# Heights on hyperelliptic curves

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

1. ### Harald HelfgottGuest

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

Harald Helfgott, Jul 8, 2003

2. ### Maarten BergveltGuest

[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