degz B) ≥ −λ4 Sd 4−r (ln S) 4−r . Set η = 1 + [5κ2 /κ1 ], where κi is as in Lemma 3.3. Then  3  ¯ p × |E|−1 × |ξ| ¯ −n ) ≥ η κ1 L1 L3 + n ln |ξ| ¯ p − λ4 Sd r−3 4−r (ln S) 4−r −η ln ||E||ξ¯ = −η ln(|B(ξ)| p p 2   r 3r 1 3 12 4−r 4−r (ln S) , ln > 4κ2 L1 L2 = 2 min 2λ Sd . ρ

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Now we can apply Lemma 2.4 to ℘ and E. For r ≥ 2, there exists an ideal J ⊂ K[x0 , x1 , x2 , x3 ] such that dimJ = dim℘ − 1, deg J ≤ ηn deg ℘, (4.5) h(J) ≤ η(nh(℘) + h(E) deg ℘ + 15νn deg ℘, (4.6) r 3r 12 ¯ p ≤ −2λ Sd 4−r (ln S) 4−r + η(nh(℘) + h(E) deg ℘ + 108νn deg ℘). (4.7) ln |J(ξ)| ¯ p =1. For r = 1, we have (4.7) in which we formally set |J(ξ)| Using the inequalities proved above, we obtain 1 3 (4.8) η(nh(℘) + h(E) deg ℘ + 108νn deg ℘) ≤ λ5 Sd 4−r (ln S) 4−r . This together with (4.7) yields r 3r ¯ p ≤ −λ12 Sd 4−r ln |J(ξ)| (ln S) 4−r . (4.9) In the case r = 1, (4.9) does not hold; therefore, in what follows, we assume r ≥ 2, (4.5) 5−r 3 and (4.3) implies deg J ≤ 12γηλ3 d 4−r (ln S) 4−r , and from (4.6) and (4.8) we have h(J) ≤ 1 3 λ5 Sd 4−r (ln S) 4−r . Thus we have 1 3 φ(J) = h(J) + deg J × ln(h(J) + deg J) ≤ 4λ5 Sd 4−r (ln S) 4−r , r−1

3r−3

r

3r

φ(J)(deg J) 5−r (ln φ(J)) 5−r ≤ λ10 Sd 4−r (ln S) 4−r . On the other hand, since dimJ = r − 2, according to the assumption of the induction, we have r−1 3r−3 r 3r ¯ p ≥ −μr−1 φ(J)(deg J) 5−r ln |J(ξ)| (ln φ(J)) 5−r ≥ −μr−1 λ10 Sd 4−r (ln S) 4−r , a contradiction with (4.9) if we choose λ2 > μr−1 . This completes the proof of the proposition. 5 Proof of Theorem For any polynomial B ∈ Z[x1 , x2 , x3 ], define the homogeneous polynomial B B(x1 /x0 , x2 /x0 , x3 /x0 ). C(x0 , x1 , x2 , x3 ) = xdeg 0 ¯ = B(P (q), Q(q), R(q)). Let I = (C) be homogeThen deg C = deg B, h(C) ≤ ν ln H(B), C(ξ) neous principal ideal in K[¯ x]. Then dimI = 2. By Lemma 2.2 we have deg I = deg C, h(I) ≤ ¯ p ≤ ln ||C|| ¯. Let d = deg B, h = 9ν 2 (ln H(B) + deg B). Then the 9ν(h(C) + deg C), ln |I(ξ)| ξ requirements of Proposition are fulfilled and we obtain ¯ p − ln |C|p − deg C ln |ξ| ¯ p ≥ ln |I(ξ)| ¯ p ln ||C|| ¯ = ln |C(ξ)| ξ

≥ −μ3 γ1 (ln H(B) + deg B ln t(B))(deg B)3 ln9 (ln H(B) + deg B ln t(B)), which yields ¯ p ≥ −μ3 γ2 (ln H(B) + deg B ln t(B))(deg B)3 ln9 (ln H(B) + deg B ln t(B)). ln |C(ξ)| (5.1) It follows from the hypothesis of Theorem that all numbers ω1 , ω2 and ω3 are algebraic over the field Q(P (q), Q(q), R(q)). If the polynomial E ∈ Q[x1 , x2 , x3 ] is such that with the number d = E(P (q), Q(q), R(q)) all numbers dωi are algebraic integers over the ring Z[P (q), Q(q), R(q)], then Norm(ddeg A A(ω1 , ω2 , ω3 )) = B(P (q), Q(q), R(q)), where B(x1 , x2 , x3 ) is a polynomial with integer coefficients. Taking into account that deg B ≤ γ3 deg A, ln H(B) ≤ γ4 ln H(A), we finally complete the proof of the Theorem. References [1] Barr´e, K., Diaz, G., Gramain, F., Philibert, G.: Une preuve de la conjecture de Mahler-Manin. Invent. Math., 124, 1–9 (1996) [2] Barr´e, K.: Measure d’approximation simultan´ee de q et J(q). J. Number Theory, 66, 102–128 (1997) [3] Nesterenko, Yu. V.: Modular functions and transcendence problems. Mat. Sb., 187(9), 65–96 (1996) [4] Nesterenko, Yu. V.: on the measure of algebraic independence of the values of Ramanujan functions. Proceedings of the Steklov Institute of Math., 218, 294–331 (1997) [5] Nesterenko, Yu. V.: Algebraic independence for values of Ramanujan functions, Introduction to algebraic independence theory, Edited by Nesterenko Yu. V. and Philippon P., Lecture Notes in Math. 1752, 27–46, Springer, Berlin, Heidelberg, New York, 2001 [6] Adams, W. W.: Transcendental numbers in the p-adic domain. Amer. J. Math., 88, 279–308 (1966) [7] Wang, T. Q.: p-adic transcendence and p-adic transcendence measures for the values of Mahler type functions. Acta Math. Sinica, English Series, 22(1), 187–194 (2006)