CHARACTERIZING INTEGERS AMONG RATIONAL NUMBERS WITH A UNIVERSAL-EXISTENTIAL FORMULA By BJORN POONEN

Characterizing Integers among Rational Numbers with a Universal-existential Formula

We prove that Z in definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points. We also give a formula, again with universal quantifiers followed by existential quantifiers, that in any number field defin...

Multivariable polynomial injections on rational numbers

For each number field k, the Bombieri-Lang conjecture for k-rational points on surfaces of general type implies the existence of a polynomial f(x, y) ∈ k[x, y] inducing an injection k × k → k.

Lectures on rational points on curves

Rational points on varieties

Large Torsion Subgroups of Split Jacobians of Curves of Genus Two or Three Everett W. Howe, Franck Leprévost, and Bjorn Poonen

We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve 15625(X4 + Y 4 + Z4)− 969...