TAILIEUCHUNG - 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. | JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci. 2014 Vol. 59 No. 7 pp. 3-8 This paper is available online at http 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. 1. Introduction 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 field C 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 field C 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 of C which is denoted by Cp . Note that by construction Cp is an algebraically closed field containing Q therefore one can analogously give the definition of p-adic transcendental numbers as follows. An element α Cp is called p-adic transcendental if P α ̸ 0 for any non-zero polynomial P T Q T . Transcendence theory in both domains C 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 .

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