Abstract Let L be a full-rank lattice in ℝ d and write + for the semigroup of all vectors with nonnegative coordinates . We call basis X positive if it is contained There are infinitely many such bases, each them spans conical S ( ) consisting integer linear combinations Such sub-semigroup , we investigate distribution gaps i.e. points ∖ ). describe some basic properties counting estimates thes...

This article generalizes the geometric quadratic Chabauty method, initiated over Q \mathbb {Q}</mml:annotation...

The starting point of this method is Falting’s article in which he proves the Mordell-Weil theorem. He remarked and Serre turned it into a working method, the fact that the equivalence of two λ-adic representations is something that can be basically determined on some finite extension of the base field (even though the representations might not factor through a finite quotient). Let Oλ be the r...

This paper assumes no background on elliptic curves and culminates with a proof of the Mordell-Weil theorem. The Riemann-Roch and Dirichlet unit theorem are recalled but used without proof, but everything else is self-contained. After some elementary properties of elliptic curves are given, the group structure is explored in detail.

of the Dissertation On some computational and analytic aspects of Chern-Weil forms

We derive the equations necessary to perform a two-descent on the Jacobians of curves of genus two with rational Weierstrass points. We compute the Mordell-Weil rank of the Jacobian of some genus two curves defined over the rationals, and discuss the practicality of using this method. Introduction Let C he a curve of genus two defined over a number field K, and J its Jacobian variety. The Morde...

Let E ! P 1 be an elliptic surface deened over a number eld K, or equivalently an elliptic curve deened over K(T). In this note we prove, assuming Tate's conjecture, that the rank of E(K(T 1=n)) is bounded by F (E)d K (n), where F (E) is an explicit constant independent of n and d K (n) is an explicit elementary function. In particular, if K \ Q(d) = Q for all djn, then d K (n) = d(n) is just t...

We show that there is a bound depending only on g, r and [K : Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g − 3. If K = Q, an explicit bound is 8rg + 33(g − 1) + 1. The proof is based on Chabauty’s method; the new ingredient is an estimate for the number of zeros of an abelian log...

Let X be a curve of genus g 2 over a number field F of degree d D ŒF W Q . The conjectural existence of a uniform bound N.g;d/ on the number #X.F / of F rational points of X is an outstanding open problem in arithmetic geometry, known by the work of Caporaso, Harris, and Mazur to follow from the Bombieri–Lang conjecture. A related conjecture posits the existence of a uniform bound Ntors; .g; d/...