Selmer Groups and Quadratic Reciprocity

In this article we study the 2-Selmer groups of number fields F as well as some related groups, and present connections to the quadratic reciprocity law in F . Let F be a number field; elements in F× that are ideal squares were called singular numbers in the classical literature. They were studied in connection with explicit reciprocity laws, the construction of class fields, or the solution of embedding problems by mathematicians like Kummer, Hilbert, Furtwängler, Hecke, Takagi, Shafarevich and many others. Recently, the groups of singular numbers in F were christened Selmer groups by H. Cohen [4] because of an analogy with the Selmer groups in the theory of elliptic curves (look at the exact sequence (2.2) and recall that, under the analogy between number fields and elliptic curves, units correspond to rational points, and class groups to Tate-Shafarevich groups). In this article we will present the theory of 2-Selmer groups in modern language, and give direct proofs based on class field theory. Most of the results given here can be found in §§ 61ff of Hecke’s book [11]; they had been obtained by Hilbert and Furtwängler in the roundabout way typical for early class field theory, and were used for proving explicit reciprocity laws. Hecke, on the other hand, first proved (a large part of) the quadratic reciprocity law in number fields using his generalized Gauss sums (see [3] and [22]), and then derived the existence of quadratic class fields (which essentially is just the calculation of the order of a certain Selmer group) from the reciprocity law. In Takagi’s class field theory, Selmer groups were moved to the back bench and only resurfaced in his proof of the reciprocity law. Once Artin had found his general reciprocity law, Selmer groups were history, and it seems that there is no coherent account of their theory based on modern class field theory. Hecke’s book [11] is hailed as a classic, and it deserves the praise. Its main claim to fame should actually have been Chapter VIII on the quadratic reciprocity law in number field, where he uses Gauss sums to prove the reciprocity law, then derives the existence of 2-class fields, and finally proves his famous theorem that the ideal class of the discriminant of an extension is always a square. Unfortunately, this chapter is not exactly bedtime reading, so in addition to presenting Hecke’s results in a modern language I will also give exact references to the corresponding theorems in Hecke’s book [11] in the hope of making this chapter more accessible. The actual reason for writing this article, however, was that the results on Selmer groups presented here will be needed for computing the separant class group of F , a new invariant that will be discussed thoroughly in [21].