Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 3-8 This paper is available onUne at http://stdb.hiiue.edu.vn
ELLIPTIC CURVES AND p-ADIC LINEAR INDEPENDENCE Pham Due Hiep
Faculty of Matfiematics, Hanoi National University of Education
Abstract. Let £ 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, jt-adic.
1. Introduction
The problem of fmding roots of a given polynomial is always a natural and big question in mathematics. It is weU-known that every polynomial with complex coefficients of positive degree has aU 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 ntmibers, that is given a G C, one may naturally ask whether there is a non-zero polynomial P in one variable with rational coefficients such that P{a) = 0?
If there exists such a P we call a algebraic, otherwise we call a (complex) franscendental. The most prominent examples of ttanscendental numbers are e (proved by C. Hermite in 1873) and TT (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 algebraicaUy closed field containing Q, therefore one can analogously give the definition of p-adic franscendental numbers as foUows.
Anelementa e Cp is called (p-adic) transcendentalif P{a) =^ 0 for any non-zero polynomial P{T) 6 Q[T].
Transcendence theory in both domains C and Cp has been smdied 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 a is a number (in C or Cp)
Received August 12, 2014. Accepted September 11, 2014.
Contact Pham Due Hiep, e-mail address: [email protected]
such that 1 and a are linearly independent over Q, then a must be transcendental. Indeed, it follows from the trivial equality: Q • 1 - 1 • Q = 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 ai,... ,an are algebraic numbers, neither 0 nor 1, suclithat log c ^ i , . . . , log a „ are linearly independent over Q, then 1, log o i , . . . , log On 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 geomefry, 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. confribution to solve Fermat's Last Theorem), but also in real Ufe (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 pp be the p-adic We ierstrass function of E. Denote by Ap the set of algebraic points of pp. Then, elements ui,...,Un S Ap are linearly independent over Q if and only ifui,...,u,n 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 A be a lattice in C, i.e. A is a set of the form A = { m a -|- n^\ m, TI € Z } where a,0EC such that a , 0 are linearly independent over E . The Weiersfrass p-function (relative to A) is defined by the series
p{z):=p{z;A):=-^+ Y " I- x ^ - - ? ) - z-^ ^ , \\Z — VJY tip-1
The function p is meromorphic on C, analytic on C \ A and periodic with period ly € A.
We also call A the lattice of periods of p . Furthermore one has ( p ' M r = 4 ( s , ( z ) f - 9 , p ( z ) - S 3 with
92 ••= 52(A) = 60G4(A), P3 := Qzi^) - 140G6(A), where the Eisenstein series of weight 2k (relative to A) are defined by
G2fc(A) := Y, '^"^^ ^'^ ^ 1-
The quantities g^, g^ are said to be the invariants of p . The Laurent expansion of p at 0 is given by
^ + E ' -
where
By induction we see that for ii> 1 there are polynomials of two variables Pn{X, Y) with coefficients in Q such that &„ = Pnig2, Pa)- In particular, if 52153 € Q then 6„ G Q for n>l.
Definition 2.1. Let K be afield. 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 I defined over K.
When A" is a subfield of C, one can characterize .E as a smooth projective curve in the projective space P ^ , namely it is defined by an equation of the form Y'^Z = AX'^ — aXZ^ - bZ^ with a,b ^ K such that a^ ^ 276^. In addition, there is a unique lattice A in C satisfying 52(A) = a and 53(A) = b. The Lie algebra of E{C) is canonically isomorphic to C, and the exponential map exp^ is given by
exp£ : C -^ E{C), z ^ {p{z) : p'{z) : 1).
One can show that A = k e r e x p ^ = {w e C; p{z -\-w) = p{z)} = 1hJ\ -F li^i with
/
'^ dr
I y/Ax^ — a.
(called fundamental periods), where ei, 62,63 are the roots of the polynomial 4x^ ~ax — b.
The map
<^ : C / A ^ £ ( C ) , z ^ {p{z) : p'{z) : 1)
induced by expjc; is a complex analytic isomorphism of complex Lie groups. We also say that the Weierstrass elliptic function relative to the lattice A is the Weierstrass elliptic function associated with E. Let End(£^) denote the ring of endomorphisms of E, and define the field of endomorphisms of E as the quotient field L := End(£') ®z Q- The map
[ ] : Z -J- E n d ( £ ) , n ^ [n]
where [n] : E —> E is the multipUcation by n, is injective. If End(£') 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 r := wi/w2 is a quadratic number, and the field L is Q ( T ) .
Pham Due 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^T - AX^ - g2XT^ - g^T^ = 0,
where 32,53 e Cp such that g^ - 27gj ^ 0. The following differential equation
admits the solutions if{z) and —^(z) which are analytic on the disk 'Dp := {z e Cp;|l/4|pmax{|g2iyM53iyn^ ^ B{rp)},
here B{Tp) := {x 6 Cp; |x|p < p " ^ } . The disk Dp is called the p-adic domain of S.
