Computing N\'eron-Tate heights of points on hyperelliptic Jacobians

It was shown by Faltings and Hriljac that the N\'eron-Tate height of a point on the Jacobian of a curve can be expressed as the self-intersection of a corresponding divisor on a regular model of the curve. We make this explicit and use it to give an algorithm for computing N\'eron-Tate heights on Jacobians of hyperelliptic curves. To demonstrate the practicality of our algorithm, we illustrate it by computing N\'eron-Tate heights on Jacobians of hyperelliptic curves of genus from 1 to 9.Comment: 13 pages. v5: As kindly pointed out by Raymond van Bommel, the height is computed in this paper with respect to twice the theta divisor, not the theta divisor itself (as written in v4

Computing generators of free modules over orders in group algebras

Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition of E[G] is explicitly computable and each component is in fact a matrix ring over a field, this leads to an algorithm that either gives an A-basis for X or determines that no such basis exists. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d=[K:E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A of O_L in E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K=E=Q.Comment: 17 pages, latex, minor revision

Constructing Galois 2-extensions of the 2-adic Numbers

Let Q_2 denote the field of 2-adic numbers, and let G be a group of order 2^n for some positive integer n. We provide an implementation in the software program GAP of an algorithm due to Yamagishi that counts the number of nonisomorphic Galois extensions K/Q_2 whose Galois group is G. Furthermore, we describe an algorithm for constructing defining polynomials for each such extension by considering quadratic extensions of Galois 2-adic fields of degree 2^{n−1}. While this method does require that some extensions be discarded, we show that this approach considers far fewer extensions than the best general construction algorithm currently known, which is due to Pauli- Sinclair based on the work of Monge. We end with an application of our approach to completely classify all Galois 2-adic fields of degree 16, including defining polynomials, ramification index, residue degree, valuation of the discriminant, and Galois group