Re: Finding a lower bound on canonical height of an elliptic curve
William Stein
<ariah.kla...@lmh.ox.ac.uk> wrote:
> Hi, William,
>
> I found a paper that describes an algorithm for computing a lower bound for
> the canonical height of points on an elliptic curve over a totally real
> number field (see attached).
Nice.
>
> Do we also need to prove that a positive lower bound exists in all cases (or
> do we already know this)?
For each individual curve there is certainly a lower bound. For our
project we do not need a uniform
bound over all curves at once, though that is an independently
interesting question.
> Either way, I think I also have a paper by
> Silverman that establishes this -- I'm just not sure about some of the
> notation in the statement of the theorem. The theorem is stated as follows:
>
> Let K/Q be a number field, let E/K be an elliptic curve, and let h: E(K) ->
> R be the canonical height on E. Then there is a constant C(E/K) > 0 s.t.
> every nontorsion point P in E(K^ab^) satisfies h(P) > C. (I believe K^ab^/K
> is defined as the maximal abelian extension).
That's true.
> My question: does the notation C(E/K) simply mean that the constant C
> depends on E/K (the elliptic curve over number field K)?
Yes.
> I'm assuming it
> does, in which case the theorem provides what we need (that there is always
> a lower bound, and so we can use the algorithm in the attached paper to find
> it in all cases).
I think this may be orthogonal to what we need. What's needed in our
project is a constant B(E/K)
such that for any point P on the curve we have
| h(P) - naive_height(P) | <= B(E/K).
-- William
>
>
> -Ariah
>
--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org
William Stein
<ariah.kla...@lmh.ox.ac.uk> wrote:
> All right, well Martin Prickett's thesis gives us two upper bounds on the
> difference between canonical and naive height (see attached). I was under
> the impression that we needed a lower bound on canonical height, but,
> regardless, it looks like we have both.
Can you start by implementing Theorem 4.4.1 of the attached thesis,
which gives a bound on the difference between the naive and canonical
height?
>
> Ariah
