Heights on hyperelliptic curves

Discussion in 'Math Research' started by Harald Helfgott, Jul 8, 2003.

  1. Dear all,

    (1) Take the hyperelliptic curve C: y^2 = f(x), where deg(f)=6. Do not assume
    that C has a rational point. Fix a point P_0 of C and embed C into its Jacobian
    correspondingly: P -> [P]-[P_0]. What relation does the canonical height
    on the Jacobian have with a naive height on C, such as, for example, h_x?
    (Define h_x by h_x((x,y))=x.) I seem to recall that we do *not* have a bounded
    difference of the form

    |h_x(P) - h(P)| <= C

    -- after all, h_x is not a naive height on the Jacobian itself. What is the
    best we can do? Can we do better than, say, h(P) << h_x(P) << h(P)?

    (2) Is there a Nagell-like algorithm taking a hyperelliptic curve

    C: y^2 = f(x), deg(f)=6

    with a rational point to a hyperelliptic curve C: y^2 = f(x), deg(f)=5?

    (3) What bounds do we have on the rank of the Selmer group of a curve of
    genus 2? For elliptic curves, we have things such as

    rank(S) <= g + 2 + e + \sum_{p a place of add. red.} (n_p - 1),

    where g depends on the ideal class group of a cubic extension F of Q associated
    with E, e is the number of places of mult. red. and n_p is the number of
    primes of F lying over K (see Brumer and Kramer, The Rank of Elliptic Curves,
    Duke Math. J., Vol 44, n 4). I have tried to get something similar for
    curves of genus 2, but it seems very tricky. Is there a result of that sort
    in the literature? (I haven't had any luck with mathscinet so far.)

    Harald Helfgott, Jul 8, 2003
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  2. [question on hyperelliptic curves of genus 2 snipped]

    I don't know the answer to your questions, but you might, for explicit
    computations and references, want to have a look at

    @article {MR91m:14045,
    AUTHOR = {Grant, David},
    TITLE = {Formal groups in genus two},
    JOURNAL = {J. Reine Angew. Math.},
    FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik},
    VOLUME = {411},
    YEAR = {1990},
    PAGES = {96--121},
    ISSN = {0075-4102},
    CODEN = {JRMAA8},
    MRCLASS = {14H40 (14K25 14L05)},
    MRNUMBER = {91m:14045},
    MRREVIEWER = {Sheldon Kamienny},
    Maarten Bergvelt, Jul 8, 2003
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