A genus one curve defined over Q which has points over Q p for all primes p may not have a
rational point. It is natural to study the classes of Q-extensions over which all such curves …

It is known that for a curve defined over $\mathbb {Q} $ of genus $ g\leq 4$, there exists a
point on the curve defined over a solvable extension of $\mathbb {Q} $. We relate points on …

Fix a non-negative integer g and a positive integer I dividing 2 g− 2. For any Henselian,
discretely valued field K whose residue field is perfect and admits a degree I cyclic …

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We study the combinatorics of constructing non-singular geometrically irreducible projective
curves that do not admit rational points over finite solvable extensions of the base field. A …

In this paper, we prove that every genus one curve C defined over the field of rational
numbers [special characters omitted] which satisfies some local conditions has a point …

We study the question of whether algebraic curves of a given genus g defined over a field K
must have points rational over the maximal abelian extension K^{ab} of K. We give:(i) an …

We show that there is a field F complete with respect to a discrete valuation whose residue
field is perfect and there is a finite Galois extension K\F such that there is no solvable Galois …

Let be a field complete with respect to a discrete valuation whose residue field is perfect of
characteristic. We prove that every smooth, projective, geometrically irreducible curve of …