Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions ofâ€¦ (More)

In the first two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic fields couldâ€¦ (More)

We prove that when all hyperelliptic curves of genus n â‰¥ 1 having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. Itâ€¦ (More)

A function f : 77, ---, 77 m is said to be congruence preserving if for all d dividing m and a,b C {0, 1 . . . . . n 1}, a = b (modd) implies f ( a ) = f (b) (modd) , In previous work, Chen definesâ€¦ (More)

In our first article [2] we developed a new view of Gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms. Moreover, we showed thatâ€¦ (More)

In 1801 Gauss laid down a remarkable law of composition on integral binary quadratic forms. This discovery, known as Gauss composition, not only had a profound influence on elementary number theoryâ€¦ (More)

In 1993, Conway formulated a remarkable conjecture regarding universal quadratic forms, i.e., integer-coefficient, positive-definite quadratic forms representing all positive integers. Based onâ€¦ (More)

Contents 1. Introduction 963 2. A game called Ã ÂµÃÂ±Âƒ-orderings 967 2.1. On Ã ÂµÃÂ±ÂŸ-removed Ã ÂµÃÂ±Âƒ-orderings 968 2.2. On Ã ÂµÃÂ±Âƒ-orderings of order â„Ž 969 3. Rings of integer-valued polynomials 969 3.1.â€¦ (More)