  The well-known analogies between number fields and function fields have led
to the transposition of many problems from one domain to the other. In this
paper, we will discuss traffic of this sort, in both directions, in the theory
of elliptic curves. In the first part of the paper, we consider various works
on Heegner points and Gross-Zagier formulas in the function field context;
these works lead to a complete proof of the conjecture of Birch and
Swinnerton-Dyer for elliptic curves of analytic rank at most 1 over function
fields of characteristic > 3. In the second part of the paper, we will review
the fact that the rank conjecture for elliptic curves over function fields is
now known to be true, and that the curves which prove this have asymptotically
maximal rank for their conductors. The fact that these curves meet rank bounds
suggests a number of interesting problems on elliptic curves over number
fields, cyclotomic fields, and function fields over number fields. These
problems are discussed in the last four sections of the paper.
