# Atelier 2015

- Welcome to Atelier PARI/GP 2015
- New features / Tasks
- Beginning in GP
- Paridroid
- Windows port
- Julia
- Cleaner Sage integration
- Zariski closure of matrix groups
- Two descent for elliptic curves over Q (version in C?)
- Point searching on cubic models and quadric intersections (3-covers of 4-covers of EC/Q)
- Hecke characters
- Elliptic curves over number fields
- L-functions (which?)
- Modular polynomials
- (Even) faster point counting on (hyper)elliptic curves over finite fields
- Central simple algebras
- Local root number of families of elliptic curves
- Markov Models with Pari
- Representations of fractional ideals of a number field
- Testing and improving parallelism with PARI/GP
- Isometries and automorphisms of quad. form.
- Applications of GP2C
- Fix Pari’s AGM of real or complex power series
- Pisot numbers and matrices
- Improve linear algebra (especially over finite fields)
- Discuss prefix uses in gp
- About modular symbols for Gamma_0(N)
- Planned features for 2.8/2.9

# Welcome to Atelier PARI/GP 2015

## New features / Tasks

## Beginning in GP

- Jean-Robert, Coline

## Paridroid

Get it into the Fdroid and Google Play app stores (Andreas and Laurent)

And FirefoxOS Marketplace too! (Luca) Note: Won’t work natively, need to compile using emscripten. Then, we can have Pari in the browser :)

## Windows port

- Check status on Cygwin64 (JPF)
- Issue on cygwin64 linked to mmap (disabling mmap fix them) (Bill A.)
- Compare perfs using Cygwin32/64 and MinGW32/64 (JPF)

## Julia

- Develop a wrapper of Pari ideal arithmetic and maximal orders in the Julia programming language (Bill Hart - Kaiserslautern)

## Cleaner Sage integration

Sage would benefit from a better integration of pari, see sage.

Vincent Delecroix, Jeroen Demeyer, Peter Bruin, Hamish, Luca.

## Zariski closure of matrix groups

- Given a finite set of integer matrix, compute (or at least approximate from below) the Zariski closure of the group generated by them (Vincent)

## Two descent for elliptic curves over Q (version in C?)

- Denis and Jeroen and Tadic
- Mark has a GP script from 2007, that is quite efficient, about 500 lines of code

## Point searching on cubic models and quadric intersections (3-covers of 4-covers of EC/Q)

- Denis + Mark

## Hecke characters

- Mark

## Elliptic curves over number fields

- Global minimal equations (François Brunault+ Tadic)

## L-functions (which?)

- Pascal + Henri + Amalia + Charlotte

## Modular polynomials

- Hamish + Luca

## (Even) faster point counting on (hyper)elliptic curves over finite fields

- JPF

## Central simple algebras

- Aurel + Denis
- see Central simple algebras

## Local root number of families of elliptic curves

- Christophe

## Markov Models with Pari

- Implementation of Forward Backward Algorithm
- Implementation of Expectation Maximization Algorithm (Kevin, Guenaelle)

## Representations of fractional ideals of a number field

- Two-element representation (Thomas Camus)

## Testing and improving parallelism with PARI/GP

- testing, evaluation and improvement (Philippe EV, Kevin, Thomas, Marie, Guenaelle)

## Isometries and automorphisms of quad. form.

- testing and evaluation of qfisom and qfauto with several families of lattices (Philippe EV, Thomas, Guenaelle, Kevin)

## Applications of GP2C

- Compilation of the GP script library Bianchi.gp with GP2C (project of Alexander Rahm - NUI Galway) (gp2c bug fixed [Bill A.])

## Fix Pari’s AGM of real or complex power series

- Example: agm(1.0, 1.0*I + x + O(x^200)) [Fredrik]

## Pisot numbers and matrices

- Write an algorithm to generate Pisot numbers of fixed degree (this is a discrete set)
- Given a Pisot number of degree d, try to find a dxd matrix which admits the given Pisot number as eigenvalue

The gp script localization.gp localizes roots of polynomials with respect to the unit disk (i.e. counts roots outside the unit disk, on the unit circle, and inside) [Christelle,Vincent].

## Improve linear algebra (especially over finite fields)

- Use standard “building blocks” like LU factorisation instead of custom functions like gauss_pivot
- Use Kronecker representation to multiply matrices over non-prime finite fields
- Strassen multiplication (preliminary version exists, but should probably be done after the above things)

(Peter Bruin)

## Discuss prefix uses in gp

- GP prefix (Pascal)

## About modular symbols for Gamma_0(N)

- test the ms functions
- check the p-adic bsd conjecture for elliptic curves
- look at the genus 2 jacobian of curves coming from the modular symbols

(Bernadette Perrin-Riou)