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 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 is (an upper bound which is implied by the Riemann Hypothesis) or (inspired by omega-results for ).

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 .

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

Define

Then the rank of is . For example, if and , then

Let be the number of divisors of . Then

so that . Can be as large as this?

Nick Katz suggests the following problem:

**Prove or Disprove:**
, where the conductor
of is
, and 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 ``'' in the above problem is finite.

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