Elliptic Curves and Analogies Between Number Fields and Function Fields

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 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 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 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.