It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. Elliptic curves, hilbert modular forms, and the hodge. 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. Lehmers conjecture for elliptic curves and abelian varieties has also been much.
On lehmers conjecture for polynomials and for elliptic curves. One can study for instance the maximal length, or the average length of such lists, when eruns over various families of elliptic curves over q. Lehmers conjecture, also known as the lehmers mahler measure problem, is a problem in. Canonical heights, elliptic divisibility sequences, elliptic curves, num. But it is very unusual conjecturally impossible for the minimal discriminant of an elliptic curve to be so much larger than its conductor. A proof of the conjecture of lehmer archive ouverte hal. An elliptic curve is defined over a field k and describes points in k 2, the cartesian product of k with itself. Goldbachs weak conjecture goldbachs conjecture is one of the oldest and bestknown unsolved problems in number theory and in all of mathematics twin prime redirect from twin prime conjecture prime conjecture, which states. Feb 01, 2004 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. 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. Points on elliptic curves elliptic curves can have points with coordinates in any. The conjecture was chosen as one of the seven millennium prize problems listed by the clay mathematics institute, which has.
Lehmer s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves. Knapp 5 knapps elliptic curves is not the book from which to learn everything about elliptic curves. Let f be a maximal, holomorphic, commutative modulus. This is an analogue of the hardylittlewood twin prime conjecture in the case of elliptic curves. Let ube the maximal open dense subscheme of cover which eis smooth. It has been proved in this case by artin and tate 20 that the lseries has a zero of order at least r, but the conjecture itself remains unproved. In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point o. Lower bounds for heights over abelian extensions and lehmer s conjecture on abelian varieties joint work with matt baker. There is an analogous conjecture for elliptic curves over function.
The lucas lehmer test for the primality of p goes as follows. Harris, bi elliptic curves and symmetric products, proc. Lehmer conjecture, schinzelzassenhaus conjecture, mahler. A lower bound for the canonical height on elliptic curves over. In 15 pollack and rubin proved the following conjecture of kobayashi. There are formulas involving special values of lfunctions of elliptic curves. Elliptic curve discrete logarithm problem ecdlp is the discrete logarithm problem for the group of points on an elliptic curve over a. Modular forms and fermats last theorem, coeditor with gary cornell and. Elliptic curves, logic, and diophantine stability b. In mathematics, the birch and swinnertondyer conjecture describes the set of rational solutions to equations defining an elliptic curve.
Motivation elliptic curves the satotate conjecture evidence returning to the elliptic curve e. The conjecture is now known to hold for elliptic curves with rank 0 and 1 due to the work of kolyvagin in 1990. Szpiros conjecture 15, which is closely related to the abc conjecture, 1. An elliptic curve test for mersenne primes benedict h. Jun 01, 2005 finally, we illustrate the arithmetic applications of these ideas by formulating a precise version of the main conjecture of iwasawa theory for an elliptic curve e over q, without complex multiplication, over the field f generated by the coordinates of all its ppower division points. Affine models for these have been given by birch 1. For a positive number n, nd a rational point with y6 0 on the elliptic curve e n. Please join the simons foundation and our generous member organizations in supporting arxiv during our giving campaign september 2327. The third smallest salem number in automorphisms of k3 surfaces oguiso, keiji, 2010. It is reasonable to ask why people believed the modular conjecture in. A number of authors have proven explicit versions of lehmer s conjecture for polynomials whose coefficients are all congruent to 1 modulo m. Lehmers conjecture for polynomials satisfying a congruence. It is well known that every triangle is everywhere superreversible. It is reasonable to ask why people believed the modular conjecture in the rst place.
