I made a diagram that illustrates most of the algorithms I wish Sage had to support my current mathematical research program. I am focusing most of my Sage development effort on ensuring that these get implemented as soon as possible. I will not depend on anybody else to help me with any of this, though if people would like to help in various ways that would be greatly appreciated. Over half of these are in the closed commercial Magma software already, which makes it much easier for me to implement them in Sage.
Research Versus the Goals of the Sage Project
A concern I have is that there might be disagreement in the Sage community about some of the design decisions that are necessary to support the implementation of my planned algorithms. For example, improving number field arithmetic to be dramatically faster requires making some sacrifices so that the implementation takes one week instead of two months (e.g., fundamentally depending on FLINT for basic arithmetic, which at least one Sage developer will be quite unhappy about). Even though I'm the leader of the Sage project, I won't just add code to Sage that a lot of people don't like. However, implementing everything planned here is a lot of work, and given the time and resources I have available, it will be impossible to fully document and test everything up to the standard necessary for inclusion in Sage. If this gets to be a nontrivial problem, I will maintain my own alternative ``fork'' of Sage, which I'll distribute separately. It will have `bleeding edge'' code that I'm developing, and I'll make full source releases available on a separate website, but I will not have to worry about constantly updating packages (like Maxima and Scipy) that are irrelevant to my research. The architecture of Sage makes creating such a fork easy, and of course I know how to manage this. In a few years, once I've built everything and complete the research I plan to do with it, I can spend the time merging back the good parts into mainline Sage. I know that several other Sage developers (e.g., Nick Alexander, David Roe, etc.) do something similar, since their personal research is so important to them, and getting code into Sage is overly difficult. It is very important that I acknowledge this tension, because currently the demands of "publishing" code in Sage -- and indeed of maintaining Sage as a general purpose system -- are so tough that they make genuine development of Sage as a research tool too difficult. Solution: create something that isn't Sage.
Here is the diagram. It is a tiny picture, but if you click on it you should get a bigger PDF that you can zoom into (if you are using Linux, download this PNG instead).
I will spend the rest of this post discussing the first few algorithms at the top of the diagram, and my motivation for implementing them. Future posts will address the rest of the algorithms.
The top of the diagram lists number fields, and the two projects involve greatly speeding up basic arithmetic in Sage with elements of number fields. This is important to my research, since I intend to do explicit comptutations with elliptic curves and modular forms over number fields, including making large tables, hence fast arithmetic is important. Arithmetic in some cases in Sage is ridiculously slow, especially for relative fields. Fixing this involves switching from using NTL for all number field arithmetic to using FLINT (via Sebastian Pancratz's new rational polynomials) for absolue fields and Singular for relative extensions. I spent some time on this project and the results are at Trac 9541, but the code there should not be used as is. Instead, I'll finish this project by extracting out the best ideas from those patches, and adding more. The main idea for relative number fields is to create a C interface (written in Cython!) to Singular for computing with transcendence degree 1 function fields over the rational numbers. The other lesson I learned when working on Trac 9541 is that it is orders of magnitude easier to delete all code related to using NTL for number fields instead of trying to simultaneously support several different models for numbers fields.
Function FieldsContinuing clockwise, the next cluster of algorithms involves function fields of transcedence degree 1 over the prime subfield, i.e., function fields of algebraic curves. Function fields are important for my research because an application of explicit Riemann-Roch spaces and divisor reduction is a key step in completing a paper on torsion points on elliptic curves over quartic fields that I'm writing with Kamienny and Stoll. Function field arithmetic, rings, ideals, polynomial factorization, norm, traces, ``Poppov reduction'', etc., are the subject of Trac 9054, which should be relatively easy to finish. The next step is to implement Florian Hesse's algorithm for Riemann-Roch spaces, by following the description in Florian's papers and talks on the topic. Next, I'll have to implement divisor reduction and computation of maximal orders (normalization of curves). I suspect that Chris Hall will help. Parallel to this, it is also a good idea to use the same code as in the number field case to make basic arithmetic in certain function fields much faster than the generic code of Trac 9054.
Elliptic Curves over Function Fields
I am also interested in code for computing with elliptic curves that are defined over function fields (usually over finite fields). This is related to work of Chris Hall, Sal Baig, and others at recent SageDays on function fields. This code will make it easier to computationally explore function field analogues of new ideas related to the Gross-Zagier formula and the BSD conjecture. The main goals are to implement algorithms for computing every quantity appearing in the Birch and Swinnerton-Dyer conjecture, including torsion, Tamagawa numbers, L-functions, Mordell-Weil groups (2-descent), and regulators. Also, I need to implement code for doing computations with Drinfeld modules, since Drinfled modules provide the analogue of modular curves and Heegner points in the function field setting.