Binary Forms, Quintic Rings and Sextic Resolvents

Abstract

This thesis is about integral binary n-ic forms and rings of rank n, with a close study of the n = 5 case. A classical construction of Birch & Merriman takes an integral binary n-ic form f(x, y) and constructsa ring R_f of rank n. If f is monic, the ring R_f is Z[x]/(f(x, 1)). If f is irreducible, R_f is an order in the field Q[x]/(f(x, 1)). Not all rings of rank n can be obtained by this construction. In the first part of this thesis, we provide two geometric and structural characterisations of the rings isomorphic to R_f for some f. In the second part of this thesis, we specialise to the case of binary quintic forms and quintic (rank 5)rings. The complete theory of quintic rings is due to Bhargava. A quintic ring R has an associated sextic (rank 6) resolvent ring S and a fundamental map \phi linking R and S. We show that the sextic resolvent ring S_f of the quintic ring R_f has a special algebraic structure. This structure characterises the quintic ring and sextic resolvent pairs (R, S) which are isomorphic to (R_f, S_f ) for some binary quintic f, and leads to a characterisation of quintic rings of the form R_f. Furthermore, this picture can be interpreted in terms of GL_2(Z)-classes of binary quintic forms. Toeach quintic ring and sextic resolvent pair (R, S), we explicitly define a 2-dimensional affine variety whose finitely many integral points correspond to the GL_2(Z)-classes of forms f such that (R_f, S_f ) is isomorphic to (R, S).