Indeed, the rst happens when 4a3 and 27b2 are unusually close. This is best illustrated in an algorithm developed by cremona 7. Under the conjecture that the gap between consecutive primes is bounded by some polynomial. Masser, counting points of small height on elliptic curves. The conjecture asserts that there is an absolute constant such that every polynomial with integer coefficients. Prime 3 5 7 11 17 np 3 8 4 17 17 ap 0 3 3 2 4 0 bp 0 0. It is reasonable to ask why people believed the modular conjecture in the. The conjecture of birch and swinnertondyer relating the analytic and geometric rank of an elliptic curve is perhaps one of the less debated and widely accepted conjectures in the. Mazur june 20, 2015 contents i the program for these lectures 2 1 around hilberts tenth problem 3 2 diophantine stability issues 4 3 the rarity of rational points, and some comments about elliptic curves over q 6 ii logic and number theory 8 4 listable sets of integers 9 5 the halting. The birch and swinnertondyer conjecture for an elliptic. In particular, let e 1q and e 2q be two nonisogenous noncm elliptic curves with respective conductors n 1 6n 2.
The viewpoint of the equation y2 x3 n2xallows one to do something striking. In section 3, using a lehmer type result for elliptic curves. There are in nature eleven other curves like the one considered by shimura which are uniformized by modular forms on ron. Koblitz conjectured a precise asymptotic for the number of primes pup to xsuch that the order of the group of points of eover fp is prime. Free elliptic curves books download ebooks online textbooks. We remark that we could replace this hypothesis by the langtrotter conjecture for elliptic curves. The lehmer inequality and the mordellweil theorem for drinfeld. Fermats method of descent, plane curves, the degree of a morphism, riemannroch space, weierstrass equations, the group law, the invariant differential, formal groups, elliptic curves over local fields, kummer theory, mordellweil, dual isogenies and the weil pairing, galois cohomology, descent by cyclic isogeny. Here n ve is the number of points of e v over the residue. In terms of mahler measure, lehmers conjecture is that there exists an. Some remarks on the main conjecture for elliptic curves.
On lehmer s conjecture for polynomials and for elliptic curves 753 suchthatforallnontorsionpointsq. Swaminathan, elliptic curve variants of the least quadratic nonresidue problem and linniks theorem, submitted. The canonical height of an algebraic point on an elliptic curve. Analogues of lehmers conjecture in positive characteristic. The dual selmer group x e is torsion over q and the characteristic ideal of x e is generated by l e as. A lower bound for the canonical height on elliptic curves. Number theory, elliptic curves, arithmetic and diophantine geometry, number theoretic aspects of dynamical systems, cryptography.
Hindry, on lehmer s conjecture for elliptic curves, s em. This question has also been much studied see 1, 2, 7, 11, 14, 20. Knapp 2 in which g2 and g3 are constants that depend on l. The main conjecture of iwasawa theory for elliptic curves. Because of the above theorem, lehmers question in the severalvariable case. Moreover, if the weak birch and swinnertondyer conjecture is true for the elliptic curve e n. In this article, we prove an analogue of goncharov and levins result for the base extension of eto an arbitrary abelian number eld. Ecpp is currently in practice the fastest known algorithm for testing the primality of general numbers, but the worstcase execution time is not known. We look at a version of lehmers conjecture involving elliptic curves and. Given an elliptic curve eq, the set of all curves e0 isogenous to eover q is.
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. If c is an elliptic curve of genus 1, then for some. As with the case of dirichlet characters, further questions naturally arise. The deeper parts of the arithmetic theory, involving complex multiplication and cohomology, are absent. We begin by recalling the usual selmer groups of an elliptic curve as well as some generalizations. We look at a version of lehmers conjecture involving elliptic curves and investigate the corresponding mahler. The characteristic power series c divides the padic lfunction l pe,k.
Salem numbers, pisot numbers, mahler measure, and graphs mckee, james and smyth, chris, experimental mathematics, 2005. Laurent lau83 proved an up to epsilon version of this conjecture under the. The gl 2 main conjecture for elliptic curves without complex. Greenbergs conjecture excludes the case of weight one forms, since certainly any form with finite image has many primes for which the local galois representation is split but. E is the elliptic curve of conductor 15 that is the. Let aeb a noetherian normal domain and ma nitely generated amodule. Selmer groups and mordellweil groups of elliptic curves over towers of function fields such an esimply as a nonisotrivial elliptic curve over c.
E i aiq i, associated to e, exhibit a modularity pattern. The birch and swinnertondyer conjecture for an elliptic curve over q 4 v 5 5 raymond van bommel abstract. If the field has characteristic different from 2 and 3 then the curve can be described as a plane algebraic curve which, after a linear change of variables. One of the fundamental results about the arithmetic of eis the celebrated theorem of mordell and weil. Mahlers measure and special values of lfunctions project euclid.
