Some applications of the p-adic analytic subgroup theorem

SOME APPLICATIONS OF THE p-ADIC ANALYTIC SUBGROUP THEOREM arXiv:1412.1248v2 [math.NT] 11 Jan 2016 CLEMENS FUCHS AND DUC HIEP PHAM Abstract. We use a p-adic analogue of the analytic subgroup theorem of Wüstholz to deduce the transcendence and linear independence of some new classes of p-adic numbers. In particular we give p-adic analogues of results of Wüstholz contained in [20] and generalizations of results obtained by Bertrand in [3] and [4]. 2010 Mathematical Subject Classification: 11G99 (14L10, 11J86) Keywords: commutative algebraic groups, transcendence theory, p-adic numbers 1. Introduction Transcendence theory is known as one of the fundamental questions in number theory. One of the first outstanding results in transcendence theory is Hermite’s proof of the transcendence of e obtained in 1873. His work was extended by Lindemann afterwards in order to prove that eα is transcendental for any non-zero algebraic number α. In particular, this showed that π is transcendental, and thus answered the problem of squaring the circle. There are many results further pushing the theory forward. Such important results are due to Gelfond, Schneider, Baker, Masser, Coates, Lang, Chudnovsky, Nesterenko, Waldschmidt, Wüstholz among others. Especially, in the 1980’s Wüstholz formulated and proved a very deep and far-reaching theorem, the so-called analytic subgroup theorem (see [21, 22] or [2]). It is known as one of the most striking theorems in the complex transcendental number theory with many applications. Wüstholz in 1983 used his analytic subgroup theorem to give a vast generalization of earlier results on linear independence in the complex domain (see [20]). To present the result, let k be a non-negative integer, n0 , n1 , . . . , nk positive integers and E1 , . . . , Ek elliptic curves defined over Q. For each i with 1 ≤ i ≤ k, denote by Li the field of endomorphisms of Ei , ℘i the Weierstrass elliptic function associated with Ei and Λi the lattice of periods of ℘i . Let u1 , . . . , un0 be non-zero elements in Q and let ui,1 , . . . , ui,ni be elements in the set Ei := {u ∈ C \ Λi ; ℘i (u) ∈ Q}, i = 1, . . . , k. Finally, let V0 be the vector space generated by the logarithms log u1 , . . . , log un0 over Q and Vi the vector space generated by ui,1 , . . . , ui,ni over Li for i = 1, . . . , k. If the elliptic curves E1 , . . . , Ek are pairwise non-isogeneous, then the dimension of the vector space V generated by 1, V0 , . . . , Vk over Q is determined by dimQ V = 1 + dimQ V0 + dimL1 V1 + · · · + dimLk Vk . It can be seen that Baker’s theorem on linear forms of logarithms of algebraic numbers is the corresponding statement of the theorem without the elliptic curves E1 , . . . , Ek . Many of the results above have been transferred into the world of p-adic numbers as well. The theory of p-adic transcendence was studied by many authors following the line of arguments in the classical complex case. This development was initiated by Mahler in the 1930’s (see [14, 15]). He obtained the p-adic analogues of the Hermite, Lindemann and Gelfond-Schneider theorems. Afterwards several p-adic analogues of known results from the complex case were proved, and new theories of the p-adic transcendence were obtained. These contributions are due to Veldkamp (cf. [18]), Günther (cf. [12]), Brumer (cf. [7]), Adams (cf. [1]), Flicker (cf. [8, 9]), Bertrand and