Explicit points on the Legendre curve III

Ulmer, Douglas

doi:10.2140/ant.2014.8.2471

Cited by 5 publications

(2 citation statements)

References 22 publications

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“…The rank of E when some power of p is −1 modulo d was first discussed in [12], and with hindsight it could have been expected to be large from considerations of "supersingularity." The results of [2] show, perhaps surprisingly, that there are many other classes of d for which high ranks occur. Our aim here is to make this observation more quantitative.…”

Section: Introductionmentioning

confidence: 91%

On Balanced Subgroups of the Multiplicative Group

Pomerance

Ulmer

2013

Springer Proceedings in Mathematics &Amp; Statistics

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A subgroup H of G = (Z/dZ) × is called balanced if every coset of H is evenly distributed between the lower and upper halves of G, i.e., has equal numbers of elements with representatives in (0, d/2) and (d/2, d). This notion has applications to ranks of elliptic curves. We give a simple criterion in terms of characters for a subgroup H to be balanced, and for a fixed integer p, we study the distribution of integers d such that the cyclic subgroup of (Z/dZ) × generated by p is balanced.

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Section: Introductionmentioning

confidence: 91%

On Balanced Subgroups of the Multiplicative Group

Pomerance

Ulmer

2013

Springer Proceedings in Mathematics &Amp; Statistics

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“…Understanding the relations among the points given by various lines will hopefully lead to a more explicit rational generating set of lower height and smaller index as found with respect to the Legendre curve in (3.1) of [10]. Further possible applications include computing Tate-Shafarevich groups, as in [11]. (3) We have considered only one family of lines on the Fermat surface.…”

Section: Properties Of the Character Summentioning

confidence: 99%

Explicit points on y2+xy−ty=x3 and related character sums

Davis

Occhipinti

2016

Journal of Number Theory

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Abstract. Let Fq denote a finite field of characteristic p ≥ 5 and let d = q + 1. Let E d denote the elliptic curve over the function field F q 2 (t) defined by the equation y 2 + xy − t d y = x 3 . Its rank is q when q ≡ 1 mod 3 and its rank is q − 2 when q ≡ 2 mod 3. We describe an explicit method for producing points on this elliptic curve. In case q ≡ 11 mod 12, our method produces points which generate a full-rank subgroup. Our strategy for producing rational points on E d makes use of a dominant map from the degree d Fermat surface over F q 2 to the elliptic surface associated to E d . We in turn study lines on the Fermat surface F d using certain multiplicative character sums which are interesting in their own right. In particular, in the q ≡ 7 mod 12 case, a character sum argument shows that we can generate a full-rank subgroup using µ d -translates of a single rational point.

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Curves and Jacobians over Function Fields

Ulmer

2014

Advanced Courses in Mathematics - CRM Barcelona

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These notes originated in a 12-hour course of lectures given at the Centre de Recerca Mathemàtica near Barcelona in February, 2010. The aim of the course was to explain results on curves and their Jacobians over function fields, with emphasis on the group of rational points of the Jacobian, and to explain various constructions of Jacobians with large Mordell-Weil rank.More so than the lectures, these notes emphasize foundational results on the arithmetic of curves and Jacobians over function fields, most importantly the breakthrough works of Tate, Artin, and Milne on the conjectures of Tate, Artin-Tate, and Birch and Swinnerton-Dyer. We also discuss more recent results such as those of Kato and Trihan. Constructions leading to high ranks are only briefly reviewed, because they are discussed in detail in other recent and forthcoming publications.These notes may be viewed as a continuation of my Park City notes [69]. In those notes, the focus was on elliptic curves and finite constant fields, whereas here we discuss curves of high genera and results over more general base fields.It is a pleasure to thank the organizers of the CRM Research Program, especially Francesc Bars, for their efficient organization of an exciting meeting, the audience for their interest and questions, the National Science Foundation for funding to support the travel of junior participants (grant DMS 0968709), and the editors for their patience in the face of many delays. It is also a pleasure to thank Marc Hindry, Nick Katz, and Dino Lorenzini for their help. Finally, thanks to Timo Keller and René Pannekoek for corrections.I welcome all comments, and I plan to maintain a list of corrections and supplements on my web page. Please check there for updates if you find the material in these notes useful.Atlanta, Unless explicitly stated otherwise, all schemes are assumed to be Noetherian and separated and all morphisms of schemes are assumed to be separated and of finite type.A curve over a field F is a scheme over F that is reduced and purely of dimension 1, and a surface is similarly a scheme over F which is reduced and purely of dimension 2. Usually our curves and surfaces will be subject to further hypotheses, like irreducibility, projectivity, or smoothness.We recall that a scheme Z is regular if each of its local rings is regular. This means that for each point z ∈ Z, with local ring O Z,z , maximal ideal m z ⊂ O Z,z , and residue field κ(z) = O Z,z /m z , we haveEquivalently, m z should be generated by dim z Z elements. A morphism f : Z → S is smooth (of relative dimension n) at z ∈ Z if there exist affine open neighborhoods U of z and V of f (z) such that f (U ) ⊂ V and a diagram

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Explicit points on the Legendre curve III

Abstract: ABSTRACT. We continue our study of the Legendre elliptic curve y 2 = x(x + 1)(x + t) over function fieldsand of finite, p-power index. We also proved the finiteness of X(E/K d ) and a class number formula:

Cited by 5 publications

References 22 publications

On Balanced Subgroups of the Multiplicative Group

On Balanced Subgroups of the Multiplicative Group

Explicit points on y2+xy−ty=x3 and related character sums

Curves and Jacobians over Function Fields

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