2
votes
3answers
317 views

Useful notion of unramified Galois representation

Let $\mathbf C(t)$ be the field of rational functions and let $\overline{\mathbf C(t)}$ be an algebraic closure. Let $G$ be the Galois group of $\overline {\mathbf C(t)}$ over $\mathbf C(t)$. Let ...
10
votes
2answers
335 views

Records for low-height points on elliptic curves over number fields

Elkies maintains a list of nontorsion points of low height on elliptic curves over Q; does anyone know of anything similar for curves over number fields? Everest and Ward give examples of points of ...
4
votes
1answer
626 views

Is there a section disjoint from 0, 1 and infinity on the projective line

Let $K$ be a number field with ring of integers $O_K$. Is there a section of $\mathbf{P}^1_{O_K}$ over $O_K$ whose image is disjoint from $0$, $1$ and $\infty$? If $K=\mathbf{Q}$ this is not possible ...
10
votes
1answer
544 views

local-global principle for units

Say that $L/K$ is a quadratic extension of number fields with $K$ totally real and $L$ totally imaginary. Then the Hasse norm theorem says that an element of $K$ that is everywhere a local norm is ...