Editing New features / Tasks

Recall: pari-2.7-stable was released in March 2014, pari-2.8-testing is
scheduled after Atelier 2015 (february ?).

# DISCUSSION FROM ATELIER 2014 & AFTERMATH ([X] = DONE!)

New mathematical features for PARI-2.8 (testing) -> PARI-2.9 (stable)

## Elliptic curves / Arithmetic geometry
    - [X] over number fields [ general infrastructure ]
    - [X] Tate's algorithm over number fields, above 2 and 3 as well ! (Nicolas?)
    - [X] global minimal equation, Weierstrass class
    - [X] qfsolve [ over Q ]
    - include ratpoint ?
    - include Denis Simon's ellQ.gp
    - descent
    - S-integral points
    - fields generated by torsion points under Galois action.
    - [X] modular equation / class polynomials (Hamish)
    - isogeny matrix [ done over Q ]
    - [X] pairings / APIP [ Jerome Milan's package ] : 
           - fix Tate self pairing
           - FpE_FpQXE_* [ 1 pt in base, 1 pt in extension ]
           - various optimized crypto pairings
    - [X] Frobenius matrix via Kedlaya's algorithm for p>2
    - Frobenius matrix via Kedlaya's algorithm for p=2
    - Genus 2 curves
    - Export local solvability of hyperelliptic equations (over number fields)
    - Arithmetic & pairings on Mumford representation for hyperelliptic
      curves Jacobians
    - Khuri-Makdisi's algorithms (Jacobians of (modular) curves) (Peter, Nicolas)

## Modular symbols
    - [X] modular symbols, merge the kb-modsym2 branch
    - support \Gamma_0(1) !
    - q-expansions for Eisenstein symbols
    - support \Gamma_1(N)
    - other coefficient modules F_q[x,y]_{k-2}, p-adic distributions
      (overconvergent symbols), etc.

## Algebraic number theory
    - dynamic nf (confer 'ell'): add new components dynamically as they are
      computed (integer basis, class group/units...)
    - [X] better rnf structures allowing maps between TWO nf [not 1 nf + 1 hack]
      (useful for CSA Aurel)
    - [X] rnfidealprimedec
    - rnfinit with list of prime to maximize at, like nfinit([pol,L]) (+rnfcertify ?) (useful for Aurel & Denis)
    - compact fundamental units (bnfinit + flag)
    - compact S-units (bnfinit + flag)
    - rewrite bnfsunit to compute directly S-units without reducing to
      fundamental units ?
    - cyclotomic units / subfields to help bnfinit (Jean-Robert)
    - [X] get rid of bnrinit(,,1): *no* function should need explicit expanded
      generators, implicit descriptions in terms of linear algebra should be
      enough.
    - rewrite lowerboundforregulator [#1572]
    - [X] relative polcompositum
    - rnfkummer for composite degrees
    - rnfkummer for prime power degrees (needed for CSA Aurel)
    - [X] fix rnfkummer (fix and merge kb-rnfkummer)
    - [X] bnrisgalois
    - [X] polsturm -> Uspensky (merge loic-Uspensky)
    - abstract / generalize ad hoc abelian groups constructions (short exact
      sequences)
    - [X] sign of algebraic number [ proven ] (useful for CSA Aurel, useful for lots of stuff Denis)
    - nfpolsturm(nf,pol,i) where 1<=i<=nf.r1 (useful for Aurel & Denis)
    - idealispower (Nicolas)
    - idealsqrtn (useful for CSA Aurel) (Nicolas)
    - nfissquare
    - non maximal orders
    - cubic / quartic / quintic fields by discriminant
    - [X] support idealstar / ideallog with nf = NULL (over Q)

- more central simple algebras: orders, ideals, localizations (Aurel)
    - Fieker-Klueners polgalois algorithm, GAP module using PARI ?
    - van Hoeij-Klueners (maximal) sufbields algorithm
    - primesnf() for the primes in a nf whose norm lie in an interval

