View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF NUMBER THEORY 35, 109-127 (1990) Algebraic Groups and Small Transcendence Degree, II ROBERT TUBBS* Department of Mathematics, University of Colorado. Campus Box 426, Boulder, Colorado 80309 Communicated by M. Waldschmidt Received December 31, 1987; revised June 20, 1989 We continue our investigation into the algebraic independence of two numbers associated with a one-parameter subgroup of a commutative algebraic group. Here we focus on numbers which are associated with either algebraic points or torsion points on the algebraic group. 0 1990 Academic press, hc. In [11] we investigated the algebraic independence of two numbers associated with a one-parameter subgroup of a commutative algebraic group defmed over an arbitrary subfield of the complex numbers. The results of [ 1 l] were rather general (see below); no hypotheses were made concerning the arithmetic nature of the values under consideration. In this paper we investigate the algebraic independence of numbers associated with algebraic and/or torsion points of a commutative algebraic group. We retain several features of [ 111, in particular our numbers are associated with a one-parameter subgroup of an algebraic group, and the algebraic group is given as the product of a general commutative algebraic group and powers of the additive and the multiplicative groups of complex numbers. Specifically, suppose that G is a commutative algebraic group of dimen- sion da 1 which is defined over an arbitrary subfield K of C. Suppose further that G=G$xG$xG*, (1) where G$’ (respectively, G$) is the maximal unipotent (respectively, multi- plicatipe) factor of G, and G2 is defined over K with dim G2 = d2. Note that d=d,,+d,+d,. Let 4: C + G(C) be an analytic homomorphism with 4(C) Zariski dense in G(C); and, let &(C) denote the tangent space of G(C) at its identity * Research supported in part by a grant from the National Science Foundation. 109 0022-314X/90 $3.00 Copyright Q 1990 by Academic Press, Inc. All rights of reproduction in any form reserved 110 ROBERT TUBBS element. Then there exists a linear map L: C -P F&C) such that 4(z) = exp, 0 L(z); here exp, is the exponential map of G, exp,: &(C) + G(C). Denote the dimension of the vector space of least dimen- sion which is defined over K and which contains L(C) by n. Assume that y,, . .. y, are Z-linearly independent complex numbers such that Y=y,Z+ ... +y,z (2) satisfies d(Y) E G(K). The general problem in [ 1 l] was to find conditions on I, IZ, d,, di , and d2 which imply that K contains at least two algebraically independent numbers. It was shown there that if Z(d-n)>I+2d,+d,+c (3) with 1 if d,=n=l &= 0 otherwise then trans dego K 2 2. Moreover, if Y n ker 4 # 0 and 21(d-n)>I+2d,+d, (4) then trans dego K > 2. One goal of this paper is to study the algebraic independence of values associated with algebraic points on a commutative algebraic group defined over Q. We will see below that in several cases relating to algebraic points the inequality (3) above with E = 0 implies that trans deg, Ka 2 even though do = n = 1. These results have several consequences concerning numbers associated with algebraic numbers or algebraic points on an elliptic curve; and, among other things provide generalizations of the so-called Brownawell-Waldschmidt Theorem (see [l, 151, or Section 2 below). To give these results a uniform formulation suppose that we have a more refined decomposition of G than in (1). More precisely, suppose that G=Gf’xG:xG;xG*, (5) where G; is a commutative algebraic group defined over K with dim G; = dg, and G,=G?xG$xG;, SMALL TRANSCENDENCE DEGREE, II 111 where Gi is defined over Q and dim G;’ = 6,. In particular, we no longer assume that G$I (resp., C$) is the maximal unipotent (resp., multiplicative) factor of G but that G$‘x G: (resp. Gd,‘x G$) is. Note that with d, = d, + 8; + 6, + 6, we still have d = do + d, + dz. Suppose that 4: C + G(C) is an analytic homomorphism with d(C) Zariski dense in G(C). Suppose moreover that &(O)EF~(K); i.e., in the notation above n = 1. With Y as in (2) above we also assume