arXiv subjects

Christopher Doris

Publications and source records attributed to Christopher Doris.

5 recordsLinked to original sources

arXiv2020

Exact $p$-adic computation in Magma

We describe a new arithmetic system for the Magma computer algebra system for working with $p$-adic numbers exactly, in the sense that numbers are represented lazily to infinite $p$-adic precision. This is the first highly featured such implementation. This has the benefits of increasing user-friendliness and speeding up some computations, as well as forcibly producing provable results. We give theoretical and practical justification for its design and describe some use cases. The intention is that this article will be of benefit to anyone wanting to implement similar functionality in other languages.

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arXiv2020

Computing the Galois group of a polynomial over a $p$-adic field

We present a family of algorithms for computing the Galois group of a polynomial defined over a $p$-adic field. Apart from the "naive" algorithm, these are the first general algorithms for this task. As an application, we compute the Galois groups of all totally ramified extensions of $\mathbb{Q}_2$ of degrees 18, 20 and 22, tables of which are available online.

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arXiv2018

ExactpAdics: An exact representation of p-adic numbers

We describe two new packages ExactpAdics and ExactpAdicsII for the Magma computer algebra system for working with p-adic numbers exactly, in the sense that numbers are represented lazily to infinite p-adic precision. This has the benefits of increasing user-friendliness and speeding up some computations, as well as forcibly producing provable results. The two packages use different methods for lazy evaluation, which we describe and compare in detail. The intention is that this article will be of benefit to anyone wanting to implement similar functionality in other languages.

math.NT↗

arXiv2018

On enumerating extensions of p-adic fields with given invariants

We give a brief re-exposition of the theory due to Pauli and Sinclair of ramification polygons of Eisenstein polynomials over p-adic fields, their associated residual polynomials and an algorithm to produce all extensions for a given ramification polygon. We supplement this with an algorithm to produce all ramification polygons of a given degree, and hence we can produce all totally ramified extensions of a given degree.

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arXiv2017

3-torsion and conductor of genus 2 curves

We give an algorithm to compute the conductor for curves of genus 2. It is based on the analysis of 3-torsion of the Jacobian for genus 2 curves over 2-adic fields.

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