Elliptic Logarithms Research Articles

Monodromy of double elliptic logarithms

We determine the relative monodromy group of abelian logarithms with respect to periods in the cases of fibered products of elliptic schemes. This gives rise to a result stronger than a theorem due to Y. André and implies in particular the algebraic independence of the logarithm of any non-torsion section and the periods. We then conjecture an analogous result for the general case of an abelian scheme of arbitrary relative dimension. This generalizes a theorem of Corvaja and Zannier which determines the said group in the case of a single elliptic scheme.

ELLIPTIC CURVES AND -ADIC ELLIPTIC TRANSCENDENCE

AbstractWe prove a necessary and sufficient condition for isogenous elliptic curves based on the algebraic dependence of p-adic elliptic functions. As a consequence, we give a short proof of the p-adic analogue of Schneider’s theorem on the linear independence of p-adic elliptic logarithms of algebraic points on two nonisogenous elliptic curves defined over the field of algebraic numbers.

A note on balancing binomial coefficients

In 2014, T. Komatsu and L. Szalay studied the balancing binomial coefficients. In this paper, we focus on the following Diophantine equation $$\binom{1}{5}+\binom{2}{5}+...+\binom{x-1}{5}=\binom{x+1}{5}+...+\binom{y}{5}$$ where $y>x>5$ are integer unknowns. We prove that the only integral solution is $(x,y)=(14,15)$. Our method is mainly based on the linear form in elliptic logarithms.

Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker’s method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker’s method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.

The complex AGM, periods of elliptic curves over C and complex elliptic logarithms

We give an account of the complex Arithmetic–Geometric Mean (AGM), as first studied by Gauss, together with details of its relationship with the theory of elliptic curves over C, their period lattices and complex parametrisation. As an application, we present efficient methods for computing bases for the period lattices and elliptic logarithms of points, for arbitrary elliptic curves defined over C. Earlier authors have only treated the case of elliptic curves defined over the real numbers; here, the multi-valued nature of the complex AGM plays an important role. Our method, which we have implemented in both MAGMA and Sage, is illustrated with several examples using elliptic curves defined over number fields with real and complex embeddings.

Upper bounds for the height of S-integral points on elliptic curves

We establish new upper bounds for the height of the S-integral points of an elliptic curve. This bound is explicitly given in terms of the set S of places of the number field K involved, but also in terms of the degree of K, as well as the rank, the regulator and the height of a basis of the Mordell–Weil group of the curve. The proof uses the elliptic analogue of Baker’s method, based on lower bounds for linear forms in elliptic logarithms.

Integral points of a modular curve of level 11

Using lower bounds for linear forms in elliptic logarithms we determine the integral points of the modular curve associated to the normalizer of a non-split Cartan group of level 11. As an application we obtain a new solution of the class number one problem for complex quadratic fields.

LINEAR FORMS IN TWO ELLIPTIC LOGARITHMS IN THE <I>p</I>-ADIC CASE

The principal aim of the present paper is to provide a new lower bound for linear forms in two p-adic elliptic logarithms. We refine a result due to Remond and Urfels. Our improvement lies in the dependence on the height of algebraic coefficients of the linear forms. Our bound is the best possible one concerning the height of the coefficients, where all the relevant constants are explicit. We use the argument that relies on the interpolation method of Laurent, on the variable change introduced by Chudnovsky, and on Faa di Bruno’s formula adapted to matrices whose elements are p-adic elliptic logarithmic functions. The bounds would be useful to determine the set of S-integer points on elliptic curves defined over a number field.

Linear forms in elliptic logarithms

Linear forms in elliptic logarithms

Approximation diophantienne sur les courbes elliptiques à multiplication complexe

Let E be an elliptic curve with complex multiplication, defined over Q . We consider linear forms on Lie( E n) with coefficients in the CM field of E . Within this framework, we present a new measure of linear independence for elliptic logarithms in (log b)(log a) n . Like recent advances in this domain (works by Ably, David, Hirata-Kohno), our result is best possible in terms of the height of the linear forms (log b) while providing a better estimate in the height of algebraic points considered (log a), removing a term in loglog a. To cite this article: M. Ably, É. Gaudron, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Interpolation Formulas and Auxiliary Functions

We prove an interpolation formula for “semi-cartesian products” and use it to study several constructions of auxiliary functions. We get in this way a criterion for the values of the exponential map of an elliptic curve E defined over Q. It reduces the analogue of Schanuel's conjecture for the elliptic logarithms of E to a statement of the form of a criterion of algebraic independence. We also consider a construction of auxiliary function related to the four exponentials conjecture and show that it is essentially optimal. For analytic functions vanishing on a semi-cartesian product, we get a version of the Schwarz lemma in which the exponent involves a condition of distribution reminiscent of the so-called technical hypotheses in algebraic independence. We show by two examples that such a condition is unavoidable.

Integral Points on Elliptic Curves Defined by Simplest Cubic Fields

Let f(X) be a cubic polynomial defining a simplest cubic field in the senseof Shanks. We study integral points on elliptic curves of the form y2 = f(X). We compute the complete list of integral points on these curves for the values of the parameter below 1000. We prove that this list is exhaustive by using the methods of Tzanakis and de Weger, together with bounds on linear forms in elliptic logarithms due to S. David. Finally, we analyze this list and we prove in the general case the phenomena that we have observed. In particular, we find all integral points on the curve when the rank is equal to 1.

