Abstracts of Talks
Abstract: This is a talk on convergence in the Sato-Tate conjecture, in particular, it is an application of Sage to studying how quickly convergence happens in the Sato-Tate conjecture; this in fact leads to a new conjecture and new questions. This is joint work with Barry Mazur.
I will outline the current status of p-adic arithmetic in Sage. In particular, I will discuss the different types of p-adics in Sage, extensions of Qp , polynomials and matrices over such local fields and their rings of integers. I will outline a number of algorithms for treating precision in polynomial and matrix computations. Come with comments on what aspects of p-adic arithmetic are most needed for your own projects.
his is a talk about some work in progress which involves identities between p-adic multipolylogs and p-adic zeta values. Some I can (almost) prove others I can't and I find pretty surprising. Mostly I'd like to have some feedback on it. It wouldn't really be that computational except for the issue of how to compute linear relations among constants in the p-adics (but the again may be this is well known--to others).
'll give a short overview of the state of affairs of modular forms in SAGE. Mostly, I'll detail what we can do, what MAGMA can do that we can't, and some speed comparisons.
The talk will present an algorithm for the computation of p-adic height pairings on hyperelliptic curves over number fields. Our work is not directly related to the work of Mazur Stein and Tate, though there are some similarities in the difficulties that occur and in the use of Kedlaya's algorithm. I will first explain where this height pairing arise, and how it decomposes into a sum of local terms at the places of the field. The most interesting is for places above the prime p. We use a description of these local terms given by Coleman and Gross that uses the theory of Coleman integration. There are two parts for the computation at these primes. One computes a certain projection from the space of meromorphic forms on the curve to its first de Rham cohomology, which can be computed using Coleman integration and the theory of the so called double index. The second involves computation of Coleman integrals, which are however mroe general than the ones computed in recent work on the subject (Gutnik, Kedlaya, ...) so there are some tricks involved which I'll explain. Finally I will discuss the situation at other primes, where there are still some delicate issues to resolve.
Robert Bradshaw and Kiran Kedlaya
We will discuss the theory of Coleman integration (as referenced in the previous talk), describe an algorithm for computing some Coleman integrals on hyperelliptic curves, and discuss (and perhaps demonstrate) how this is implemented in SAGE. Besides the application to computing p-adic heights, there are also potential applications to finding torsion and rational points on curves over number fields (also due to Coleman); we will say a bit about these too.
will cover libSINGULAR (with some examples, to show that the code isn't as scary as people believe), a bit of PolyBoRi, maybe something about CoCoALib, some benchmarks, and stuff we desperately need and don't know how to get. Obviously, I would throw in some benchmarks and stuff. Actually, I would like to add some (unrelated) slides on the state of sparse linear algebra over finite fields. That would cover what package can do it (not many!) and how SAGE is in that area (surprisingly good it seems for now) and what is done to improve it. Also, someone in the audience might have some input on William's echelon via solve idea adapted to this setting.
FLINT is a C library, in the very early stages of development, which has the aim of extending the state of the art in core arithmetic computations and eventually algebraic number theory. We will briefly discuss progress that has been made so far in polynomial and integer arithmetic, including a new variant/implementation of Mulder's recursive polynomial "short division" algorithm which we have worked up for doing faster polynomial division.
In this controversial report, I will describe the "right" way to discover coercion and explain why the idea of using base extension is wrong.
As Sage has gained more and more developers and users we need some system to deal with bugs, enhancements and other issues. As every other large open and closed source projects has to deal with the same problems there is a large body of open as well as closed software available to manage those issues. One of those software packages is the open source software trac. Sage has had a trac installation for over a year now, but until recently failed to take full advantage of its many capabilities. This talk will try to recommend some rules/best practices to make the use of Sage's trac installation more efficient and homogeneous. In addition this presentation as well as hopefully a lively discussion afterwards will result in documentation that can be added to the Sage manual, increasing the quality of the bug reports as well as giving some clear and concise recommendations on how to report issues and thereby allowing developers more time on fixing bugs instead of chasing down issues.
CoCoALib is a GPLed C++ library for computations in commutative
algebra. It emphasizes a clean design and maintainable, cross platform
code. Besides offering basic mathematical objects like rings, fields,
ideals and modules it also offers a classical Buchberger implementation
for Gr"obner bases as well as ideal and module operations. ApCoCoALib is
also a GPLed library that builds on top of CoCoALib and implements
algorithms for approximate commutative algebra. This talk will explore
algorithms and features of both libraries that will (hopefully) be
interesting for Sage to give an incentive to finish the integration of
[Ap]CoCoALib into Sage. The first step to integrate the libraries
has already been done in March 2007 thanks to work by Martin Albrecht
and Michael Abshoff. The talk should cover the following topics:
* History of CoCoA Systems & CoCoALib
* A Shorter History of ApCoCoALib
* Complementary goals of CoCoALib and ApCoCoALib
* Interesting bits in CoCoALib
* Interesting bits in ApCoCoALib
* The Future?
I will discuss a recent project (joint with K. Lauter and W. Stein) on computing Heegner points defined over higher ring class fields. This computation is interesting because it allows us to verify a conjecture of Kolyvagin for specific elliptic curves. Kolyvagin's conjecture has nontrivial consequences towards the Birch and Swinnerton-Dyer conjecture for elliptic curves of high analytic rank. I will outline the subtleties of the computations, explain why the proposed algorithm works and say how it is implemented in SAGE.