others. Also a p-adic analogue of the Wüstholz’ analytic subgroup theorem has been worked out (see [16] and the paper [11] by the authors which revisited that result and which is based on 1 2 C. FUCHS AND D. H. PHAM the results of [10]; observe that this statement has already been mentioned by Bertrand in [5]). Before we go on, we state the p-adic analogue of the analytic subgroup theorem of Wüstholz, since it will be the main tool for the results below. As usual, we denote by Cp the completion of Qp with respect to the p-adic absolute value. Let G be a commutative algebraic group defined over Q and Lie(G) denote the Lie algebra of G. Then the set G(Cp ) of Cp -points of G is a Lie group over Cp whose Lie algebra is given by Lie(G(Cp )) = Lie(G) ⊗Q Cp . It is known that there is the p-adic logarithm map logG(Cp ) : G(Cp )f → Lie(G(Cp )) where G(Cp )f is the set of x ∈ G(Cp ) for which there exists a strictly increasing sequence (ni ) of integers such that xni tends to the unity element of G(Cp ) as i tends to ∞ (see [6, Chapter III, 7.6]). We denote by Gf (Q) := G(Cp )f ∩ G(Q) the set of algebraic points of G in G(Cp )f . The following statement is the p-adic analytic subgroup theorem: Theorem 1.1. Let G be a commutative algebraic group of positive dimension defined over Q and let V ⊆ Lie(G) be a non-trivial Q-linear subspace. For any γ ∈ Gf (Q) with 0 6= logG(Cp ) (γ) ∈ V ⊗Q Cp there exists an algebraic subgroup H ⊆ G of positive dimension defined over Q such that Lie(H) ⊆ V and γ ∈ H(Q). In this paper we give some applications of the p-adic analytic subgroup theorem. It is known that in the complex domain many results on linear independence and transcendence are deduced from Wüstholz’ analytic subgroup theorem. We shall show that it is also possible to obtain p-adic analogues of such results by using the p-adic analytic subgroup theorem. We will not give proofs of results which already appeared in the literature but instead concentrate on applications which do not exist in the literature so far. 2. Results We start by giving a p-adic analogue of the result of Wüstholz in [20] mentioned in the introduction. With the notations as above, but in the p-adic domain we let ℘p,i be the (LutzWeil) p-adic elliptic function associated with Ei (it is given by the same formal power series as the Weiterstrass ℘-function in the complex case, however the domain on which it is analytic depends on p), and vi,1 , . . . , vi,ni elements in the set Ep,i := {0} ∪ {u ∈ Dp,i \ {0}; ℘p,i (u) ∈ Q} where Dp,i is the p-adic domain of Ei for i = 1, . . . , k (see [19] and [13]). We have the following result. Theorem 2.1. Let V0 be the vector space generated by the p-adic logarithms Logp (u1 ), . . . , Logp (un0 ) over Q, and Vi the vector space generated by vi,1 , . . . , vi,ni over Li for i = 1, . . . , k. If the elliptic curves E1 , . . . , Ek are pairwise non-isogeneous, then the dimension of the vector space V generated by 1, V0 , . . . , Vk over Q is determined by dimQ V = 1 + dimQ V0 + dimL1 V1 + · · · + dimLk Vk . Following [20] we turn to abelian varieties. Let A be an abelian variety defined over Q and denote by a the Lie algebra of A. It is well-known that A(Cp ) has naturally the structure of a Lie group over Cp . By [6, Chapter III, 7.2, Prop. 3], there is an open subgroup Ap of Lie(A(Cp )) = a ⊗Q Cp on which the p-adic exponential map expA(Cp ) is defined and expA(Cp ) : Ap → expA(Cp ) (Ap ) ⊂ A(Cp ) is an