$S$-integral points on elliptic curves - Notes on a paper of B. M. M. de Weger

In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and p-adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.

Formes linéaires de logarithmes de points algébriques sur une courbe elliptique de type $CM$

We give a lower bound for a linear form in elliptic logarithms of algebraic points on an elliptic curve with complex multiplication and which is defined over Q ¯. The dependence of this bound on the height of this linear form is optimal up to constant.

Elliptic binomial diophantine equations

The complete sets of solutions of the equation(nk)=(mℓ)\binom {n}{k} = \binom {m}{\ell }are determined for the cases(k,ℓ)=(2,3)(k,\ell ) = (2,3),(2,4)(2,4),(2,6)(2,6),(2,8)(2,8),(3,4)(3,4),(3,6)(3,6),(4,6)(4,6),(4,8)(4,8). In each of these cases the equation is reduced to an elliptic equation, which is solved by using linear forms in elliptic logarithms. In all but one case this is more or less routine, but in the remaining case ((k,ℓ)=(3,6)(k,\ell ) = (3,6)) we had to devise a new variant of the method.

Solving Elliptic Diophantine Equations Avoiding Thue Equations and Elliptic Logarithms

We determine the solutions in integers of the equation y2 = (x + p)(x2 + p2) for p = 167, 223, 337, 1201. The method used was suggested to us by Yu. Bilu, and is shown to be in some cases more efficient than other general purpose methods for solving such equations, namely the elliptic logarithms method and the use of Thue equations.

Integral points on elliptic curves over number fields

In recent years there has been an interest in using elliptic logarithms to find integral points on elliptic curves defined over the rationals, see [23], [17], [6] and [12]. This has been partly due to work of David [5], who gave an explicit lower bound for linear forms in elliptic logarithms. Previously, integral points on elliptic curves had been found by Siegel's method; that is, a reduction to a set of Thue equations which could be solved, in principle, by the methods in [19]. For examples of this method see [3], [7], [16], [18], [21], [22] and [8]. Other techniques can be used to find all integral points in some special cases, see, for instance, [14].

On Sums of Consecutive Squares

In this paper we consider the problem of characterizing those perfect squares that can be expressed as the sum of consecutive squares where the initial term in thus sum isk2. This problem is intimately related to that of finding all integral points on elliptic curves belonging to a certain family which can be represented by a Weierstraß equation with parameterk. All curves in this family have positive rank, and for those of rank 1 a most likely candidate generator of infinite order can be explicitly given in terms ofk. We conjecture that this point indeed generates the free part of the Mordell–Weil group and give some heuristics to back this up. We also show that a point which is modulo torsion equal to a nontrivial multiple of this conjectured generator cannot be integral. Forkin the range 1⩽k⩽100 the corresponding curves are closely examined, all integral points are determined and all solutions to the original problem are listed. It is worth mentioning that all curves of equal rank in this family can be treated more or less uniformly in terms of the parameterk. The reason for this lies in the fact that in Sinnou David's lower bound of linear forms in elliptic logarithms—which is an essential ingredient of our approach—the rank is the dominant factor. Also the extra computational effort that is needed for some values ofkin order to determine the rank unconditionally and construct a set of generators for the Mordell–Weil group deserves special attention, as there are some unusual features.

Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations

In a recently published paper [ST], R.J. Stroeker and the author describe in detail – with examples included – a general method for computing all integer solutions of a Weierstrass equation, which defines over Q an elliptic curve. A little later, J.Gebel, A.Petho and H.G.Zimmer worked independently on similar lines and solved a number of impressive numerical examples (see [GPZ]); for some history of the main ideas underlying that method see the Introduction in [ST]. The main advantage of the method, which combines the Arithmetic of Elliptic Curves with the Theory of Linear Forms in Elliptic Logarithms , is that it is very easily – almost mechanically – applicable, once one knows a Mordell-Weil basis for the elliptic curve associated with the given equation. In this way, one can solve, with reasonable effort, many elliptic diophantine equations that are “of the same type”; see an interesting application in [Str], where fifty elliptic equations are solved. On the other hand, the (theoretically) ineffective part of the method (i.e. the computation of a Mordell-Weil basis), which may be very easy but also very difficult or, in exceptional cases, even impossible, is well separated from the rest steps of the method, so that, if necessary, one can turn for it to the specialists, without asking from them that they be involved in anything else related to the specific equation he wants to solve, but the computation of the Mordell-Weil basis. In this paper we extend the aforementioned method to the class of quartic elliptic equations , i.e. we describe a general practical method for computing explicitly all integral solutions of equations of the form V 2 = Q(U), where Q(U) ∈ Z[U ] is a quartic polynomial ∗e-mail: tzanakis@math.uoc.gr †I thank Dr. Dick Nickalls who pointed me out a number of computational inaccuracies in Examples 2,4 and 7.

Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms

In order to compute all integer points on a Weierstras equation for an elliptic curve E/Q, one may translate the linear relation between rational points on E into a linear form of elliptic logarithms. An upper bound for this linear form can be obtained by employing the Neron-Tate height function and a lower bound is provided by a recent theorem of S. David. Combining these two bounds allows for the estimation of the integral coefficients in the group relation, once the group structure of E(Q) is fully known. Reducing the large bound for the coefficients so obtained to a manageable size is achieved by applying a reduction process due to de Weger. In the final section two examples of elliptic curves of rank 2 and 3 are worked out in detail.