Elliptic curves and p-Adic linear independence

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 ∈ C p 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 instance, if α is a number (in C or Cp) such 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 results in this direction is due to A. Baker. Namely, in 1967 he proved the following theorem (see [1])

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JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 3-8 This paper is available online at 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 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 Hiep such 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 results in 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 that logα1, . . . , logαn are linearly independent over Q, then 1, logα1, . . . , logαn are linearly independent over Q. J. Coates extended the Baker’s method to the p-adic case in 1969 (see [5]). It is natural to think of similar problems in the language of arithmetic algebraic geometry, in particular, for the elliptic curves over number field. Such a theory is called elliptic linear independence 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 also in real life (e.g. in cryptography). The aim of this paper is to formulate and prove a new result on elliptic linear independence 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 the p-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 are linearly 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 detailed theory). 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 overR. TheWeierstrass ℘-function (relative to Λ) is defined by the series ℘(z) := ℘(z; Λ) := 1 z2 + ∑ w∈Λ\{0} ( 1 (z − w)2 − 1 w2 ) . 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)− g3 with 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∈Λ\{0} ω−2k, ∀k ≥ 1. 4 Elliptic curves and p-adic linear independence The quantities g2, g3 are said to be the invariants of ℘. The Laurent expansion of ℘ at 0 is given by ℘(z) = 1 z2 + ∞∑ n=1 bnz 2n where b1 = g2 20 , b2 = g3 28 , bn = 3 (2n+ 3)(n− 2) n−2∑ k=1 bkbn−k−1, ∀n ≥ 3. By induction we see that for n ≥ 1 there are polynomials of two variables Pn(X,Y ) with coefficients in Q such that bn = Pn(g2, g3). In particular, if g2, g3 ∈ Q then bn ∈ Q for n ≥ 1. Definition 2.1. Let K be a field. We call E an elliptic curve defined over K if E is a projective algebraic group of dimension 1 defined over K, i.e. an abelian variety of dimension 1 defined over K. WhenK is a subfield of C, one can characterize E as a smooth projective curve in the projective space P2K , namely it is defined by an equation of the form Y 2Z = 4X3 − aXZ2 − bZ3 with a, b ∈ K such that a3 ̸= 27b2. In addition, there is a unique lattice Λ in C satisfying g2(Λ) = a and g3(Λ) = b. The Lie algebra of E(C) is canonically isomorphic to C, and the exponential map expE is given by expE : C→ E(C), z 7→ (℘(z) : ℘′(z) : 1). One can show that Λ = ker expE = {ω ∈ C;℘(z + w) = ℘(z)} = Zω1 + Zω2 with ωi := ∫ ∞ ei dx√ 4x3 − ax− b (i = 1, 2) (called fundamental periods), where e1, e2, e3 are the roots of the polynomial 4x3−ax−b. The map ϕ : C/Λ→ E(C), z 7→ (℘(z) : ℘′(z) : 1) induced by expE is a complex analytic isomorphism of complex Lie groups. We also say that the Weierstrass elliptic function relative to the lattice Λ is the Weierstrass elliptic function associated with E. Let End(E) denote the ring of endomorphisms of E, and define the field of endomorphisms of E as the quotient field L := End(E) ⊗Z Q. The map [ ] : Z→ End(E), n 7→ [n] where [n] : E → E is the multiplication by n, is injective. If End(E) is strictly larger than Z then we say that E has complex multiplication (this is the so-called CM case or CM type). In this case the quotient τ := ω1/ω2 is a quadratic number, and the field L is Q(τ). 5 Pham Duc Hiep In the p-adic domain, it is known that there is a p-adic analogue of elliptic function which was constructed by E. Lutz and A. Weil (see [7] and [10]). Let E be an elliptic curve defined over Cp of the form Y 2T − 4X3 − g2XT 2 − g3T 3 = 0, where g2, g3 ∈ Cp such that g32 − 27g23 ̸= 0. The following differential equation y′(z) = ( 1− g2 4 y4(z)− g3 4 y6(z) )1/2 , y(0) = 0 admits the solutions φ(z) and −φ(z) which are analytic on the disk Dp := { z ∈ Cp; |1/4|pmax{|g2|1/4p , |g3|1/6p }z ∈ B(rp) } , here B(rp) := {x ∈ Cp; |x|p < p− 1 p−1}. The disk Dp is called the p-adic domain of E. Put ℘p := φ−2, we get ℘′2p = 4℘ 3 p − g2℘p − g3. This leads to the following definition. Definition 2.2. We call ℘p the (Lutz-Weil) p-adic elliptic function associated with the elliptic curve E. The function ℘p(z) is analytic on Dp \ {0}, and can be represented by the p-adic power series ℘p(z) = 1 z2 + ∞∑ n=1 Pn(g2, g3)z 2n with the polynomials Pn given as in the complex case above. 