## L-functions (Pascal / Henri)
    - [X] Dirichlet character
    - [X] Hecke character
    - [X] L(s,\chi), at least for deg <=2
    - [X] Artin L-functions

## Multiprecision:
     - Change libpari prec variable to be in bits instead of words.
     - merge the new-t_REAL branch
     - go through transcendental functions and include rigorous/faster
       algorithms
     - hooks to optionally link with mpfr / mpc higher lever routines
     - asymptotically fast Flx_resultant [ see Flint ]

## Parallelisation, use parallel interface internally:
     - CRT
     - [X] polmodular
     - factorint
     - znlog
     - SEA
     - bnfinit
     - ...

## GP
    -  [X] 0xDEADBEEF  (integers in hexadecimal)
    -  [X] expose the iterator associated to forxxx() functions, e.g.
       forprime [ vectorprime(), sumprime(), prodprime(), prime ideals... ]
       partition / forpartion [ ... ]
       certain subsets (e.g. sets with m elements, words of Hamming
       weight k...)

-  forprimestep(p = a, {b}, Mod(c,d), ...)
    -  forfactored
    - [X] fold
    - [X] move "useful" functions from gp.c -> libpari (e.g. handling of
        \[a-z] shortcuts)
    -  V[-1] (= V[#V-1]) ?
    -  V[a..b] when a > b ? (= [])

[CONTROVERSIAL: cool but marginally useful compared to effort]
    - "dictonary arguments" (aka named parameters)
       f( disc := 10, p := 7, len := 18 )

instead of f(len, disc, p) = ; f(18,10,7)  \\ imagine 42 arguments

## Misc
     - Factorisation of bivariate polynomials (bifactor script)
     - Baker-Davenport
     - Generic Newton method
     - Abel-Jacobi map

## Technical stuff / internals
    - cleanup input/output (don't change global pari_infile, etc.)
    - [Configure] let gcc try to find its libraries first [ don't start by
      overriding with /usr/lib, /usr/lib64, etc ]
    - properly tune Flx operations wrt p and degree.
    - cleanup entree*
            - remove built-in hashtable and use generic one
            - don't overload "value" / remove "useless" struct members
    - parser must only create polynomial variables when creating a t_POL/t_SER
    - separate valuation for t_SER / t_PADIC (servalp / valp)
    - rename all gerepile* functions -> gc_*
    - fix DLLDFLAGS on OS/X [#1623]

# LEFTOVERS FROM ATELIER 2013

## Short hacks
     - Lambert W [ for x >= -1/e ! ]
     - primepi for large arguments
     - inverse gamma / inverse erfc
     - LinearRecurrence [ via Mod(x, T(x))^N ]
     - Hurwitz zeta
     - Bell numbers
     - sumrat \sum F(n),        F rational function
              \sum (-1)^n F(n), F rational function
              \prod F(n))
     - sumeulerrat,  \sum_{p prime} F(p^s)
                  \prod_{p prime} F(p^s)
     - tools for p-adic analysis: Newton polygon (slopes, # of zeros), Amice
       transform and interpolation
     - vecprod (= \prod_i v[i]), see vecsum

## Long term projects 
     -  change t_SER format (-> faster + easier to maintain)
         s[0] = type | lg
         s[1] = valuation 
         s[2] = precp
         s[2] = t_POL + functions operating on t_POL mod X^n [ RgXn_... ]
     -  t_REAL format
     -  stat functions ? -> Gaussian vectors

- Elliptic curves over finite fields: [proposed by Damien Robert]
        1) Basic
        - geometric points over an extension
        - base change
        - Various models, morphisms between them, e.g. from / to Weierstrass
        2) Isogenies
        - isogeny graphs
        - If $\phi_l(j,j')=0$, compute the isogeny corresponding to E, E'
        - Isogeny class of an elliptic curve
        - Endomorphism ring
        3) Misc
        - Weil restriction
        - Symplectic basis for l-torsion

Description of changes: