From last year: - The PARI library: * short hacks 1) Lambert W 3) sqrtnint 5) primepi for large arguments 6) inverse gamma / inverse erfc 7) LinearRecurrence [ via Mod(x, T(x))^N ] 8) Hurwitz zeta 9) Bell numbers 10) sumrat \sum F(n), F rational function \sum (-1)^n F(n), F rational function \prod F(n)) 11) sumeulerrat, \sum_{p prime} F(p^s) \prod_{p prime} F(p^s) 13) tools for p-adic analysis: Newton polygon (slopes, # of zeros), Amice transform and interpolation * long term projects 3) number field structures [ cycgen / matalpha ] 4) Pascal Molin's complex L function package 5) Dirichlet characters ? 6) change t_SER format s[0] = type | lg s[1] = valuation s[2] = precp s[2] = t_POL faster t_SERdd 7) t_REAL format 8) 2-descent over Q, then number fields [ Denis Simon's scripts ] 9) pairings / APIP [ Jerome Milan's package ] 10) modular symbols [ generalize modsym.gp to weight k >= 2 and Gamma_1(N) ] 11) Frobenius matrix via Kedlaya's algorithm 13) Denis Simon's qfsolve package 14) stat functions ? -> Gaussian vectors 15) Elliptic curves over finite fields: [proposed by Damien Robert] 0) Field of definition OK Elliptic curve / Fq - geometric points over an extension - base change 1) Arithmetic - Various models, morphisms between them, e.g. from / to Weierstrass OK: Weil / Tate pairing OK: Pick a random point on a curve - Arithmetic & pairings on Mumford representation for hyperelliptic curves Jacobians 2) Invariants: OK: j-invariant OK: create curve with given j - compute twists 3) l-torsion OK: Division polynomials - Symplectic basis for l-torsion 4) Isogenies - modular polynomials $\phi_l$ (for l prime, say). - isogeny graphs - VĂ©lu formulas (starting from the half-sum of x-coordinates in kernel H, compute E/H) - If $\phi_l(j,j')=0$, compute the isogeny corresponding to E, E' - Isogeny class of an elliptic curve 5) Structure OK: point counting OK: group structure (Z/n1Z + Z/n2Z) - Endomorphism ring 6) Misc. OK: discrete log - Weil restriction GENUS 2 curves - same for genus 2 ===================================================================== elliptic curves: over Q: S-integral points : fields generated by torsion points under galois action. over number field: factorisation of bivariate polynomials: Baker-Davenport number fields: S-unit sign of algebraic number idealispower/better idealfactor nfissquare Iterators: forrat(q,h1,h2,f(q)) Generic Newton method Non maximal orders Sum/Product of a vector Abel-Jacobi map Quaternion algebra (Aurel, Jeroen, Christophe Delaunay) Quadratic fields/Cubic fields/quintic fields Parallelisation.