There are genus one curves over Q of every odd index

William A. Stein


The index of a genus one curve X over a field K is the smallest degree of an extension L of K such that X(L) is nonempty. Let K be a number field. We prove that for every integer r not divisible by 8, there is a genus one curve X over K of index r. Our proof involves an analysis of Kolyvagin's Euler system of Heegner points combined with explicit computations on the modular curve X0(17).

Here is the PDF version of the paper that appeared in Volume 547 (2002) of Crelle's Journal.