We study the problem of constructing and enumerating, for any integers , number fields of degree whose ideal class groups have “large" -rank. Our technique relies fundamentally on Hilbert’s irreducibility theorem and results on integral points of bounded degree on curves.
We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge’s theorem valid for higher-dimensional varieties, generalizing a uniform version of Runge’s theorem due to Bombieri. We then take up the study of how Runge’s method may be expanded by taking advantage of certain coverings. We prove both a result for arbitrary curves and a more explicit result for superelliptic curves. As an application of our...
It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field and any finite set of places of , one can effectively compute the set of isomorphism classes of hyperelliptic curves over with good reduction outside . We show here that an extension of this result to an effective Shafarevich conjecture for of hyperelliptic curves of genus would imply an effective version of Siegel’s theorem for integral points on hyperelliptic...
Page 1