isomorphism. Its inverse is the restriction of the p-adic logarithmic map to expA(Cp ) (Ap ). We denote by AQ the set {α ∈ Ap ; expA(Cp ) (α) ∈ A(Q)}. SOME APPLICATIONS OF THE p-ADIC ANALYTIC SUBGROUP THEOREM 3 Let R be a ring operating on a. We say that elements u1 , . . . , um in a are linearly independent over R if whenever we have a relation r1 u1 + · · · + rm um = 0 with r1 , . . . , rm in R, then it implies that r1 = · · · = rm = 0. Finally, let End(A) denote the ring of endomorphisms of A and End(a) the ring of endomorphisms of a (defined over Q). We prove the following theorem which again is the p-adic analogue of a result of Wüstholz in [20]. Theorem 2.2. Assume that A is a simple abelian variety. Let m be a positive integer, and u1 , . . . , um elements of AQ such that they are linearly independent over End(A). Then u1 , . . . , um are linearly independent over End(a). We now generalize a result of Bertrand from p-adic transcendence. To describe the results n below, as usual, we identify the Lie algebra a with the vector space Q , here n := dim A. Under this identification it follows that AQ is a subset of n Lie(A(Cp )) = a ⊗Q Cp = Q ⊗Q Cp = Cnp . We say that A is an abelian variety of RM type (or with real multiplication) if there are a totally real number field F of degree n and an embedding F ֒→ End(A) ⊗Z Q (hence an abelian variety of CM type is also of RM type). The following result was given by Bertrand in 1979 (see [4]): If A is a simple abelian variety of RM type defined over Q, then all coordinates of u are transcendental for any non-zero element u in AQ . We shall use the p-adic analytic subgroup theorem to give a generalization of this theorem for general simple abelian varieties. That is, we prove the following result. Theorem 2.3. If A is a simple abelian variety of positive dimension defined over Q, then all coordinates of u are transcendental for any non-zero element u in AQ . It is known that there are homomorphisms from the additive group Cp to the multiplicative group C∗p extending the p-adic exponential function (see [17, Chapter 5, 4.4]). Let ϕp be such a homomorphism. We get the following result which can be seen as a generalization of Corollary 2 in [3]. Theorem 2.4. If A is a simple abelian variety defined over Q of dimension n > 0, then the elements ϕp (αu1 ), . . . , ϕp (αun ) are transcendental for any non-zero element u = (u1 , . . . , un ) in AQ and for any non-zero element α ∈ Q. In the next section we give the proofs of the statements as consequence of the p-adic analytic subgroup theorem. 3. Proofs 3.1. Proof of Theorem 2.1. The proof is very similar to the one given in [20]. First of all, one may assume without loss of generality that n0 = dimQ V0 and ni = dimLi Vi for 0 ≤ i ≤ k. Let ri be the p-adic valuation of ui , and define vi := p−ri ui for i = 1, . . . , n0 . Then vi are algebraic numbers in U(1) := {x ∈ Cp ; |x|p = 1} and Logp (vi ) = Logp (ui ), ∀i = 1, . . . , n0 . We therefore have to show that the elements 1, Logp (v1 ), . . . , Logp (vn0 ), u1,1 , . . . , u1,n1 , . . . , uk,1 , . . . , uk,nk are linearly independent over Q. In fact, assume on the contrary that they are not linearly independent over Q. This means that there exists a non-zero linear form l in n := 1+n0 +· · ·+nk variables with coefficients in Q such that l(1, Logp (v1 ), . . . , Logp (vn0 ), u1,1 , . . . , u1,n1 , . . . , uk,1 , . . . , uk,nk ) = 0. Let V be the Q-vector space