Higher Descent on Pell Conics. I. from Legendre to Selmer

The theory of Pell’s equation has a long history, as can be seen from the huge amount of references collected in Dickson [Dic1920], from the two books on its history by Konen [Kon1901] and Whitford [Whi1912], or from the books by Weber [Web1939], Walfisz [Wal1952], Faisant [Fai1991], and Barbeau [Bar2003]. For the better part of the last few centuries, the continued fractions method was the undisputed method for solving a given Pell equation, and only recently faster methods have been developed (see the surveys by Lenstra [Len2002] and H.C. Williams [Wil2002]). This is the first in a series of articles which have the goal of developing a theory of the Pell equation that is as close as possible to the theory of elliptic curves: we will discuss 2-descent on Pell conics, introduce Selmer and Tate-Shafarevich groups, and prove an analogue of the Birch–Swinnerton-Dyer conjecture. In this article, we will review the history of results that are related to this new interpretation. We will briefly discuss the construction of explicit units in quadratic number fields, and then deal with Legendre’s equations and the results of Rédei, Reichardt and Scholz on the solvability of the negative Pell equation. The second article [Lem2003a] is devoted to instances of a “second 2-descent” in the mathematical literature from Euler to our times, and in [Lem2003b] we will discuss the first 2-descent and the associated Selmer and Tate-Shafarevich groups from the modern point of view.