Towards an exact adaptive algorithm for the determinant of a rational matrix

In this paper we propose several strategies for the exact computation of the determinant of a rational matrix. First, we use the Chinese Remaindering Theorem and the rational reconstruction to recover the rational determinant from its modular images. Then we show a preconditioning for the determinant which allows us to skip the rational reconstruction process and reconstruct an integer result. We compare those approaches with matrix preconditioning which allow us to treat integer instead of rational matrices. This allows us to introduce integer determinant algorithms to the rational determinant problem. In particular, we discuss the applicability of the adaptive determinant algorithm of [9] and compare it with the integer Chinese Remaindering scheme. We present an analysis of the complexity of the strategies and evaluate their experimental performance on numerous examples. This experience allows us to develop an adaptive strategy which would choose the best solution at the run time, depending on matrix properties. All strategies have been implemented in LinBox linear algebra library

Computing local p-adic height pairings on hyperelliptic curves

We describe an algorithm to compute the local component at p of the Coleman-Gross p-adic height pairing on divisors on hyperelliptic curves. As the height pairing is given in terms of a Coleman integral, we also provide new techniques to evaluate Coleman integrals of meromorphic differentials and present our algorithms as implemented in Sage

Computing topological zeta functions of groups, algebras, and modules, II

Building on our previous work (arXiv:1405.5711), we develop the first practical algorithm for computing topological zeta functions of nilpotent groups, non-associative algebras, and modules. While we previously depended upon non-degeneracy assumptions, the theory developed here allows us to overcome these restrictions in various interesting cases.Comment: 33 pages; sequel to arXiv:1405.571

Computing fundamental domains for the Bruhat-Tits tree for GL2(Qp), p-adic automorphic forms, and the canonical embedding of Shimura curves

We describe an algorithm for computing certain quaternionic quotients of the Bruhat-Tits tree for GL2(Qp). As an application, we describe an algorithm to obtain (conjectural) equations for the canonical embedding of Shimura curves.Comment: Accepted for publication in LMS Journal of Computation and Mathematic

Solving Sparse Integer Linear Systems

We propose a new algorithm to solve sparse linear systems of equations over the integers. This algorithm is based on a $p$-adic lifting technique combined with the use of block matrices with structured blocks. It achieves a sub-cubic complexity in terms of machine operations subject to a conjecture on the effectiveness of certain sparse projections. A LinBox-based implementation of this algorithm is demonstrated, and emphasizes the practical benefits of this new method over the previous state of the art