Consider a smooth, geometrically irreducible, projective curve of genus $g\ge 2$ defined over number field degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, theorem Faltings. We show that is bounded only in terms $g$, $d$ and Mordell–Weil rank curve's Jacobian, thereby answering affirmative question Mazur. In addition we obtain uniform bounds, $g$ $d$, fo...

1 The Tate{Lichtenbaum pairing In the paper F-R] it is shown how the Tate pairing on Abelian varieties in Licht-enbaum`s version can be used to relate the discrete logarithm in the group J m (F q) of m{torsion points of the Mordell-Weil group of the Jacobian J of a curve over a nite eld F q to the discrete logarithm in F q if q ? 1 is divisible by m. 1 More precisely the main result of F-R] can...

A technique of descent via 4-isogeny is developed on the Jacobian of a curve of genus 2 of the form: Y 2 = q1(X)q2(X)q3(X), where each qi(X) is a quadratic defined over Q. The technique offers a realistic prospect of calculating rank tables of Mordell-Weil groups in higher dimension. A selection of worked examples is included as illustration.

Given two elliptic curves defined over a number field K, not both with j-invariant zero, we show that there are infinitely many D ∈ K × with pairwise distinct image in K × /K × 2 , such that the quadratic twist of both curves by D have positive Mordell-Weil rank. The proof depends on relating the values of pairs of cubic polynomials to rational points on another elliptic curve, and on a fiber p...

We give an overview over recent results concerning rational points on hyperelliptic curves. One result says that ‘most’ hyperelliptic curves of high genus have very few rational points. Another result gives a bound on the number of rational points in terms of the genus and the Mordell-Weil rank, provided the latter is sufficiently small. The first result relies on work by Bhargava and Gross on ...

Let E be an elliptic curve defined over Q. Let Γ be a free subgroup of rank r of E(Q). For any prime p of good reduction, let Γp be the reduction of Γ modulo p and Ep be the reduction of E modulo p. We prove that if E has CM then for all but o(x/ log x) of primes p ≤ x, |Γp| ≥ p r r+2 + , where (p) is any function of p such that (p)→ 0 as p→∞. This is a consequence of two other results. Denote ...

Let E → C be an elliptic surface, defined over a number field k, let P : C → E be a section, and let l be a rational prime. We bound the number of points of low algebraic degree in the l-division hull of P at the fibre Et. Specifically, for t ∈ C(k) with [k(t) : k] ≤ B1 such that Et is non-singular, we obtain a bound on the number of Q ∈ Et(k) such that [k(Q) : k] ≤ B2, and such that lQ = Pt, f...

We study a subgroup of the Shafarevich-Tate group of an abelian variety known as the visible subgroup. We explain the geometric intuition behind this subgroup, prove its finiteness and describe several techniques for exhibiting visible elements. Two important results are proved one what we call the visualization theorem, which asserts that every element of the Shafarevich-Tate group of an abeli...