# Sage Days in Le Teich 2023-2024 ## Anderson motives *Participants*: Xavier Caruso + Joseph Musleh *Short Description*: Anderson motives are a generalization of Drinfeld modules, allowing for more algebraic constructions (e.g. tensor products). We plan to implement them in SageMath. ## The Burnside ring of permutation representations - [Issue #35475](https://github.com/sagemath/sage/issues/35475) *Participants*: Martin Rubey - help (especially mathematically) greatly appreciated *Short Description*: The Burnside ring is the ring of group actions of a finite group, see https://en.wikipedia.org/wiki/Burnside_ring. The main difficulty in implementing this ring is that there is no normal form for the objects. ## Sage integration of Pari functionalities *Participants*: Aurel Page + some Sage programmer (volunteer needed!): Marc Masdeu + F. Chapoton + S. Labbé *Short Description*: Some functionalities of Pari are currently only available in Sage through explicit call of the gp function. It would be nice to implement better integration (with nice Sage classes, doc and tests). I think this would benefit both Pari and Sage. Examples: * construction of class fields (bnrclassfield); in fact, even ray class groups are missing (bnrinit and friends) (see [this issue]()) * L-functions (lfuninit and friends) (doc missing [here](https://doc.sagemath.org/html/en/reference/lfunctions/index.html#l-functions) but see [PR #37166](https://github.com/sagemath/sage/pull/37166)) * Hecke Grossencharacters (gcharinit and friends) * central simple algebras over number fields (alginit and friends) in particular, improve quaternion algebras: * construct from ramification (see [this PR](https://github.com/sagemath/sage/pull/37189)) * compute maximal order * compute mod p splitting * note: there are improvements to the ramification computations in progress in [this PR](https://github.com/sagemath/sage/pull/37164) and [that one](https://github.com/sagemath/sage/pull/37173) * enumeration of number fields with given Galois group and discriminant (nflist) * modular forms via the trace formula (mfinit and friends) * S-units of number fields in product representation (bnfunits) (see [this issue](https://github.com/sagemath/sage/issues/31754)) * Flatter algorithm for LLL reduction (qflll) [done](https://github.com/sagemath/sage/pull/37207) (allows algorithm='pari' in LLL, but will need a more recent pari version to have Flatter) * use `nfsplitting` in Sage's `splitting_field` ## Central simple algebras of Pari *Participants*: Aurel Page *Short Description*: Pari has functionalities for central simple algebras, but some useful features are missing. I would like to work on some of them: * splitting at infinity, LLL-reduction of the integral basis * orders * include James Rickards code for fundamental domains ## An implementation of the Gentry-Szydlo algorithm that works *Participants*: Guilhem Mureau, Alice Pellet-Mary *Short Description*: We need to run the Gentry-Szydlo algorithm for some project, but we were unable to find an existing implementation that works for our regime of parameters, so we will try to make a nice implementation that other people can reuse. (The point of the Gentry-Szydlo algorithm is to recover an element $g$ in a CM number field, when given as input a basis of the principal ideal generated by $g$ and the relative norm $g \overline{g}$) ## Jupyter kernel XEUS-GP for PARI/GP *Participants*: Bill Allombert *Short Description*: XEUS-GP is a Jupyter kernel for PARI/GP using the C++ XEUS protocol instead of python, which allows direct access to the PARI C library interface. [Link](https://pari.math.u-bordeaux.fr/cgi-bin/gitweb.cgi?p=xeus-gp.git;a=summary) ## Prepare a CongruenceGroup type x *Participants*: Razvan Barbulescu *Short Description*: SageMath has the congruence groups Gamma, Gamma0 and Gamma1 but not the more general ones. The CongruenceGroup type could include a methods called level, gens, genus, width and one day modular_curve. ## Add efficient computation of p-adic Schottky theta functions *Participants*: Marc Masdeu *Short Description*: There are new algorithms to compute p-adic periods (via theta functions) of Schottky groups in polynomial time, given a set of generators. This is now part of the **darmonpoints** package, but since it has no other dependencies it would be nice to have it well documented and distributed with Sage. [Link](https://github.com/mmasdeu/darmonpoints) ## ThetAV *Participants*: Andreas Pieper *Short Description*: **ThetAV** is a sage library that aims to include functionality for abelian varieties using algebraic theta nullvalues, more particularly the calculation of the group law and isogenies. [Link](https://github.com/anna-somoza/thetAV) ## Round2 algorithm for function fields *Participants*: Fabrice Drain *Short Description*: We plan to implement the Round2 algorithm for normalizing orders in function fields in order to improve the performance of maximal order computation in SageMath. ## Volume contribution of Stable trees *Participants*: Vincent Delecroix, Kai Fu *Short Description*: The project is to compute the volume contribution of different stable trees to the Masur-Veech volume in the genus zero case. * compute the distribution of the number of cylinders (expectation, variance, higher moments...) and compare it with a Poisson distribution * generate all stable trees with a given number of legs * compute the volume contribution of stable trees with variables * compute the distribution of the diameter of stable