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.
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Elliptic curves and p-adic linear independence JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 3-8 This paper is available online at http://stdb.hnue.edu.vn 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 andbig question in mathematics. It is well-known that every polynomial with complexcoefficients of positive degree has all roots in the complex field C and in particular, sodoes every polynomial with rational coefficients of positive degree. Dually, to study the arithemetic of complex numbers, that is given α ∈ C, onemay naturally ask whether there is a non-zero polynomial P in one variable with rationalcoefficients 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 (provedby C. Hermite in 1873) and π (proved by F. Lindemann in 1882). Apart from the complexfield C, there is another important field, the so-called (complex) p-adic number field (firstdescribed by K. Hensel in 1897) for each prime number p. Namely, it is a p-adic analogueof C which is denoted by Cp . Note that by construction, Cp is an algebraically closed fieldcontaining Q, therefore one can analogously give the definition of p-adic transcendentalnumbers as follows. An element α ∈ Cp is called (p-adic) transcendental if P (α) ̸= 0 for any non-zeropolynomial P (T ) ∈ Q[T ]. Transcendence theory in both domains C and Cp has been studied and developedby many authors. In order to investigate the theory more deeply, one can naturally put theproblem in the context of linear independence. For instance, if α is a number (in C or Cp )Received August 12, 2014. Accepted September 11, 2014.Contact Pham Duc Hiep, e-mail address: phamduchiepk6@gmail.com 3 Pham Duc Hiepsuch that 1 and α are linearly independent over Q, then α must be transcendental. Indeed,it follows from the trivial equality: α · 1 − 1 · α = 0. One of the most celebrated resultsin this direction is due to A. Baker. Namely, in 1967 he proved the following theorem(see [1]).Theorem 1.1 (A. Baker). If α1 , . . . , αn are algebraic numbers, neither 0 nor 1, such thatlog α1 , . . . , log αn are linearly independent over Q, then 1, log α1 , . . . , log αn are linearlyindependent over Q. J. Coates extended the Baker’s method to the p-adic case in 1969 (see [5]). It isnatural to think of similar problems in the language of arithmetic algebraic geometry, inparticular, for the elliptic curves over number field. Such a theory is called elliptic linearindependence theory. Note that the theory of elliptic curves plays a very important role,not only in pure mathematics (e.g. contribution to solve Fermat’s Last Theorem), but alsoin real life (e.g. in cryptography). The aim of this paper is to formulate and prove a new result on elliptic linearindependence over p-adic fields which is given by the following theorem.Theorem 1.2. Let E be an elliptic curve defined over a number field and let ℘p be thep-adic Weierstrass function of E. Denote by Ap the set of algebraic points of ℘p . Then,elements u1 , . . . , un ∈ Ap are linearly independent over Q if and only if u1 , . . . , un arelinearly independent over the field of endomorphisms of E.2. The arithemtic of elliptic curves In this section, we briefly recall the theory of elliptic curves (see [9] for detailedtheory). Let Λ be a lattice in C, i.e. Λ is a set of the form Λ = {mα + nβ; m, n ∈ Z}where α, β ∈ C such that α, β are linearly independent over R. The Weierstrass ℘-function(relative to Λ) is defined by the series 1 ∑ ( 1 1 ) ℘(z) := ℘(z; Λ) := 2 + − . z (z − w)2 w2 w∈Λ\{0}The function ℘ is meromorphic on C, analytic on C \ Λ and periodic with period w ∈ Λ.We also call Λ the lattice of periods of ℘. Furthermore one has (℘′ (z))2 = 4(℘(z))3 − g2 ℘(z) − g3with g2 := g2 (Λ) = 60G4 (Λ), g3 := g3 (Λ) = 140G6 (Λ),where the Eisenstein series of weight 2k (relative to Λ) are defined by ∑ G2k (Λ) := ω −2k , ∀k ≥ 1. ω∈Λ\{0}4 Elliptic curves and p-adic linear independenceThe quantities g2 , g3 are said to be the invariants of ℘. The Laurent expansion of ℘ at 0 isgiven by 1 ∑∞ ℘(z) = 2 + bn z 2n z n=1where g2 g3 3 ∑ n−2 b1 = , b2 = , bn = bk bn−k−1 , ∀n ≥ 3. 20 28 (2n + 3)(n − 2) k=1By induction we see that for n ≥ 1 there are polynomials of two variables Pn (X, Y ) withcoefficients in Q such that bn = Pn (g2 , g3 ). In parti ...