THE EXPLICIT MORDELL CONJECTURE FOR FAMILIES OF CURVES

Forum of Mathematics Sigma ◽

10.1017/fms.2019.20 ◽

2019 ◽

Vol 7 ◽

Author(s):

SARA CHECCOLI ◽

FRANCESCO VENEZIANO ◽

EVELINA VIADA

Keyword(s):

Classical Method ◽

Rational Points ◽

Computer Search ◽

General Shape ◽

Sharp Estimates ◽

Families Of Curves ◽

Large Families ◽

Mordell Conjecture

In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method bases on some explicit and sharp estimates for the height of such rational points, and the bounds are small enough to successfully implement a computer search. As an evidence of the simplicity of its application, we present a variety of explicit examples and explain how to produce many others. In the appendix our method is compared in detail to the classical method of Manin–Demjanenko and the analysis of our explicit examples is carried to conclusion.

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V.29 Rational Points on Curves and the Mordell Conjecture

The Princeton Companion to Mathematics ◽

10.1515/9781400830398.720 ◽

2010 ◽

pp. 720-722

Keyword(s):

Rational Points ◽

Mordell Conjecture

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An Explicit Manin-Dem’janenko Theorem in Elliptic Curves

Canadian Journal of Mathematics ◽

10.4153/cjm-2017-045-0 ◽

2018 ◽

Vol 70 (5) ◽

pp. 1173-1200 ◽

Cited By ~ 1

Author(s):

Evelina Viada

Keyword(s):

Elliptic Curves ◽

Rational Points ◽

Sharp Bound ◽

Explicit Bound ◽

Minimal Dimension ◽

Mordell Conjecture

AbstractLet be a curve of genus at least 2 embedded in E1 × … × EN, where the Ei are elliptic curves for i = 1, . . . , N. In this article we give an explicit sharp bound for the Néron–Tate height of the points of contained in the union of all algebraic subgroups of dimension < max(), where is the minimal dimension of a translate (resp. of a torsion variety) containing .As a corollary, we give an explicit bound for the height of the rational points of special curves, proving new cases of the explicit Mordell Conjecture and in particular making explicit (and slightly more general in the CM case) the Manin–Dem’janenko method for curves in products of elliptic curves.

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Rational Points on Certain Families of Curves of Genus at Least 2

Proceedings of the London Mathematical Society ◽

10.1112/plms/s3-55.3.465 ◽

1987 ◽

Vol s3-55 (3) ◽

pp. 465-481 ◽

Cited By ~ 9

Author(s):

Joseph H. Silverman

Keyword(s):

Rational Points ◽

Families Of Curves

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An effective Chabauty–Kim theorem

Compositio Mathematica ◽

10.1112/s0010437x19007243 ◽

2019 ◽

Vol 155 (6) ◽

pp. 1057-1075 ◽

Cited By ~ 2

Author(s):

Jennifer S. Balakrishnan ◽

Netan Dogra

Keyword(s):

Rational Points ◽

Higher Rank ◽

Mordell Conjecture

The Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a non-abelian generalisation of Coleman’s effective Chabauty theorem.

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Arithmetic variation of fibers in families of curves. Part I: Hurwitz monodromy criteria for rational points on all members of the familiy.

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crll.1990.409.106 ◽

1990 ◽

Vol 1990 (409) ◽

pp. 106-137

Keyword(s):

Rational Points ◽

Families Of Curves

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Uniform Bound for the Number of Rational Points on a Pencil of Curves

International Mathematics Research Notices ◽

10.1093/imrn/rnz248 ◽

2019 ◽

Author(s):

Vesselin Dimitrov ◽

Ziyang Gao ◽

Philipp Habegger

Keyword(s):

Number Field ◽

Rational Points ◽

Parameter Family ◽

Uniform Bound ◽

Torsion Points ◽

Uniform Bounds ◽

The Family ◽

Projective Curves ◽

Mordell Conjecture

Abstract Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell–Weil rank of the fiber’s Jacobian. Our proof uses Vojta’s approach to the Mordell Conjecture furnished with a height inequality due to the 2nd- and 3rd-named authors. In addition we obtain uniform bounds for the number of torsion points in the Jacobian that lie in each fiber of the family.

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Orientation of actin filaments during motion in motility assay

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100136635 ◽

1995 ◽

Vol 53 ◽

pp. 52-53

Author(s):

J. Borejdo ◽

S. Burlacu

Keyword(s):

Actin Filaments ◽

Thin Filament ◽

Striated Muscle ◽

Myosin Head ◽

Classical Method ◽

Polarization Of Fluorescence ◽

Single Protein ◽

Intracellular Organelles ◽

Change Orientation ◽

Active Entity

Polarization of fluorescence is a classical method to assess orientation or mobility of macromolecules. It has been a common practice to measure polarization of fluorescence through a microscope to characterize orientation or mobility of intracellular organelles, for example anisotropic bands in striated muscle. Recently, we have extended this technique to characterize single protein molecules. The scientific question concerned the current problem in muscle motility: whether myosin heads or actin filaments change orientation during contraction. The classical view is that the force-generating step in muscle is caused by change in orientation of myosin head (subfragment-1 or SI) relative to the axis of thin filament. The molecular impeller which causes this change resides at the interface between actin and SI, but it is not clear whether only the myosin head or both SI and actin change orientation during contraction. Most studies assume that observed orientational change in myosin head is a reflection of the fact that myosin is an active entity and actin serves merely as a passive "rail" on which myosin moves.

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Gaussian ϕ(ρz) curves: Comparison with other models

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100134557 ◽

1990 ◽

Vol 48 (2) ◽

pp. 192-193

Author(s):

Alberto Riveros ◽

Gustavo Castellano

Keyword(s):

Good Description ◽

Bulk Sample ◽

Incident Electron Energy ◽

Power Mean ◽

General Shape ◽

Ionization Cross ◽

X Ray ◽

Mass Absorption ◽

Absorption Correction ◽

Image Position

X ray characteristic intensity Ii , emerging from element i in a bulk sample irradiated with an electron beam may be obtained throughwhere the function ϕi(ρz) is the distribution of ionizations for element i with the mass depth ρz, ψ is the take-off angle and μi the mass absorption coefficient to the radiation of element i.A number of models has been proposed for ϕ(ρz), involving several features concerning the interaction of electrons with matter, e.g. ionization cross section, stopping power, mean ionization potential, electron backscattering, mass absorption coefficients (MAC’s). Several expressions have been developed for these parameters, on which the accuracy of the correction procedures depends.A great number of experimental data and Monte Carlo simulations show that the general shape of ϕ(ρz) curves remains substantially the same when changing the incident electron energy or the sample material. These variables appear in the parameters involved in the expressions for ϕ(ρz). A good description of this function will produce an adequate combined atomic number and absorption correction.

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