trees (compute moments) ## Better integration of FFLAS-FFPACK routines for linear algebra over Z/nZ *Participants*: Vincent Neiger, +others welcome *Short Description*: Currently, the C++ library FFLAS-FFPACK is called by SageMath for basic operations on matrices over Z/nZ, for n up to ~27 bits. In 2023, work has been done to update the integration of this library and to fix some performance issues (see https://github.com/marizee/GSoC-2023 for more details). The goal is to follow up notably by working on the integration of recent updates of Gaussian elimination in FFLAS-FFPACK: PLUQ decomposition, rank profile-preserving echelonization... It should also be discussed whether Sage's LU method could/should call such fast tools: currently, Sage calls FFPACK's LU for the reduced row echelon form, but for Sage's own LU method it uses general-purpose (fairly slow) python code. ## Generalized Gröbner bases for ideals in Laurent polynomial rings, + polytopal affinoid algebras *Participants*: Legrand Lucas, anyone welcome *Short Description*: I plan to expand the capabilities of the LaurentPolynomial class by integrating generalized monomial orders and the ability to compute Gröbner bases with respect to such orders, following the Pauer-Unterkircher 1999 paper. I already have partial code for that. Additionally, I aim to use the enhanced LaurentPolynomial class to create a new class called PolytopalAffinoidAlgebra. This new class will mirror the functionalities of the TateAlgebra class, but for polytopal affinoid algebras this time. ## Interval Exchange Transformations with Flips on multiple circles *Participants*: Magali Jay, anyone welcome *Short Description*: Sage handles IETs on an interval : in particular, one can compose them, and induce them on a smaller interval (by Rauzy-Veech induction). The goal is to develop a code to do the same thing with IETs defined on a circle and even on several circles. I am specifically interested in a class of IETs with flips, where all intervals are flipped and all rotated by a same angle. I began working on this project in September. For now, I have a few functions for one circle but none for several. ## Unicellular maps *Participants*: Ivan Yakovlev *Short Description*: I would like to incorporate some of the code I have written into the `surface_dynamics` package. The code concerns surgeries on unicellular maps (Chapuy's slicing/gluing of trisections) which allow for their uniform random generation in fixed genus and number of vertices (via Chapuy-Feray-Fusy bijection). If time permits, I would like to implement some other algorithms for random generation of maps. ## Simplicial systems and their properties *Participants*: Charles Fougeron + David Siukaev *Short Description*: We compute explicitly the density of the absolutely continuous invariant measures for two Markov MCF algorithms: Selmer algorithm and Triangle map. To do this, we use the language of simplicial systems and construct a fixed dual graph in order to find invariant domains for these algorithms. Finally, using the same approach, we want to estimate the second Lyapunov exponent for the given algorithms. ## More on hypergeometric motives *Participants*: Frédéric Chapoton *Short Description*: I have code for one additional useful method on the existing class of hypergeometric motives, that should be incorporated in Sage. I would also like to write another method in the same class that creates a polytope. [Merge request](https://github.com/sagemath/sage/pull/37190) and [Merge request](https://github.com/sagemath/sage/pull/37187) ## The alt-Tamari lattices and friends *Participants*: Clément Chenevière *Short Description*: The alt-Tamari lattices are posets related to the Tamari and nu-Tamari lattices, which are implemented in Sage. I would like to add these new posets to Sage, to enable using general methods and to make them available. ## Basis of morphisms for Drinfeld modules *Participants*: Joseph Musleh *Short Description*: There is currently a pull request open for implementing a method to compute a basis for the space of morphisms between pairs of Drinfeld Fq[x]-modules over various coefficient rings. We intend to get this pull request ready for final review and merge. The PR is available here: https://github.com/sagemath/sage/pull/35386. ## Hyperelliptic curve functionality *Participants*: Alex Best *Short Description*: Hyperelliptic curves are still missing a lot of very basic functionality in Sage. Continuing from last year it would be great to rectify this! ## Certification of Sage outputs *Participants*: Alex Best *Short Description*: It would be great if more of the algorithms implemented in Sage had an option to also produce a certificate for a claimed result. I'm interested in exploring what low hanging fruit there is for letting end users more easily ask for such certificates, and improving the consistency of the library with respect to such interactions. ## Approximating Hausdorff dimension for dynamically defined Cantor sets *Participants*: Julia Slipantschuk *Short Description*: Let $E_A$ be a set of all $x\in(0,1)$ such that continued fraction of each $x$ has digits in a finite set $A$, e.g $A = \{1,2,11\}$. There is an algorithm to approximate the Hausdorff dimension of such set $E_A$ based on spectral approximation of so-called transfer operators. I have an implementation of this code in Python using FLINT. I would like to integrate this implementation into sage. I have no Sage experience, so support with Sage would be great.