A note on pépin’s counter examples to the hasse principle for curves of genus 1

Abstract

In a series of articles published in the C.R. Paris more than a century ago, T. Pépin announced a list of "theorems" concerning the solvability of diophantine equations of the type ax 4 + by 4 = z 2 . In this article, we show how to prove these claims using the structure of 2-class groups of imaginary quadratic number fields. We will also look at a few related results from a modern point of view.

Some families of non-congruent numbers

“…The use of genus theory in this connection was suggested by the proofs of Pépin's conjectures in [20]. This concludes our discussion of the φ-part of X( E/Q).…”

“…Birch and Swinnerton-Dyer [2], Razar [32], Lagrange [19,20], Wada [39] and Nemenzo [27]), employ the Cassels pairing (see e.g. Aoki [1], Bölling [3], Cassels [5], and McGuinness [26]), compare the Selmer groups Sel (ψ) ( E/Q) and Sel (2) (E/Q) as done by Kramer [18] (essentially, the methods mentioned so far are all equivalent to the classical second 2-descent), or use the method usually attributed to Lind [25] but actually going back (in a slightly different context) to Rédei [33] and Dirichlet [8] (I learned this technique from Stroeker & Top [38] and used it in [22] and [21]). In this paper, we continue to use this last method; as we shall see, it will allow us to obtain results that are stronger than those provided by simple second 2-descents.…”

Parametrizing Algebraic Curves

“…The case z = r 2 leads, for example, to the quartic curve y 2 = r 4 − 3r 2 + 3. By studying these quartics arising from y 2 = x 3 + 1 Euler found all rational points on this elliptic curve (see [11] for an exposition of Euler's proof).…”

Effectivity of Brauer–Manin obstructions

“…Counterexamples to the Hasse principle appeared as early as 1880 [Pép80]. (For a discussion, and proofs of the claims that appeared at that time, see [Lem03].) By the early 1940's it was well established that genus 1 curves may fail to satisfy the Hasse principle [Lin40], [Rei42].…”