Moduli Spaces for Rings and Ideals

The association of algebraic objects to forms has had many important applications in number theory. Gauss, over two centuries ago, studied quadratic rings and ideals associated to binary quadratic forms, and found that ideal classes of quadratic rings are exactly parametrized by equivalence classes of integral binary quadratic forms. Delone and Faddeev, in 1940, showed that cubic rings are parametrized by equivalence classes of integral binary cubic forms. Recently, Bhargava has showed that quartic rings (with cubic resolvents) are parametrized by classes of pairs of integral ternary quadratic forms, and that quintic rings (with sextic resolvents) are parametrized by quadruples of integral alternating quinary forms. Bhargava has also studied ideals in quadratic and cubic rings, and found that they are associated to pairs of 2 by 2 and 3 by 3 integral matrices. Birch, Merriman, Nakagawa, Corso, Dvornicich, and Simon have all studied rings associated to binary forms of degree n for any n, but it has not previously been known which rings, and with what additional structure, are associated to binary forms. In this thesis, we explain exactly what algebraic structures are parametrized by binary n-ic forms, for all n. The algebraic data associated to an integral binary n-ic form includes a rank n ring, an ideal class for that ring, and a condition on the ring and ideal class that comes naturally from geometry. We also give a different story for what is parametrized by integral binary quartic forms, namely, binary quartic forms parametrize quartic rings with a monogenic cubic resolvent. We further show that classes of pairs of n by n matrices parametrize the ideal classes of rings associated to binary n-ic forms. In fact, we prove these parametrizations when any base scheme replaces the integers, and show that the correspondences between forms and the algebraic data are functorial in the base scheme. We also give geometric constructions of the rings and ideals from the forms that parametrize them. This geometric approach allows us to also give a statement of Gauss composition, the parametrization of ideal classes of quadratic rings by binary quadratic forms, over an arbitrary base scheme. We give an analog of Bhargava’s parametrization of quartic rings over an arbitrary base scheme, including a geometric construction of a quartic ring from a pair of ternary quadratic forms that works even in degenerate cases and commutes with base change. We also give a subspace of pairs of ternary quadratic forms that parametrizes quartic rings with quadratic subrings, which includes orders in quartic fields whose Galois closure has Galois group D4.