2-Selmer groups, 2-class groups, and the arithmetic of binary forms

Abstract

This thesis concerns the arithmetic of binary forms, with a particular emphasis on studying 2-class groups of the rings and fields defined by such forms, as well as 2-Selmer groups of hyperelliptic curves and related Diophantine objects. The keystone result of this thesis is an orbit parametrization of square roots of the ideal class of the inverse different of rings defined by binary forms (see Chapter 1). This parametrization, which can be construed as a new integral model of a "higher composition law" discovered by Bhargava and generalized by Wood, was the missing ingredient needed to solve a range of open problems concerning distributions of class groups, Selmer groups, and more. For instance, in this thesis, we apply the parametrization to bound the average size of the 2-class group in families of number fields defined by binary n-ic forms, where n >= 3 is an arbitrary integer, odd or even (see Chapter 2). We also apply it to prove that most "superelliptic equations" of the form z^2 = f(x,y), where f is an integral binary form of odd degree, have a Brauer--Manin obstruction to having a primitive integer solution (see Chapter 3). Moreover, in joint work with Bhargava and Shankar, we use the parametrization to prove that when locally soluble genus-1 curves over Q are ordered by height, the average size of the 2-Selmer group is at most 6, and further that when elliptic curves over Q are ordered by height, the second moment of the size of the 2-Selmer group is at most 15, confirming well-known conjectures of Poonen and Rains (see Chapter 4). Finally, in joint work with Shankar, Siad, and Varma, we use the parametrization to determine the average size of the 2-class group in families of monogenic degree-n orders, enumerated by height; for this application, we develop a new method for counting integral orbits having bounded invariants and satisfying local conditions that lie in the cusps of fundamental domains for coregular representations (see Chapter 5).