The Geometric Bogomolov Conjecture for Curves of Small Genus
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Abstract The Bogomolov conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov conjecture for all curves of genus at most 4 over a function field of characteristic zero. We recover the known result for genus-2 curves and in many cases improve upon the known bound for genus-3 curves. For many curves of genus 4 with bad reduction, the conjecture was previously unproved. Keywords: Primary 11G30Secondary 14G40, 11G50Bogomolov conjecturecurves of higher genusfunction fieldsmetric graphsKeywords:
Function field
Zero (linguistics)
Using a geometric approach involving Riemann surface orbifolds, we provide lower bounds for the genus of an irreducible algebraic curve of the form $E_{A,B}:\, A(x)-B(y)=0$, where $A, B\in\mathbb C(z)$. We also investigate "series" of curves $E_{A,B}$ of genus zero, where by a series we mean a family with the "same" $A$. We show that for a given rational function $A$ a sequence of rational functions $B_i$, such that ${\rm deg}\, B_i \rightarrow \infty$ and all the curves $A(x)-B_i(y)=0$ are irreducible and have genus zero, exists if and only if the Galois closure of the field extension $\mathbb C(z)/\mathbb C(A)$ has genus zero or one.
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The exceptional zero conjecture relates the first derivative of the $p$-adic $L$-function of a rational elliptic curve with split multiplicative reduction at $p$ to its complex $L$-function. Teitelbaum formulated an analogue of Mazur and Tate's refined (multiplicative) version of this conjecture for elliptic curves over the rational function field $\FQ(T)$ with split multiplicative reduction at two places $\fp$ and $\infty$, avoiding the construction of a $\fp$-adic $L$-function. This article proves Teitelbaum's conjecture up to roots of unity by developing Darmon's theory of double integrals over arbitrary function fields. A function field version of Darmon's period conjecture is also obtained.
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The exceptional zero conjecture relates the first derivative of the p-adic L-function of a rational elliptic curve with split multiplicative reduction at p to its complex L-function. Teitelbaum formulated an analogue of Mazur and Tate's refined (multiplicative) version of this conjecture for elliptic curves over the rational function field 𝔽q(T) with split multiplicative reduction at two places 𝔭 and ∞, avoiding the construction of a 𝔭-adic L-function. This article proves Teitelbaum's conjecture up to roots of unity by developing Darmon's theory of double integrals over arbitrary function fields. A function field version of Darmon's period conjecture is also obtained.
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Function field
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Computations on the Birch and Swinnerton-Dyer conjecture for elliptic curves over pure cubic extensions Celine Maistret The Birch and Swinnerton-Dyer conjecture remains an open problem. In this thesis, we propose to give numerical evidence toward this conjecture when restricted to elliptic curves over pure cubic extensions. We present the general conjecture for elliptic curves over number fields and detail each arithmetic invariants involved. Assuming the conjecture holds, for given elliptic curves E over specific number fields K, we compute the order of the Shafarevich-Tate group of E(K).
Sato–Tate conjecture
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We consider algebraic affine and projective curves of Edwards \cite{E, SkOdProj} over a finite field $\text{F}_{p^n}$. Most cryptosystems of the modern cryptography \cite{SkBlock} can be naturally transform into elliptic curves \cite{Kob}. We research Edwards algebraic curves over a finite field, which at the present time is one of the most promising supports of sets of points that are used for fast group operations \cite{Bir}. New method of counting Edwards curve order over finite field was constructed. It can be applied to order of elliptic curve due to birational equivalence between elliptic curve and Edwards curve. We find not only a specific set of coefficients with corresponding field characteristics, for which these curves are supersingular but also a general formula by which one can determine whether a curve $E_d[\mathbb{F}_{p^n}]$ is supersingular over this field or not. The embedding degree of the supersingular curve of Edwards over $\mathbb{F}_{p^n}$ in a finite field is investigated, the field characteristic, where this degree is minimal, was found. The criterion of supersungularity of the Edwards curves is found over $\mathbb{F}_{p^n}$. Also the generator of crypto stable sequence on an elliptic curve with a deterministic lower estimate of its period is proposed.
Supersingular elliptic curve
Edwards curve
Jacobian curve
Twists of curves
Tripling-oriented Doche–Icart–Kohel curve
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We consider algebraic affine and projective curves of Edwards \cite{E, SkOdProj} over a finite field $\text{F}_{p^n}$. Most cryptosystems of the modern cryptography \cite{SkBlock} can be naturally transform into elliptic curves \cite{Kob}. We research Edwards algebraic curves over a finite field, which at the present time is one of the most promising supports of sets of points that are used for fast group operations \cite{Bir}. New method of counting Edwards curve order over finite field was constructed. It can be applied to order of elliptic curve due to birational equivalence between elliptic curve and Edwards curve. We find not only a specific set of coefficients with corresponding field characteristics, for which these curves are supersingular but also a general formula by which one can determine whether a curve $E_d[\mathbb{F}_{p^n}]$ is supersingular over this field or not. The embedding degree of the supersingular curve of Edwards over $\mathbb{F}_{p^n}$ in a finite field is investigated, the field characteristic, where this degree is minimal, was found. The criterion of supersungularity of the Edwards curves is found over $\mathbb{F}_{p^n}$. Also the generator of crypto stable sequence on an elliptic curve with a deterministic lower estimate of its period is proposed.
Supersingular elliptic curve
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The Birch and Swinnerton-Dyer Conjecture is a well known mathematics problem in the area of Elliptic Curve. One of the crowning moments is the paper by Andrew Wiles which is difficult to understand let alone to appreciate the conjecture. This paper surveys the background of the conjecture treating the ranks of the elliptic curves over the field of rational numbers. Then we present major results like the theorems of Mordell and Mazur leading us to the current state of the conjecture.
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The Birch and Swinnerton-Dyer (BSD) conjecture and Goldfeld conjecture are fundamental problems in the arithmetic of elliptic curves. The congruent number problem (CNP) is one of the oldest problems in number theory which is, for each integer $n$, to find all the rational right triangles of area $n$. It is equivalent to finding all rational points on the elliptic curve $E^{(n)}: ny^{2}=x^{3}-x$. The BSD conjecture for $E^{(n)}$ solves CNP, and Goldfeld conjecture for this elliptic curve family solves CNP for integers with probability one. In this article, we introduce some recent progress on these conjectures and problems.
Sato–Tate conjecture
Supersingular elliptic curve
Rational number
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Twisted cubic
Stable curve
Algebraic variety
Family of curves
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