defined by n V := {v ∈ Q ; l(v) = 0}, 4 C. FUCHS AND D. H. PHAM and consider the commutative algebraic group G given by G = Ga × Gnm0 × E1n1 × · · · × Eknk . Then G is defined over Q and the Lie algebra Lie(G) = Lie(Ga ) × Lie(Gm )n0 × Lie(E1 )n1 × · · · × Lie(Ek )nk which is identified with Q 1+n0 +···+nk n = Q . This shows that Lie(G(Cp )) = Lie(G) ⊗Q Cp = Cnp . We write abbreviately expi and logi for the p-adic exponential and logarithm map associated with Ei respectively for each i = 1, . . . , n. One has G(Cp )f = Cp × U(1)n0 × (E1 (Cp ))n1 × (Ek (Cp ))nk and the p-adic logarithm map logG(Cp ) : G(Cp )f → Cnp is determined by (idCp , (Logp )n0 , logn1 1 , . . . , lognk k ). Let γ ∈ G(Cp )f be the algebraic point given by One gets
1, v1 , . . . , vn0 , exp1 (u1,1 ), . . . , exp1 (u1,n1 ), . . . , expk (uk,1 ), . . . , expk (uk,nk ) . logG(Cp ) (γ) = (1, Logp (v1 ), . . . , Logp (vn0 ), u1,1 , . . . , u1,n1 , . . . , uk,1 , . . . , uk,nk ) which is a non-zero element in V ⊗Q Cp . We apply the p-adic analytic subgroup theorem to G, V and γ to obtain an algebraic subgroup H of G of positive dimension defined over Q such that γ is in H(Q) and Lie(H) is contained in V . Let H 0 be the connected component of H then H 0 = H−1 × H0 × · · · × Hk where H−1 is an algebraic subgroup of Ga , H0 is an algebraic subgroup of Gm and Hi is an algebraic subgroup of Eini for 1 ≤ i ≤ k. We have dim H 0 = dim H = dimQ Lie(H) ≤ dimQ V = n − 1. Since γ ∈ H and the first coordinate of γ is 1 it follows that H−1 = Ga . This means that at least one of the algebraic groups H0 , . . . , Hk is contained in the corresponding factor with positive codimension. If this happens for the algebraic group H0 , i.e. H0 is a proper algebraic subgroup of Gm , then the Lie algebra of H0 is defined by non-zero linear forms with integer coefficients. This means that the elements Logp (v1 ), . . . , Logp (vn0 ) are linearly dependent over Q since (v1 , . . . , vn0 ) ∈ H0 , i.e. dimQ V0 < n0 , a contradiction. Hence there must be at least one i ∈ {1, . . . , k} such that dim Hi ≤ dim(Eini ) − 1 = ni − 1. Using the same arguments as in the proof of [2, Theorem 6.2], we conclude that the elements ui,1 , . . . , ui,ni are linearly dependent over Li . This is, dimLi Vi < ni , a contradiction, and the theorem follows.  3.2. Proof of Theorem 2.2. Suppose that there are elements r1 , . . . , rm not all zero in End(a) such that r1 u1 + · · · + rm um = 0. Let V be the vector space defined by V := {v = (v1 , . . . , vm ) ∈ an ; r1 v1 + · · · + rm vm = 0}. Then V is a Q-linear subspace of am . We consider the abelian variety G := Am and the element γ := (expA(Cp ) (u1 ), . . . , expA(Cp ) (um )). Then γ is an algebraic point of m G(Cp )f = A(Cp )m f = A(Cp ) , and logA(Cp ) (γ) = (u1 , . . . , um ) is a non-zero element in V ⊗Q Cp . The p-adic analytic subgroup theorem then shows that there exists an algebraic subgroup H of G of positive dimension defined SOME APPLICATIONS OF THE p-ADIC ANALYTIC SUBGROUP THEOREM 5 over Q such that γ ∈ H(Q) and Lie(H) is a subspace of V . Since A is simple, H is isogeneous to Ak for a certain integer k < m. One can therefore define a projection π from G onto H. We get the corresponding tangent map dπ : Lie(G) → Lie(H). Since γ ∈ H(Q) it follows that the point (u1 , . . . , um ) = logA(Cp ) (γ) = logH(Cp ) (γ) belongs to Lie(H). On the other hand, we may identify Lie(G) and Lie(H) with am and ak , respectively. Then