Recall: pari-2.9-stable was released in 2016, pari-2.10-testing is scheduled after Atelier 2017 . # DISCUSSION FROM ATELIER 2016 & AFTERMATH ([X] = DONE!) New mathematical features for PARI-2.10 (testing) -> PARI-2.11 (stable) [X] done in PARI-2.16 ## Elliptic curves [X] - include ratpoint ? [X] - include Denis Simon's ellQ.gp [X] - descent - S-integral points - fields generated by torsion points under Galois action. [X] - canonical height of points on elliptic curves over number fields - improve the interface for complex periods of elliptic curves [X] - Falting height of elliptic curves [X] - 2-descent on elliptic curves [X] - isogeny matrix [ done over Q; not as easy as expected over K ] [X] - ellbsd to check the BSD hypothesis over number fields or p-adically ## Hyperelliptic curves / Arithmetic geometry - 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) ## 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...) [X] - rnfinit with list of prime to maximize at, like nfinit([pol,L]) (+rnfcertify ?) (useful for Aurel & Denis) [X] - compact fundamental units (bnfinit + flag) [X] - compact S-units (bnfinit + flag) [X] - rewrite bnfsunit to compute directly S-units without reducing to fundamental units ? [X] - cyclotomic units / subfields to help bnfinit (Jean-Robert) generators, implicit descriptions in terms of linear algebra should be enough. [X] - 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 [X] - rnfidealprimedec for infinite places - rewrite lowerboundforregulator [#1572] [X] - rnfkummer for composite degrees [X] - rnfkummer for prime power degrees (needed for CSA Aurel) Challenges: 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 sequences) [X] - nfpolsturm(nf,pol,i) where 1<=i<=nf.r1 (useful for Aurel & Denis) [X] - idealispower (Nicolas) [X] - idealsqrtn (useful for CSA Aurel) (Nicolas) [X] - nfissquare - non maximal orders [X] - 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 ? [X] - van Hoeij-Klueners (maximal) subfields algorithm - nfprimes for the primes in a nf whose norm lie in an interval [X] - asymptotically fast Flx_resultant [ see Flint ] - use truncated polynomials for power series instead of re-implementing the wheel ## 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 ## Parallelisation, use parallel interface internally: [X] - CRT [X] - factorint - SEA - bnfinit - ... ## GP [X] - forprimestep(p = a, {b}, Mod(c,d), ...) [X] - forfactored ## Elementary and analytic number theory [X] - ECPP (Jared) - prime counting function primepi for large arguments - primorial ## Misc [X] - Factorisation of bivariate polynomials (bifactor script) - Baker-Davenport - Generic Newton method - Abel-Jacobi map - merge K2.gp [X] - Brew recipe for OSX ## Technical stuff / internals - cleanup input/output (don't change global pari_infile, etc.) [X] - [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 [X] - separate valuation for t_SER / t_PADIC (servalp / valp) - rename all gerepile* functions -> gc_* # LEFTOVERS FROM ATELIER 2013 ## Short hacks [X] - Lambert W [ for x >= -1/e ! ] - inverse gamma / inverse erfc [X] - LinearRecurrence [ via Mod(x, T(x))^N ] [X] - Hurwitz zeta - Bell numbers [X] - sumrat \sum F(n), F rational function \sum (-1)^n F(n), F rational function \prod F(n)) [X] - 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 (= \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