Bucur and kedlaya apply their results to nding an upper bound for the smallest prime p n 1n 2 at which a pe 1 and a pe 2 are nonzero and of opposite sign. Add a section on szpiros conjecture and the abc conjecture. An elliptic curve test for mersenne primes benedict h. The local lehmer inequality for drinfeld modules introduction. In the general version of the main theorem, we must use line bundles which play. Jun 05, 2007 pdf file 292 kb article info and citation. The elliptic analogof lehmers conjecture says thathp.
Elliptic curves, hilbert modular forms, and the hodge. Introduction to elliptic curves to be able to consider the set of points of a curve cknot only over kbut over all extensionsofk. On lehmers conjecture for polynomials and for elliptic curves. Harris, bi elliptic curves and symmetric products, proc. Dyer and there is a deep conjecture known as the elliptic lehmer problem, see. The elliptic analogof lehmer s conjecture says thath. The theory of elliptic curves was essential in andrew wiles proof of fermats last theorem. Download citation on lehmer s conjecture for polynomials and for elliptic curves a number of authors have proven explicit versions of lehmer s conjecture for polynomials whose coefficients are.
Conjectures on distributions of primes associated with elliptic curves 29 3. Analogues of lehmers conjecture in positive characteristic. Rocky mountain journal of mathematics project euclid. A gentle introduction to elliptic curve cryptography. We will concentrate on the algebraic structures of groups, rings, and elds.
The question of stabilization of the euler factors of c t1. We then describe the mov attack, which is fast for. Moduli spaces and arithmetic dynamics, crm monograph series 30, ams, 2012. An explicit manindemjanenko theorem in elliptic curves volume 70 issue 5. Ek is nontorsion and is the image of a krational point under a nontrivial krational isogeny, then say that p is magni. Theorem let fbe a holomorphic newform of weight 2 with rational fourier coe. In section2, inspired by the model developed by lang and trotter in 19, we employ a probabilistic model to propose an explicit conjecture on the distribution of primes with two given traces for two. Computational problems involving the group law are also used in many cryptographic applications. In section 3, using a lehmer type result for elliptic curves. Earlier resultson lehmer s conjecture for the convenience of the reader, in this section we summarize some of the known results regarding lehmer s conjecture for number. Silverman 29 wieferichs criterion and the abcconjecture, j. On zagiers conjecture for base changes of elliptic curves. On the conjecture of birch and swinnertondyer for an. December 21, 2012 communicated by stephen lichtenbaum abstract.
Salem numbers, pisot numbers, mahler measure, and graphs mckee, james and smyth, chris, experimental mathematics, 2005. On a conjecture for the distributions of primes associated with elliptic curves jeremy porter for an elliptic curve eand xed integer r, lang and trotter have conjectured an asymptotic estimate for the number of primes p xsuch that the trace of frobenius a pe r. Elliptic curves and analogies between number fields and. Denote by mk the set of places of k and let s mk be a nonempty finite subset. Lehmer s conjecture, also known as the lehmer s mahler measure problem, is a problem in number theory raised by derrick henry lehmer. Letuscheckthisinthecase a 1 a 3 a 2 0 andchark6 2,3. The canonical height of an algebraic point on an elliptic curve core. An explicit manindemjanenko theorem in elliptic curves. The dual selmer group x e is torsion over q and the characteristic ideal of x e is generated by l e as ideals of q. This base point uniquely determine a structure of algebraic group on e.
Isogenies and endomorphism rings of elliptic curves ecc. The best known general result towards this conjecture is obtained in 5 see also section 2. Satotate conjecture and densities pertaining to lehmertype questions. Let aeb a noetherian normal domain and ma nitely generated amodule. This means that one should make sure that the curve one chooses for ones encoding does not fall into one of the several classes of curves on which the problem is tractable. The canonical height of an algebraic point on an elliptic curve. In particular, let e 1q and e 2q be two nonisogenous noncm elliptic curves with respective conductors n 1 6n 2. An elliptic curve ekis the projective closure of a plane a ne curve y2 fx where f2kx is a monic cubic polynomial with distinct roots in k. The height appears in basic conjectures such as birchswinnertondyer and there is a deep conjecture known as the elliptic lehmer problem, see hs90, concerning lower bounds for. The search of a proof of lehmer s conjecture has led many authors to develop various technics for obtaining a minoration of the mahler measure, or equivalently the weil.
The arithmetic of elliptic curves, springerverlag, gtm 106, 1986. Modular forms, which are analytic functions on h2 satisfying a functional equation with respect to the modular group, have farreaching connec. Hindry, on lehmer s conjecture for elliptic curves, s em. A number of authors have proven explicit versions of lehmer s conjecture for polynomials whose coefficients are all congruent to 1 modulo m. Advanced topics in the arithmetic of elliptic curves, springerverlag, gtm 151, 1995.
This question has also been much studied see 1, 2, 7, 11, 14, 20. Correct numerous typographical and minor mathematical errors. A lower bound for the canonical height on elliptic curves over abelian extensions. In order to speak about cryptography and elliptic curves, we must treat ourselves to a bit of an algebra refresher. Ecpp is currently in practice the fastest known algorithm for testing the primality of general numbers, but the worstcase execution time is not known.
Fishers part iii course on elliptic curves, given at cambridge university in lent term, 20. Correct, clarify, and simplify the proofs of some results. On the conjecture of lehmer, limit mahler measure of trinomials and asymptotic expansions jeanlouis vergergaugry dedicated to the memory of pierre liardet. Click here for a cv and complete list of publications books. Computational problems involving the group law are also used in many cryptographic applications, and in. Smith msc thesis, department of mathematics and applied mathematics, the university of the western cape the aim of this dissertation is to provide an exposition of the birch and swinnertondyer conjecture, considered by many to be one of. The canonical height of an algebraic point on an elliptic.
