Sage Days 51
An edition of the Sage Days titled Algorithms in Arithmetic Geometry will take place at the Lorentz Center in Leiden from 22-26 July 2013.
For further information, see the workshop webpage or contact one of the organizers: Peter Bruin Maarten Derickx Michiel Kosters
Schedule
Proposed projects
There will be four main (proposed) projects during this week.
Finite fields. First, operations in finite fields of cardinality larger than 2^{16} can be drastically sped up; one solution is available for testing at Trac #12142. (See also a discussion on the sage-nt list.) Faster finite fields will mean that the algorithms in the other projects will also be significantly faster. Second, more functionality for embeddings between finite fields is needed; see Trac tickets #8335, #11938, #13214. Is it also possible to implement `standard models' of finite fields?
Enhancing the function field functionality of Sage. In particular, it is important to have algorithms for computing with Jacobians of algebraic curves. It is desirable to implement two different frameworks, each with its own advantages: one developed by F. Hess (Computing relations in divisor class groups of algebraic curves over finite fields, PDF) and the other developed by K. Khuri-Makdisi (Asymptotically fast group operations on Jacobians of general curves, arXiv:0409209). This project is motivated by the following two projects.
Working on practical implementations of the algorithms from Edixhoven, Couveignes, Bosman, de Jong, and Merkl, Computational Aspects of Modular Forms and Galois Representations (book, webpage) and Bruin, Modular curves, Arakelov theory, algorithmic applications (thesis, PDF) for computing Galois representations over finite fields attached to modular forms.
Working on computing semi-stable models of curves over local fields. The goal is a practical implementation of the algorithms that follow from the new proof of Deligne and Mumford's stable reduction theorem in: K. Arzdorf and S. Wewers, A local proof of the semistable reduction theorem, arXiv:1211.4624. This is worked out for superelliptic curves (y^n = f(x)) with n not divisible by the residue characteristic) by I. Bouw and S. Wewers, Computing L-functions and semistable reduction of superelliptic curves, arXiv:1211.4459.