# Atelier 2017b

# Previous Ateliers:

2015 (Bordeaux), 2016 (Grenoble), 2017 (Lyon)

# Welcome to Atelier PARI/GP 2017b (Clermont)

## Tutorials

## New features

### Genuine Bianchi modular forms for Pari/GP :

Alexander D. Rahm wants to find out if there is interest in a script for the Pari/GP scripts library, computing dimensions of genuine Bianchi modular forms. Details will be given in his talk on Thursday morning, summarized here :

http://math.uni.lu/~rahm/genuine_Bianchi_modular_forms_in_PARI-GP.txt

### interface issues

- mffromell (from hell?) -> mfell…
- mfcoefs: inconsistent with polcoeff

- mfseries(mf,n,{v}) ?
- mfcoeffs
- mfexpansion (since mfcuspexpansion)

- mftolfun -> lfunmf
mfinit/mfdim : change flag order to new, old, cusp, eisen, full

- znchargalois -> chargalois
lfuninit(lmath,[t]) -> lfuninit(lmath,t)

## Tasks

- Doctesting-2017b (Karim Belabas)
- Everything and anything (Karim, Bill, Henri)
- Doctesting done. Improved my own code on p-adic modular forms with the new functions. Now running some tests for a conjecture of mine relative to minimal slopes of p-adic modular forms for p=5. (Dino D.) Lots of progress!
- Rational points on elliptic and hyperelliptic curves over Q_p (Devika), going to work with a script by Denis for descent (which allowed to find some examples)
- 2-descent on hyperelliptic curves over Q (Nicolas M., Pascal) Nicolas has implemented 2-descent for hyperelliptic curves in the simplest case. The plan is to use it to do Chabauty style work and determine all rational points. Group law on an hyperelliptic curves in project.
- Clean and maybe improve my own code on p-adic modular forms with the new functions (Dino D.)
- ECPP (Jared, Jean-Pierre)

- currently experimenting in plafrim
- looking to implement Galois decomposition to be able to get roots for class polynomials which are b-smooth where b is small (b <= 32)
- now the branch is improved and workable (test it!)

- PARI-GNUMP (Andreas E., Fredrik) autotools support is ready a preliminary interface with arb has been written; the exact semantics needs to be decided, the full build-system is done and working, some progress (e.g. can take tuple as input…)
- Generalise ellisomat to curves over number fields (Nicolas B.) Bill has extended ellisomat with an extra argument corresponding to the prime isogeny degree. Inclusion of Nicolas’s script is on-going, CM curves still pose problems, as well as the computation of minimal polynomials of generators of principal ideals. still combining both algorithms, Bill has now added to PARI/gp an (improved) version of Nicolas’s method.
- Tannakian symbols and multiplicative functions (Torstein, Andreas H.) some functions have been translated from Sage to GP and are running faster now an algorithm detecting linear reduction was written (compare it to what already exists in PARI?), optimizations were performed
- Improve GP scripts for inclusion into PARI/GP: points on elliptic curves over Q using 2-descent, also discussing with Devika on a connected issue (Denis) the code (ell2descent) has been imported into a git repository on the pari server, an open problem is to determine linear combinations of points with finally small coefficients: use real embeddings or modular arithmetic
- documentation of modular symbols and forms (Bernadette), working on congruences (with François)
- statistics of Frobenius actions of curves over finite fields (Florent) I consider the family E_t: y^2=f(x)(x-t) over Fq. Here f is a monic integral separable polynomial of degree 2g. I am interested in statistics for the numerator L_{f,t} of the zeta function of E_t over Fq. It is a polynomial of degree 2g and maximal Galois group over Q isomorphic to W_{2g} (the ‘signed’ permutations of pairs of roots in S_{2g}). Results: overwhelming evidence that the typical Galois group is W_{2g}. Very few exceptions. Next task: what are the other Galois groups that can be realized in this way. What is the occurence of these ‘pathologic’ groups? What are the groups that do NOT occur as the Galois group of some L_{f,t}? Some exceptions were found.
- revive the git branch of the Wednesday talk (Pascal) there are bugs, in the process of being collected (and maybe resolved), will start to work on Hecke characters
- encrypted matrix multiplication with Paillier cryptosystem (Matthieu)
- fundamental units of totally complex quartic fields (祝辉林) progress is being made for special number fields
- iterators over fundamental discriminants (Jared) now a new branch jared-forfunddisc, you can look at it
- functions for modular curves (Antonin)
- dimensions of Bianchi modular forms (Alexander) more details to be expected during the Thursday talk, waiting for user’s feedback and expectations!
- Finding the rational expression for the generating function of a given linearly recursive sequence. Application: Recognizing reciprocity laws (Olav, Magnus) code is being implemented in GP, no special problems to report
- Computing Meissel-Mertens constants for abelian number fields (Alex M., Preben) the code for quadratic number fields has been ported from Sage to GP and is faster now work for higher degree number fields is on-going, constants computed with higher precision, still work in progress, but progress
- Congruences of modular forms (François), some examples were computed, now working on their interpretation (with Bernadette)
- Read again the documentation on the ray class group and p-rationality (Razvan)
- Translating a project related to elliptic curves in PARI from Sage (Sudarshan)