Lehmer s conjecture, also known as the lehmer s mahler measure problem, is a problem in number theory raised by derrick henry lehmer. On zagiers conjecture for base extensions of elliptic curves. The average elliptic curve has few integral points. The analog of lehmer conjecture for elliptic curves and abelian varieties asks for a good lower bound. 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. Lehmer also computed the a using the above product expansion for all primes p lehmer found many primes p for which a 0.
It should be noted that if the existence conjecture is. Geometry of points over of small height part ii georgia tech math. We conjecture that the list provided on each row is complete. There is a positive constant cek such that for all p. Dyer and there is a deep conjecture known as the elliptic lehmer problem, see. In section 3, using a lehmer type result for elliptic curves from 5, we prove the. Upper and lower bounds for divisibility sequences on tori, elliptic curves, and abelian varieties and the relation of such bounds to vojtas conjecture. The canonical height of an algebraic point on an elliptic curve emis. Baker imrn 29 2003 15711582, we prove that there is a constant cek0 so that every nontorsion point p. In this paper, we show that the birch and swinnertondyer conjecture for a certain elliptic curve over q 4 v 5 is equivalent to the same conjecture for a certain pair of hyperelliptic curves of genus 2 over q. If langs conjecture holds then the number of prime power terms b nb 1 is bounded independently of the curve, the point and the degree of the isogeny. The iwasawa main conjecture for supersingular elliptic curves is conjecture 1. Galois case of the elliptic lehmer problem, combining kummer theory and massers result.
Click here for a cv and complete list of publications books. On lehmer s conjecture for polynomials and for elliptic curves. Some remarks on the main conjecture for elliptic curves with. Iwasawa main conjecture for supersingular elliptic curves. The number of points on an elliptic curve modulo a prime ii.
Swaminathan, benfords law for traces of frobenius on newforms with complex multiplication, ramanujan journal, accepted. While the second happens when 4a3 and 27b2 are unusually close padically. Mar 21, 2020 after all, taken to its logical conclusion it would imply not only lehmer s conjecture but also combined with elkies theorem that all elliptic curves are cm. Tunnells theorem provides an unconditional method of proving a squarefree integer nis not congruent, and a. We also discuss a number of numerical examples where one can actually prove the conjecture see 1 for further results in this direction. Moduli spaces and arithmetic dynamics, crm monograph series 30, ams, 2012. Advanced topics in the arithmetic of elliptic curves, springerverlag, gtm 151, 1995. If the elliptic curve e has complex multiplication, then the analogue of dobrowolskis result holds. Assume k q and b is an elliptic divisibility sequence generated by a magni. For elliptic curves with complex multiplication approximately half the al are zero, on the other.
It is a generalpurpose algorithm, meaning it does not depend on the number being of a special form. A partial result towards this conjecture is obtained in 5. Selmer groups and mordellweil groups of elliptic curves. Ek is nontorsion and is the image of a krational point under a nontrivial krational isogeny, then say that p is magni. Click here for pdf and powerpoint slides of selected presentations. Polynomials with small elliptic mahler measure via genetic algorithms. Primality testing using elliptic curves journal of the acm.
Curiously, although this conjecture could have been formulated in the last century, its discovery was based on modern research in the theory of function. We prove a similar result for polynomials fx that are divisible in zmzx by a polynomial of the form. The conjecture is now known to hold for elliptic curves with rank 0 and 1 due to the work of. 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.
Lehmers polynomial is known to have the lowest mahler measure for all polynomials. We can look at lehmer question in the context of elliptic curves. The modularity theorem, once known as the taniyamashimuraweil conjecture, states that every elliptic curve e over q is a modular curve, that is to say, its hasseweil zeta function is the lfunction of a modular form of weight 2 and level n, where n is the conductor of e an integer divisible by the same prime numbers as the. 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. These curves as well as those obtained from them by rational transformations of the variables are the elliptic curves of the title. Venjakobs 44 work on commutative local isomorphisms.
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