that 4(Y) c G(K). We note that since d(C) is Zariski dense in G(C) it follows that d,+6,< 1. We begin with a result which concerns the situation where all of the points under consideration are associated with algebraic points on G2. Throughout, this paper rr : G + Gz will be the canonical projection mapping. THEOREM 1. If dim G, > 6, and ~0 i(yi) E G,(Q) (1 <i<l) then l(d- l)>Z+2(dg+6,)+(d, +6,) (7) implies that trans deg, K > 2. The next theorem involves only one algebraic point, but requires additional hypotheses on G. THEOREM 2. Suppose that d, = 0 and d > d, + 2. If ~~~(Y~)EGz(Q) then l(d-l)>I+26,+(d,+6,) (8) implies that trans deg, K 2 2. These theorems can be seen to be refinements of the results of [ 111 in that inequality (7) or (8) reduces to (3) with E = 0 when we take dg = 0 and G2 = G;. In a slightly different direction we also consider values associated with torsion points on G2 in the decomposition of G given by (5) and drop the hypothesis that G2 is defined over Q. If y, , . Y,, are Z-linearly independent complex numbers with 112 ROBERTTUBBS then there exist nonzero integers n,, . n,, such that ni yi E ker(7t 0 #), l<i<h. Since rz 0 4 ; C -P G2(C) is an analytic homomorphism, whose image is not trivial, it follows that h < 2. We take the cases h = 1 or h = 2 into account in our next theorems. THEOREM 3. Assume that G is defined over K with d, = d, = 0 and that 4 and Y are as before with do = n = 1. Suppose further that 26, + 6, > 3. If q5(Y) E G(K) and ~~~(YI)~GzWL~ then 12 2 implies that trans dego K> 2. When h = 2 in the above notation, n 0 $(z) is a doubly periodic function of a single complex variable. Consequently we may as well assume that G2 = E is an elliptic curve. THEOREM 4. Assume that G = G, x G, x E with n = 1 and E defined over K, and that $ is as above. Suppose that y,, y2 are Z-linearly independent with ~~4(vi)~W%,s, i= 1,2. If 4(yi) E G(K), i= 1, 2, then trans dego K > 2. To conclude this introduction we thank Michel Waldschmidt for his many helpful comments on this project and the referee of an earlier version of this paper. 1. APPLICATIONS Let &I(Z) be a Weierstrass elliptic function with lattice of periods 9 = Zo, + Zo, and with invariants g,, g,. Let U(Z)’ denote the associated Weierstrass sigma function and let h(z) = d(z), f(z)=~3(z) &e), g(z) = a3(z) P'(Z). SMALL TRANSCENDENCEDEGREE,II 113 Then p(z) = (h(z), j(z), g(z)) gives a parametrization of an elliptic curve E defined over Q(gz, g3). COROLLARY 1. Suppose that u and v are Z-linearly independent complex numbers with p(u) E E(K),,,, . Then at least two of the finite values among a, g,, n, u, v, e(+)“, 64~) are algebraically independent. ProojI Let G = G, x G, x E and define 4 : C --f G(C) by d(z) = (1, 2, 1, e(+)‘, p(z)). The algebraic independence of the functions z, e(ni’u)z, and p(z) over C implies that d(C) is Zariski dense in G(C). Thus the hypotheses of Theorem 3 are satisfied with y, = u and y2 = v. For example, suppose that g, and g, are algebraic. If c1 is an algebraic number, a # 0, 1, such that (log a)/zi# Q, put v = (u/xi) log a in Corollary 1. Then at least two of z log a 4 n, log a, p ( > are algebraically independent. In the next three applications we retain the hypothesis that g, and g, are algebraic. COROLLARY 2. Suppose that u,, u2, u3 are Z-linearly independent complex numbers with p(u,) E E(Q) f or i = 1,2,3. Then for any fl# 0 at least two of are algebraicaify independent, Proof: Take G and 4 as in the proof of Corollary 1, with e@ replacing e(ni’u)r. The hypotheses of Theorem 1 are satisfied and the corollary follows. For g(z) as above let r=oz/w 1 ; p(z) has complex multiplications precisely when t is imaginary quadratic. Put K, =Q(z) in this case and K, = Q otherwise. The significance of K, lies in the fact that p(z) and @(az) are algebraically dependent functions over C(z) exactly when a E K,. 114 ROBERT TUBBS COROLLARY 3. Suppose that @[z) has complex multiplication and that g, and g, are algebraic. Then at least two of ml,e,e’ are algebraically independent. Proof: Apply Theorem 4 to G = G, x G, x E with y, = 1, yZ = r, and d(z) = (1, z, 1, ez9P(Q~z)).