3. Main result We are now interested in the linear independence of elliptic functions. The elliptic analogue of Baker’s theorem on the linear independence of logarithms was first proved by Masser in 1974 (see [8]) in the CM case. Masser and Bertrand in 1980 completed this for the non-CM case (see [4]). To describe the theorems below, we put A := Λ ∪ {u ∈ C \ Λ;℘(u) ∈ Q}; recall that L denotes the field End(E)⊗Z Q. Theorem 3.1 (Bertrand-Masser). If elements u1, . . . , un ∈ A are linearly independent over L, then 1, u1, . . . , un are linearly independent over Q. In the p-adic domain, Bertrand in 1976 formulated and proved a p-adic analogue for elliptic curves with complex multiplication (see [3]) which deals with the homogeneous case. Similar to the complex case, we denote the set of algebraic points of the p-adic Weierstrass function ℘p by Ap := {0} ∪ {u ∈ Dp \ {0};℘p(u) ∈ Q}. 6 Elliptic curves and p-adic linear independence Theorem 3.2 (Bertrand). Assume that E has complex multiplication. If elements u1, . . . , un ∈ Ap are linearly independent overL, then u1, . . . , un are linearly independent over Q. We generalize Theorem 3.2 to arbitrary elliptic curves which is given by the following theorem. Theorem 3.3. Elements u1, . . . , un ∈ Ap are linearly independent over Q if and only if they are linearly independent over L. Proof. The implication ”⇒” is trivially true because L is a subfield of Q. We prove the implication “⇐”. Suppose that u1, . . . , un are linearly dependent over Q. This means that there is a non-zero linear form in n variables l(X1, . . . , Xn) = a1X1 + · · ·+ anXn with a1, . . . , an ∈ Q such that l(u1, . . . , un) = 0. Let G = En be the direct product of n-copies of the elliptic curve E. Then G is commutative and defined over Q. The Lie algebra Lie(G) is identified with Qn, hence Lie(G(Cp)) = Lie(G)⊗Q Cp = Cnp . We have G(Cp)f = E(Cp)n, and the p-adic logarithm map of G is given by logG(Cp) : C n p → Lie(G(Cp)), (z1, . . . , zn) 7→ (logE(Cp)(z1), . . . , logE(Cp)(zn)). Since ℘p(u1), . . . , ℘p(un) are in Q, the point γ := (expE(Cp)(u1), . . . , expE(Cp)(un)) is an algebraic point of G(Cp)f . The value of logG(Cp)(γ) is( logE(Cp) ( expE(Cp)(u1) ) , . . . , logE(Cp) ( expE(Cp)(un) )) = (u1, . . . , un) ̸= 0 since u1, . . . , un are linearly independent over L. Let V := {v ∈ Qn; l(v) = 0} be theQ-vector space defined by l. We find that logG(Cp)(γ) is a non-zero point in V⊗QCp. The p-adic analytic subgroup theorem (see [6]) says that there is an algebraic subgroup H of G of positive dimension defined over Q such that γ ∈ H(Q) and Lie(H) ⊆ V . It is known that H is isogeneous to Em for some positive integer m ≤ n. Using the same argument as in the proof of Theorem 6.2 in [2], there is an element π of the algebra of endomorphisms EndQ(G) := End(G)⊗Z Q 7 Pham Duc Hiep of G given by the projection from G to H . One can identify elements of EndQ(G) to be square matrices of order n× n with coefficients in End(E)⊗Q Z = L. This means that the endomorphism φ := idG − π can be written as an n × n matrix A with entries in L. On the other hand H is a proper subgroup of G since dimQ Lie(H) ≤ dimQ V = n− 1 < n = dimQ Lie(G). Then the matrix A is non-zero. The subgroup H is isogeneous to Em, and this shows that the Lie algebra of H can be identified with that of Em. The Lie algebra of H then is the kernel of the endomorphism of Lie(G) given by the matrix A. Further since γ ∈ H one gets (u1, . . . , un) = logG(Cp)(γ) ∈ Lie(H)⊗Q Cp. This means that A(u1, . . . , un) = 0. Since A is non-zero this provides a non-trivial linear dependence of the elements u1, . . . , un over L. This contradiction proves the theorem. REFERENCES [1] A. Baker, 1966. Linear forms in the logarithms of algebraic numbers I, II, III, IV, Mathematika 13, 204-216; 1967, Mathematika, 14, 102-107; 1967, Mathematika 14, 220-228; 1968, Mathematika 15, 204-216. [2] A. Baker and G. Wu¨stholz, 2007. Logarithmic forms and Diophantine geometry, New Mathematics Monographs, Vol. 9, Cambridge University Press. [3] D. Bertrand, 1976. Problèmes arithmétiques liés à l’exponentielle p-adique sur les courbes elliptiques, C.R. Acad. Sci. Paris Sér. A-B 282, No. 24, Ai, A1399-A1401. [4] D. Bertrand and D. Masser, 1980. Linear forms in elliptic integrals. Invent. Math. 58, No. 3, 283-288. [5] J. Coates, 1969. The effective solution of some diophantine equations. PhD Dissertation, University of Cambridge. [6] C. Fuchs and D. H. Pham, 2014. The p-adic analytic subgroup theorem revisited, preprint. [7] E. Lutz, 1937. Sur l’équation Y 2 = AX3 − AX − B dans les corps p-adiques. J. reine angew. Math. 177, 238-247. [8] D. Masser, 1975. Elliptic functions and transcendence. Lecture notes in Mathematics 437 Springer-Verlag, Berlin. [9] J. H. Silverman, 1986. The arithmetic of elliptic curves. Springer-Verlag, New York. [10] A. Weil, 1936. Sur les fonctions elliptiques p-adiques. Note aux C.R. Acad. Sc. Paris 203, 22-24. 8
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