the point (u1 , . . . , um ) is in the kernel of the linear map idLie(G) − dπ which can be written as an m × m matrix with entries in End(A) (since the algebra of endomorphisms End(G) of G is represented on the Lie(G) by the matrix algebra Mm (End(A))). In other words, the image of (u1 , . . . , um ) under this matrix is zero. Note that this matrix is non-zero (since k < m), hence there is at least one column of its is non-zero. We have thus shown that there is a non-trivial dependence relation over End(A) among the elements u1 , . . . , um , or equivalently, u1 , . . . , um are linearly dependent over End(A). This contradiction proves the theorem.  3.3. Proof of Theorem 2.3. Denote by n the dimension of A. Suppose on the contrary that there is a non-zero element u = (u1 , . . . , un ) ∈ AQ with ui algebraic over Q for some i ∈ {1, . . . , n}. Let G = Ga × A be the direct product of the additive group Ga with A. Then G is commutative and defined over Q with Lie(G) = Lie(Ga ) × Lie(A) = Q n+1 This gives . Lie(G(Cp )) = Lie(G) ⊗Q Cp = Cn+1 p We have G(Cp )f = Cp × A(Cp ) and the p-adic logarithm map logG(Cp ) is idCp × logA(Cp ) . Let V be the Q-vector space defined by {(v0 , . . . , vn ) ∈ Q n+1 ; v0 − vi = 0} and γ the algebraic point (ui , expA(Cp ) (u)). We observe that γ is non-zero since u is non-zero, and furthemore we have 
 logG(Cp ) (γ) = idCp (ui ), logA(Cp ) expA(Cp ) (u) = (ui , u1 , . . . , un ) is an element in V ⊗Q Cp . Applying the p-adic analytic subgroup theorem, we obtain an algebraic subgroup H of G of positive dimension defined over Q such that γ ∈ H(Q) and Lie(H) ⊆ V . Note that V is proper in Lie(G), and this shows that H is proper in G. Since the abelian variety A is simple, H must be either of the form Ga × {e} or {0} × A where e is the identity element of A. If H = Ga × {e} then expA(Cp ) (u) = e, i.e. u = 0, a contradiction. If H = {0} × A then n the Lie algebra Lie(H) = {0} × Q . This contradicts with the condition Lie(H) ⊆ V , and the theorem is proved.  3.4. Proof of Theorem 2.4. Assume on the contrary that there exists i ∈ {1, . . . , n} such that ϕp (αui ) ∈ Q. There is a sufficiently large positive integer r such that w := pr αui ∈ B(rp ), where B(rp ) denotes the ball {x ∈ Cp ; |x|p < rp } with rp := p−1/(p−1) . Hence X wk r ep (w) := = ϕp (w) = ϕp (pr αui ) = ϕp (αui )p k! k≥1 is also in Q. Let G = Gm × A be the direct product of the multiplicative group Gm with A. Then G is commutative and defined over Q with Lie(G) = Lie(Gm ) × Lie(A) = Q n+1 . This implies that Lie(G(Cp )) = Lie(G) ⊗Q Cp = Cn+1 . p We have G(Cp )f = Gm (Cp )f × A(Cp )f = U(1) × A(Cp ), 6 C. FUCHS AND D. H. PHAM and the p-adic logarithm map logG(Cp ) is the product Logp × logA(Cp ) . Let V be the Q-vector space defined by n+1 {(v0 , . . . , vn ) ∈ Q ; v0 − pr αvi = 0} and γ the algebraic point (ep (w), expA(Cp ) (u)). We see that γ is non-zero since u is non-zero, and furthermore 

 logG(Cp ) (γ) = logp ep (w) , logA(Cp ) expA(Cp ) (u) = (pr αui , u1 , . . . , un ) is a non-zero element in V ⊗Q Cp . Using the p-adic analytic subgroup theorem, we obtain an algebraic subgroup H of G of positive dimension defined over Q such that γ ∈ H(Q) and Lie(H) ⊆ V . Note that V is proper in Lie(G), and this shows that H is proper in G. Since the abelian variety A is simple, H must be either of the form Gm × {e} or {1} × A where e is the identity element of A. If H = Gm × {e} then expA(Cp ) (u) = e, i.e. u = 0, a contradiction. n If H = {1} × A then the Lie algebra Lie(H) = {0} × Q . This contradicts with the condition Lie(H) ⊆ V , and the theorem is proved.  