Density of rational points on elliptic fibrations
نویسندگان
چکیده
منابع مشابه
Rational Points on Elliptic Surfaces
x.1. Elliptic Surfaces Deenition. An elliptic surface consists of a smooth (projective) surface E, a smooth (projective) curve C, and a morphism : E ?! C such that almost all bers E t = ?1 (t) are (smooth projective) curves of genus 1. In addition, we will generally assume that our elliptic surfaces come equipped with an identity section 0 : C ?! E which serves as the identity element of the gr...
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The following divisors in the space Sym12 P1 of twelve points on P 1 are actually the same: (A) The possible locus of the twelve nodal fibers in a rational elliptic fibration (i.e. a pencil of plane cubic curves); (B) degree 12 binary forms that can be expressed as a cube plus a square; (C) the locus of the twelve tangents to a smooth plane quartic from a general point of the plane; (D) the bra...
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Let Ef : y 2 = x + f(t)x, where f ∈ Q[t] \ Q, and let us assume that deg f ≤ 4. In this paper we prove that if deg f ≤ 3, then there exists a rational base change t 7→ φ(t) such that there is a non-torsion section on the surface Ef◦φ. A similar theorem is valid in case when deg f = 4 and there exists t0 ∈ Q such that infinitely many rational points lie on the curve Et0 : y 2 = x + f(t0)x. In pa...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2007
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa129-1-5