Put pp := tfi^"^, we get p'^ = 4p^ - g2pp - 53- This leads to the following definition.
Definition 2.2. We call pp the (Lutz-Weil) p-adic elliptic function associated with the elliptic curve E.
The function pp{z) is analytic on Dp \ {0}, and can be represented by the p-adic power series
Pp(^) = ^ + E^"(52,53)^^"
^ n=l
with the polynomials Pn given as in the complex case above.
3. Main result
We are now interested in the Hnear independence of elliptic functions. The elliptic analogue of Baker's theorem on the linear independence of logarithms was first proved by Masserin 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 U {u e C \ A; p{u) C Q};
recall that L denotes the field End(£;) ®z Q.
Theorem 3.1 (Bertrand-Masser). If elements u i , . . . , tx„ e A are linearly independent over L, then 1, u i , . . . , u^ are linearly independent over Q.
In the p-adic domain, Bertiand in 1976 formulated and proved a p-adic analogue for elliptic curves with complex multipUcation (see [3]) which deals witii the homogeneous case. Similar to the complex case, we denote the set of algebraic points of the p-adic Weierstrass function pp by
Ap := {0} U {u € Dp \ {0}; Pp{u) e Q}.
Theorem 3.2 (Bertrand). Assume that E has complex multiplication. If elements ui,...,UneApare linearly independent over L, tJien u i , . . . , u^ are linearly independent over Q.
We generalize Theorem 3.2 to arbitrary elUptic curves which is given by the following theorem.
Theorem 3.3. Elements Ui,... ,u„ t 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 u i , . . . , Un are Hnearly dependent over Q. This means that there is a non-zero linear form in n variables
l{Xu . . . , X„) = aiXx + •••+ anXn with a i , . . . , a„ G Q such that
l[U\, . . . ,Un) = 0.
Let G = £^" 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 Q , hence
Lie(G(Cp)) = Lie(G) % C p = CJ.
We have G{Cp)f = E{CpY, and the p-adic logarithm map of G is given by
logG(Cp) • Cp -^ Lie(G(Cp)), (21,..., 2^) H-)- (log£(Cp)(2i), • • • - '^<^EE{<c^){zn))- Since pp(ui), • • •, pp(un) are in Q, the point
7 := (exp£(Cp)(ui), • • • ,exp£(Cp)(wn)) isanalgebraicpointof G(Cp)/.The valueof logG(Cp)(7) is
(log£(Cp) (exp£(Cp,(ui)),... ,logB(Cp) (exp£(Cp)(wn))) = (wi,.. -, w„) 7^ 0 since u^,..., Un, are linearly independent over L. Let
V •={ve'^\l{v) = Qi)
be the Q-vector space defined by I. We find that logG(Cp} (7) is a non-zero point in V®^p.
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 7 e i/(Q) and Lie(i7) C V, It is known that H is isogeneous to E"^ for some positive integer m < n. Using the same argument as in the proof of Theorem 6.2 in [2], there is an element TT of the algebra of endomorphisms
EndQ(G) :=End(G)®sQ
Pham Due Hiep
of G given by the projection from G to H. One can identify elements of EndQ(G; '" "^
square matrices of order n xn with coefficients in End(£) giQ Z = L.
This means that the endomorphism ^5 := idc - TT can be written as an « x n matrix A with entries in i . On the other hand ff is a proper subgroup of G since
dimgLie(H) < dim^ V = ii - 1 < n = dimQLie{G).
Then the matrix A is non-zero. The subgroup H is isogeneous to £ " , and this shows that the Lie algebra of H can be identified with fliat of £"". The Lie algebra of H then is the kernel of die endomorphism of Lie(G) given by die matrix yl. Further since 7 e H one gets
This means that
A{ui,... ,u„) = 0 .
Since A is non-zero this provides a non-trivial linear dependence of the elements
Ui,...,Un over L. This contradiction proves the theorem. D
REFERENCES
[1] A. Baker, 1966. Linear forms in the logarithms of algebraic numbers I, II, III, IV, Madiematika 13, 204-216; 1967, Mathematika, 14,102-107; 1967, Mathematika 14, 220-228; 1968, Madiematika 15, 204-216.
[2] A. Baker and G. Wiistholz, 2007. Logarithmic forms and Diophantine geometry.
New Mathematics Monographs, Vol. 9, Cambridge University Press.
[3] D. Bertrand, 1976. Problemes arithmetiques lies a Vexponentielle p-adique sur les courbes elliptiques, CR. Acad. Sci. Paris Ser. 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 Vequation Y^ = AX'' - AX - B dans les corps p-adiques. J.
reine angew. Madi. 177, 238-247.
[8] D. Masser, 1975. Elliptic functions and transcendence. Lecture notes in Mathematics 437 Springer-Verlag, Berlin.
[9] J. H. Silverman, 1986. 77ic arithmetic of elliptic curves. Springer-Verlag, New York.
[10] A. Weil, 1936. Sur lesfonctions elliptiques p-adiques. Note aux CR. Acad. Sc. Paris 203, 22-24.
