New features

Recall: pari-2.9-stable was released in 2016, pari-2.10-testing is scheduled after Atelier 2017 .


New mathematical features for PARI-2.10 (testing) -> PARI-2.11 (stable)

Elliptic curves / Arithmetic geometry

- include ratpoint ?
- include Denis Simon's
- descent
- S-integral points
- fields generated by torsion points under Galois action.
- canonical height of points on elliptic curves over number fields
- improve the interface for complex periods of elliptic curves
- Falting height of elliptic curves
- 2-descent on elliptic curves
- isogeny matrix [ done over Q; not as easy as expected over K ]
- Frobenius matrix via Kedlaya's algorithm for p=2
- Genus 2 curves; in particular complex periods, Tamagawa numbers and endomorphism rings
- 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)
- ellbsd to check the BSD hypothesis over number fields or p-adically

Modular symbols

- proper normalisation of new symbols, integrality conditions
- 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...)
- 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)
  generators, implicit descriptions in terms of linear algebra should be
- regulator/units and class number/class group for special cases of high degree fields
  (abelian fields, say), using different methods from bnfinit
- nfinit for abelian fields using adapted bases
- rnfidealprimedec for infinite places
- rewrite lowerboundforregulator [#1572]
- rnfkummer for composite degrees
- rnfkummer for prime power degrees (needed for CSA Aurel)
      nf = nfinit(y);
      p2 = idealprimedec(nf,2)[1];
      p3 = idealprimedec(nf,3)[1];
      al = alginit(nf,[4,[[p2,p3],[1/4,3/4]],[0]],0);
      al = alginit(nf,[9,[[p2,p3],[1/9,8/9]],[0]],0);
      nf = nfinit(y^2+3);
      p3 = idealprimedec(nf,3)[1];
      p7 = idealprimedec(nf,7)[1];
      al = alginit(nf,[4,[[p3,p7],[1/4,3/4]],[]],0);
      nf = nfinit(y^2+1);
      p2 = idealprimedec(nf,2)[1];
      p5 = idealprimedec(nf,5)[1];
      al = alginit(nf,[9,[[p2,p5],[1/9,8/9]],[]],0);
- abstract / generalize ad hoc abelian groups constructions (short exact
- 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
- rnfsplitting
- unify and make accessible the Finke-Pohst implementations

- more central simple algebras (Aurel)
    - localisation (local splitting, localisation of an element, compute local invariants in general, additive strong approx)
    - lattices (inter,add,left/right mul,conj,index,subset,left/right order,norm,twoelt,inv,primedec,factor)
    - orders (pmaximal, connecting ideal)
    - advanced (class sets, multiplicative strong approx, generators of ideals, units)
    - algsimpledec: return simple algebras in alginit format
    - T2 norm and LLL-reduce the basis
    - global splitting and isomorphism
- Fieker-Klueners polgalois algorithm, GAP module using PARI ?
- van Hoeij-Klueners (maximal) subfields algorithm
- nfprimes for the primes in a nf whose norm lie in an interval


 - 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
 - hooks to optionally link with mpfr / mpc higher lever routines
 - asymptotically fast Flx_resultant [ see Flint ]
 - use truncated polynomials for power series instead of re-implementing the wheel

Parallelisation, use parallel interface internally:

 - CRT
 - factorint
 - SEA
 - bnfinit
 - ...


-  forprimestep(p = a, {b}, Mod(c,d), ...)
-  forfactored

Elementary and analytic number theory

 - ECPP (Jared)
 - prime counting function primepi for large arguments
 - primorial


 - Factorisation of bivariate polynomials (bifactor script)
 - Baker-Davenport
 - Generic Newton method
 - Abel-Jacobi map
 - merge
 - Brew recipe for OSX

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
- separate valuation for t_SER / t_PADIC (servalp / valp)
- rename all gerepile* functions -> gc_*


Short hacks

 - Lambert W [ for x >= -1/e ! ]
 - 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

[X] - vecprod (= _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