On the Rank of Elliptic Curves
نویسندگان
چکیده
Notation. K will always denote a number eld. xI. Ranks of Families of Elliptic Curves For our purposes, a family of elliptic curves will be given by an equation E : y 2 = x 3 + A(T)x + B(T); A(T); B(T) 2 KT]; (T) = 4A(T) 3 + 27B(T) 2 6 = 0: We will always assume that E is non-split (i.e., is not obtained by base extension from a curve deened over K). The rank Generic Rank = Rank E(K(T)) is the rank of the family as an elliptic curve over the function eld K(T), or equivalently the rank of its group of sections as an elliptic surface over P 1 K. It is diicult to produce families of high rank, especially for K = Q. Mestre has given a construction which allows one to construct families of at least moderately high rank.
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