Solvable points on genus-one curves over local fields

Let $F$ be a field complete with respect to a discrete valuation whose residue field is perfect of characteristic $p\gt 0$. We prove that every smooth, projective, geometrically irreducible curve of genus one defined over $F$ with a non-zero divisor of degree a power of $p$ has a solvable point over $F$.

Solvable Points on Projective Algebraic Curves

We examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective curves without solvable points of genus g, when g is at least 40, over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus 0, 2, 3 or 4 defined over any field has a solvable point. Finally we prove that every genus 1 curve defined over a local field of characteristic zero with residue field of characteristic p has a divisor of degree prime to 6p defined over a solvable extension.

EXACTLY SOLVABLE POINTS IN THE MODULI SPACE OF HETEROTIC N=2 STRINGS

We investigate the subset of exactly solvable (0, 4) worldsheet supersymmetric string vacua contained in a recent class of Gepner-like (0, 2) superconformal models. The identification of these models with certain points of enhanced gauge symmetry on K3×T2 can be achieved completely. Furthermore, we extend the construction of in general (0, 2) supersymmetric exactly solvable models to the case where also a nontrivial part of the vector bundle is embedded into the hidden E8 gauge group. For some examples we explicitly calculate the enhanced gauge symmetries and show that they open up the way to interesting branches of the N=2 moduli space. For some of these models candidates of type-II dual descriptions exist.