The maximal rank of an elliptic curve as a function of its conductor

Many people believe that ranks of elliptic curves are unbounded. If so, what is the maximal rank of an elliptic curve as a function of its conductor? This question is related to the maximal size of the argument of $L_E(1/2)$ for the (normalized) L-function associated with an elliptic curve E. (See the article on the maximal size of $S(T)$.)

If one were to make a guess, the two possibilities which may seem most natural are that the maximal rank of a curve of conductor $N$ is $\log N/\log \log N$ (an upper bound which is implied by the Riemann Hypothesis) or $\sqrt{\log N/\log \log N}$ (inspired by omega-results for $S(T)$).

In the function field case the answer is the larger of these two as recent work of Ulmer [ arXiv:math.NT/0109163] shows. Ulmer conjectures that the larger bound is occasionally achieved for elliptic curves over $Q$.

An elementary version of this problem was formulated by Penney and Pomerance ([MR 51 #12862] and [MR 51 #12861]) for curves of the form

$$E: y^2=x^3+ax^2+bx.$$

Define

$$A=\{n: n\mid b \hbox{ and } n+b/n+a \hbox{ is a square}\}.$$

Then the rank of $E$ is $\gg \log \vert A\vert$. For example, if $a=17$ and $b=-105$, then

$$A=\{-21,-15,-7,-3,-1,5,7,15,35,105\}.$$

Let $d(b)$ be the number of divisors of $b$. Then

$$\limsup \frac{d(b)}{2^{\log b/\log \log b}}=1,$$

so that $\log \vert A\vert\ll \log b/\log \log b$. Can $\log \vert A\vert$ be as large as this?

Nick Katz suggests the following problem:

Prove or Disprove: $rank(E) \le C \sum e_i$, where the conductor of $E$ is $\prod p_i^{e_i}$, and $C$ is the maximal rank of an elliptic curve with prime conductor.

It is conjectured by Brumer and Silverman [ MR 97e:11062] that the number of of elliptic curves of prime conductor is infinite. So it is not clear that the constant ``$C$'' in the above problem is finite.

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