4. Acknowledgments The first author was supported by Austrian Science Fund (FWF): P24574. The second author was supported by grant PDFMP2 122850 funded by the Swiss National Science Foundation (SNSF). References [1] W. Adams, Transcendental Numbers in the p-adic domain, American Journal of Mathematics 88 (1966), no. 2, 279-308. [2] A. Baker and G. Wüstholz, Logarithmic forms and Diophantine geometry, New Mathematics Monographs, vol. 9, Cambridge University Press, 2007. [3] D. Bertrand, Sous-groupes à un paramètre p-adique de variétés de groupe, Invent. Math. 40 (1977), no. 2, 171-193. [4] D. Bertrand, Problèmes locaux, Société Mathématique de France, Astérisque 69-70 (1979), 163-189. [5] D. Bertrand, Lemmes de zèros et nombres transcendants, Séminaire Bourbaki Vol. 1985/86, Astérisque 145146 (1987), 21-44. [6] N. Bourbaki, Elements of Mathematics. Lie groups and Lie algebras. Part I: Chapters 1-3. English translantion., Actualities scientifiques et industrielles, Herman. Adiwes International Series in Mathematics. Paris: Hermann, Publishers in Arts and Science; Reading, Mass.: Addison-Wesley Publishing Company. XVII, 1975. [7] A. Brumer, On the units of algebraic numbers fields, Mathematika 14 (1967), 121-124. [8] Y. Z. Flicker, Transcendence theory over local fields, PhD dissertation, University of Cambridge, 1978. [9] Y. Z. Flicker, Linear forms on arithmetic Abelian varieties: ineffective bounds, Mémoires de la S. M. F. 2e série, 1980, 41-47. [10] C. Fuchs and D. H. Pham, Commutative algebraic groups and p-adic linear forms, Acta Arith. 169 (2015), 115-147. [11] C. Fuchs and D. H. Pham, The p-adic analytic subgroup theorem revisited, p-Adic Number, Ultrametric Analysis and Applications 7 (2015), 143-156. [12] A. Günther, Über transzendente p-adische Zahlen. I, J. reine Angew. Math. 192 (1953), 155-166. [13] E. Lutz, Sur l’équation Y 2 = AX 3 − AX − B dans les corps p-adiques, J. reine angew. Math. 177 (1937), 238-247. [14] K. Mahler, Ein Beweis der Transzendenz der P -adischen Exponentialfunktion, J. reine angew. Math. 169 (1932), 61-66. [15] K. Mahler, Über transzendente P -adische Zahlen, Compositio Math. 2 (1935), 259-275. [16] T. Matev, The p-adic analytic subgroup theorem and applications, http://arxiv.org/pdf/1010.3156v1.pdf. [17] A. M. Robert, A course in p-adic analysis, GTM 198, Springer-Verlag, 2000. [18] G. Veldkamp, Ein Transzendenz-Satz für p-adische Zahlen, J. London Math. Soc. 15 (1940), 183-192. [19] A. Weil, Sur les fonctions elliptiques p-adiques, Note aux C.R. Acad. Sc. Paris 203 (1936), 22-24. [20] G. Wüstholz, Some remarks on a conjecture of Waldschmidt, Diophantine approximations and transcendental numbers, Progress in Mathematics 31, Birkhäuser Boston, Boston, MA, (1983), 329-336. [21] G. Wüstholz, Algebraische Punkte auf Analytischen Untergruppen algebraischer Gruppen, Ann. of Math. 129 (1989), 501-517. [22] G. Wüstholz, Multiplicity estimates on group varieties, Ann. of Math. 129 (1989), 471-500. SOME APPLICATIONS OF THE p-ADIC ANALYTIC SUBGROUP THEOREM Clemens Fuchs Department of Mathematics University of Salzburg Hellbrunnerstr. 34 5020 Salzburg Austria Email: [email protected] Duc Hiep Pham University of Education Vietnam National University, Hanoi 144 Xuan Thuy, Cau Giay, Hanoi Vietnam Email: [email protected] 7