From: Joseph H. Silverman
Subject: Re: Heights of E(Q) generators
Dear Iftikhar,
If I had to make a guess, I'd guess that there are absolute constants
c1 and c2 such that on "most" rank 1 elliptic curves E/Q, the smallest
nontorsion point P in E(Q) satisfies
h(P) > c1*|Disc(E)| + c2.
This is the logarithmic canonical height, so the bound would be
exponential. The next question would be what one means by "most".
It wouldn't surprise me if "most" means that the probability of
having
h(P) < c1*|Disc(E)| + c2
is O(|Disc(E)|^{-c3}), but maybe it could be subexponential (I'd be
quite surprised if it were polynomial).
BTW, experiments are generally _very_ misleading, because when looking
at collections of elliptic curves with small coefficients (e.g., less
than 10^6), one often includes families of curves with points whose
heights are polynomial. This effect should die away for truly large
coefficients, but no one can really do the requisite calculations in
those huge ranges. Also, there are some effects where the error term
is something like O(1/loglog|Disc|), so one literally can't do
computations in which the error term is negligible.
Offhand I don't have a reference for the analogy between unit
equations (like Pell's equation) and elliptic curves. But the more
general analogy is between algebraic tori, which are twists of
products of multiplicative groups, and abelian varieties. These are
two of the three basic types of commutative algebraic groups and they
have many similarities, although of course there are differences, too,
since abelian varieties are projective (i.e. complete), while tori are
affine (hence noncomplete).
Anyway, you have an interesting idea, but I'd be surprised if it was
better than subexponential, while I wouldn't be at all surprised if it
is fully exponential.
Yours truly,
Joe Silverman
From: Iftikhar Burhanuddin
Subject: Heights of E(Q) generators
Professor Silverman,
In a paper of ours (which is attached), we came up with a reduction
between a subproblem of factoring and the problem of computing a
generator of E(Q), where E is a member of a family of conjectural rank
1 elliptic curves.
For the reduction to be polynomial time, we conjectured that the
height of the generator is upper bounded by a polynomial of degree at
most 3 (in log of the discriminant). We have realized that the data we
collected cannot be used to heuristically support the conjecture.
My question is whether it is possible for a polynomial bound (as
opposed to Lang's exponential bound) to exist. In his Conjectured
Diophantine Estimates paper, Lang says: "It would be interesting to
have a precise statistical analysis of the possibility of a polynomial
estimate". We fail to grasp what he meant by this statement.
If a polynomial bound is not possible, then an explanation might be helpful.
The people I've spoken to refer to Pell's equation and the size of
fundamental units in the real quadratic number field case. This
analogy seems tenuous as we do not see any explicit connection and
have not come across any literature which elaborate on this analogy.
Your feedback will be greatly appreciated.
Regards,
Iftikhar.
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