Rational elliptic surfaces and the trigonometry of tetrahedra

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...

Integral Models of Extremal Rational Elliptic Surfaces

Miranda and Persson classified all extremal rational elliptic surfaces in characteristic zero. We show that each surface in Miranda and Persson’s classification has an integral model with good reduction everywhere (except for those of type X11( j), which is an exceptional case), and that every extremal rational elliptic surface over an algebraically closed field of characteristic p > 0 can be o...

Rational Points on Certain Elliptic Surfaces

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...

Gravitational Instantons from Rational Elliptic Surfaces

Construction of Rational Elliptic Surfaces with Mordell-Weil Rank 4