EUDML

Let $ℓ \geq 7$ be a prime, $C$ be the non-singular projective curve defined over $ℚ$ by the affine model $y (1 - y) = x^{ℓ}$ , $\infty$ the point of $C$ at infinity on this model, $J$ the Jacobian of $C$ , and $φ : C \to J$ the albanese embedding with $\infty$ as base point. Let $\bar{ℚ}$ be an algebraic closure of $ℚ$ . Taking care of a case not covered in [], we show that $φ (C) \cap J_{tors} (\bar{ℚ})$ consists only of the image under $φ$ of the Weierstrass points of $C$ and the points $(x, y) = (0, 0)$ and $(0, 1)$ , where $J_{tors}$ denotes the torsion points of $J$ .

Some remarks on almost rational torsion points

John Boxall; David Grant — 2006

Journal de Théorie des Nombres de Bordeaux

For a commutative algebraic group $G$ over a perfect field $k$ , Ribet defined the set of almost rational torsion points $G_{tors, k}^{ar}$ of $G$ over $k$ . For positive integers $d$ , $g,$ we show there is an integer $U_{d, g}$ such that for all tori $T$ of dimension at most $d$ over number fields of degree at most $g$ , $T_{tors, k}^{ar} \subseteq T [U_{d, g}]$ . We show the corresponding result for abelian varieties with complex multiplication, and under an additional hypothesis, for elliptic curves without complex multiplication. Finally, we show that except for an explicit finite...

On the classification of 3-dimensional non-associative division algebras over $p$ -adic fields

Abdulaziz Deajim; David Grant — 2011

Journal de Théorie des Nombres de Bordeaux

Let $p$ be a prime and $K$ a $p$ -adic field (a finite extension of the field of $p$ -adic numbers $ℚ_{p}$ ). We employ the main results in [12] and the arithmetic of elliptic curves over $K$ to reduce the problem of classifying 3-dimensional non-associative division algebras (up to isotopy) over $K$ to the classification of ternary cubic forms $H$ over $K$ (up to equivalence) with no non-trivial zeros over $K$ . We give an explicit solution to the latter problem, which we...

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Formal groups in genus two.

A generalization of Jacobi's derivative formula to dimension two.

Coates-Wiles Towers in Dimension Two.

Units from 3- and 4-torsion on jacobians of curves of genus 2

A proof of quintic reciprocity using the arithmetic of y² = x⁵ + 1/4

Some product formulas for theta functions in one and two variables

The cuspidal torsion packet on hyperelliptic Fermat quotients

Some remarks on almost rational torsion points

On the classification of 3-dimensional non-associative division algebras over $p$ -adic fields