Let E be an elliptic curve over Q with positive rank and trivial torsion structure. Is there any sort of upper bound (conjectural or unconditional) on the regulator of E in terms of the conductor of E? (For lower bound we have Lang's conjecture.)

In any specific case, you can get a conjectural upper bound from the leading coefficient of the $L$-function at $s=1$. Is that what you are after or are you looking for a theoretical uniform bound? If the latter, then dependent on what? It is very unlikely that there is a constant uniform bound that works for all curves.
– Alex B.Nov 11 '10 at 5:12

I was looking for an upper bound in terms of the conductor or the discriminant of E. I've been trying to figure out if I can give some sort of a bound for $\lim_{s\rightarrow 1}L^{(r)}(s)$ using the conductor or the discriminant of the elliptic curve, but I don't know if that's reasonable or not.
– SorooshNov 12 '10 at 19:05

There is Lang's conjecture on the regulator times size of Sha. See his book "Survey of Diophantine Geometry", pp. 99, Chapter 3 section 6, conjecture 6.3. If you don't have access to the book (and can't search google), I can post it here.
– Dror SpeiserNov 12 '10 at 20:10

2

$L^r(1)$ can be bounded by complex analysis, in particular, if you don't care about optimal results, use the Phragmen Lindelof theorem. I think it should give $L(1)$ is less than $N^{1/4}$ times some power of $\log N$. Each deriviative introduces another log via a Cauchy derivative formula, among other methods. More explicitly, $L(s)$ is bounded by $(\log N)^{2?}$ on the $s=3/2$ line via a limiting Euler product (or do it on $s=3/2+1/\log N$), and then by the functional equation, you get a bound $\sqrt N$ times this on the $s=1/2$ line, and apply convexity to get $N^{1/4}$ in the middle.
– JunkieFeb 28 '11 at 12:30