Christophe Ritzenthaler

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Please send comments and bugs to ritzenth(at)iml.univ-mrs.fr

Invariants and reconstruction of genus 3 hyperelliptic curves

This is a work in collaboration with Reynald Lercier. It has been implemented in Magma (v.2.17). You can download the .tgz here.

Short description: It provides (in characteristic not equal to 2,3,5 or 7) the computation of a set of generators for the ring of invariants (the Shioda invariants) and construct a hyperelliptic curve from given Shioda invariants. It also gives the geometric automorphism group of the curve (as an abstract group). Note that this works also for family of curves.
The construction takes place (in most cases) over the field of moduli when there is no obstruction.

Changes: Thanks to new ideas using covariants we hope to implement soon the construction over the field of moduli when there is no obstruction in all cases and to compute the obstruction (in particular when the automorphism group is (Z/2Z)^2).
A future version will also include the computations of twists over finite fields.

References:
  • The preprint Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects.
  • The slides Invariants and hyperelliptic curves: geometry, arithmetic and algorithmic aspects.

    • Invariants and reconstruction of genus 2 hyperelliptic curves

    This is a work in collaboration with R. Lercier. It has been implemented in Magma (v.2.13) and is part of the current Magma distribution. You can also download the .tgz here.

    Short description: It provides (in all characteristics --especially the difficult cases 2,3 and 5--) the computation of a set of generetors for the invariants and construct a curves from given invariants. It also gives the geometric automorphism group of the curve (as an abstract group). Note that this works also for family of curves.
    The construction takes place over the field of moduli when there is no obstruction and compute all twists (not only quadratic) over a given finite field.

    AGM algorithm for non-hyperelliptic curves of genus 3

    This is a work in collaboration with M. Fouquet and P. Gaudry. It has been implemented in Magma (v.2.09). You can download the .tgz here.

    Short description: This program performs the computation of points on a genus 3 ordinary non hyperelliptic curve over F_2^N thanks to the so-called AGM method. The folder also contains a program written by S. Flon, R. Oyono and myself to make addition in the Jacobian of a plane smooth quartic.

    Changes: this program does not work with the new version of MAGMA. You'll have to make changes in the definitions of the p-adic spaces. It is not completely optimized (the loops are much slower than Lercier and Lubicz's ones).

    Tests: The computation of the initial theta was done thanks to the present program but the loop process was carried out by Lercier using their program.
    curve over F_2^N computation of the initial theta constants computation of the product of the pi_i with a precision of 10 N computation of the minimal polynomial with LLL Signe ; Frobenius polynomial
    N=100   65 s  1mn 2s 4s answer
     N=5002  34000 s more or less 2 weeks
    		 1mn 27000s answer

    References :
  • Point counting on genus 3 non hyperelliptic curves, Algorithmic Number Theory 6th International Symposium, ANTS VI, University of Vermont 13-18 June 2004, Proceedings.
  • AGM method for non hyperelliptic curves of genus 3.


    • Short programs:

    Magma programs to check the computations of  `Fast computation of isomorphisms of hyperelliptic curves and explicit descent' with Reynald Lercier and Jeroen Sijsling.
  • For Sections 1.5, 2.3.1 and 2.3.2: the programs, associated data and the package for genus 3 hyperelliptic curves.
  • For Section 2.4: the general descent program; the resultin equation (8Mo) and the program to descend a particular example.

    • Two programs related to the article on intersection of a line and a quartic :
  • Computation of the correspondence curve in characteristic 2 (magma 2.13).
  • Flexes in characteristic 3 (Maple 11).

    • A program (collaboration with P. Trebuchet) in MAGMA which tests if a plane curve over a finite field is absolutely irreducible. It is based on E. Kaltofen article : Fast parallel absolute irreductibility testing, J. Symb. Comp. 1, (1985), 57-67.

    A program in MAGMA which computes the p-rank of the modular curves X(N).

    A basic program in MAGMA for computing the number of points on an elliptic curve with AGM as suggested by J.-F. Mestre.