Springer new york berlin heidelberg hong kong london milan paris tokyo. Earlier resultson lehmers conjecture for the convenience of the reader, in this section we summarize some of the known results regarding lehmers conjecture for number. Lehmer calculated that there are exactly 27 primes p 0. The iwasawa main conjecture for supersingular elliptic curves is conjecture 1. If the elliptic curve e has complex multiplication, then the analogue of dobrowolskis result. The height appears in basic conjectures such as birchswinnerton. The elliptic analogof lehmer s conjecture says thathp. Silverman, on lehmers conjecture for elliptic curves. Elliptic curves are curves defined by a certain type of cubic equation in two variables.
Table 1 summarizes the history oflowerbounds forhp where c denotes apositive constant thatdependsonek. A lower bound for the canonical height on elliptic curves. I have made them public in the hope that they might. The analog of lehmer conjecture for elliptic curves and abelian varieties has also been much studied see 1, 2, 11, 16. Since our aim is cryptographic applications of elliptic curves, most of the time k will be a finite field. When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to comprise all nonsingular cubic curves. We assume some familiarity with the theory of elliptic curves. Wiles theorem and the arithmetic of elliptic curves. The birch and swinnertondyer conjecture for elliptic curves. The third smallest salem number in automorphisms of k3 surfaces oguiso, keiji, 2010. On elliptic curves, the abc conjecture, and polynomial. Lehmers conjecture for elliptic curves and abelian varieties has also been much. The birch and swinnertondyer conjecture for elliptic curves d.
Elliptic curves, hilbert modular forms, and the hodge conjecture. The survey by tim dokchitser that has already been linked to describes the proof. It is reasonable to ask why people believed the modular conjecture in the. Elliptic curves, second edition dale husemoller springer springer new york berlin heidelberg hong kong london milan paris tokyo. The analog of lehmer s conjecture for elliptic curves and abelian varieties asks for a good lower bound for the canonical height of a nontorsion point of the abelian variety. Let ek be an elliptic curve defined over a number field, let h be the canonical height on e, and let kabk be the maximal abelian extension of k. We survey the known results and open questions related to the lehmer problem. Baker imrn 29 2003 15711582, we prove that there is a constant cek0 so that every nontorsion point p. Lehmers conjecture ii article pdf available in journal of the mathematical society of japan 623 january 2009 with 41 reads how we measure reads.
Detailed information and an extensive bibliography for the former is given on the lehmer conjecture website 15. Isogenies on elliptic curves definitions4 66 notations we fix a perfect fieldk. The dual selmer group x e is torsion over q and the characteristic ideal of x e is generated by l e as. The analog of lehmers conjecture for elliptic curves and abelian varieties asks for a good lower bound for the canonical height of a nontorsion point of the abelian variety. Add a chapter devoted to algorithmic aspects of elliptic curves, with an emphasis on those features that are used in cryptography. Lehmer s conjecture lehmer s mahler measure problem. In sections 4 and 5 we move beyond rank one and explain that the rank conjecture holds for elliptic curves over function. Historically, the cmpoints on modular curves were rst used by heegner 43 in his work on the class number problem for imaginary quadratic elds.
The set of rational solutions to this equation has an extremely interesting structure, including a group law. It is a generalpurpose algorithm, meaning it does not depend on the number being of a special form. The height appears in basic conjectures such as birchswinnertondyer and there is a deep conjecture known as the elliptic lehmer problem, see hs90, concerning lower bounds for the height. Number theory 30 1988, 226237 30 recent and not so recent developments in the arithmetic theory of elliptic curves, nieuw archief voor wiskunde 7 1989, 5370 31 elliptic curves of bounded degree and height, proc.
An elliptic curve e is a smooth complete curve of genus 1 with a base point 0e. Elliptic curves, hilbert modular forms, and the hodge conjecture don blasius september 8, 2004 1 introduction 1. Let ek be an elliptic curve defined over a number field, let h be the canonical height on e, and let k ab k be the maximal abelian extension of k. Lehmer s conjecture for elliptic curves and abelian varieties has also been much.
Elliptic curves notes for the 20045 part iii course 28012005 16032005. The lucas lehmer test for the primality of p goes as follows. Mason in his paper diophantine equations over function fields chapter vi, theorem 14 and voloch in explicit p. Lehmers conjecture, also known as the lehmers mahler measure problem, is a problem in. Conjecture of lehmer, asymptotic expansions of the mahler measure ii we will obtain an e.
The proof relies on the euler system of elliptic units, and the results and methods of ru3 which also went into the proof of the main conjecture for cm elliptic curves at ordinary primes. The conjecture asserts that there is an absolute constant such that every polynomial with integer coefficients. Below, we describe the baby step, giant step method, which works for all curves, but is slow. In section 3, using a lehmertype result for elliptic curves from 5, we prove the. Rational points on elliptic curves, with john tate, springerverlag, utm, 1992. On elliptic curves, the abc conjecture, and polynomial threshold functions abstract we present a number of papers on topics in mathematics and theoretical computer science. Isogenies on elliptic curvesdefinitions4 66 notations we fix a perfect fieldk. Lehmer s conjecture ii article pdf available in journal of the mathematical society of japan 623 january 2009 with 41 reads how we measure reads.
Let ek be an elliptic curve over a number field k and. Survey on lehmer problems institute of mathematics and. Number theory, elliptic curves, arithmetic and diophantine geometry, number theoretic aspects of dynamical systems, cryptography. Using beilinsons theorem on a suitable modular curve, we prove a weak version of zagiers conjecture for. Lehmers conjecture, also known as the lehmers mahler measure problem, is a problem in number theory raised by derrick henry lehmer.
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