Class field theory, its three main generalisations, and applications

Class Field Theory (CFT) is the main achievement of algebraic number theory of the 20th century. Its reach, beauty and power, stemming from the first steps in algebraic number theory by Gaus, have substantially influenced number theory. Shafarevich wrote: 'Weil was undoubtedly right when he asserted, in the preface to the Russian edition of his book on number theory 1 , that since class field theory pertains to the foundation of mathematics, every mathematician should be as familiar with it as with Galois theory. Moreover, just like Galois theory before it, class field theory was reputed to be very complicated and accessible only to experts ... For class field theory, on the other hand, there is a wide range of essentially different expositions, so that it is not immediately obvious even whether the subject is the same'. 2 Weil's opinion has proved to be quixotic: these days even some number theorists are not familiar with the substance of CFT. This text reviews the enduring process of discovering new branches of CFT and its generalisations. Many of such developments were complicated at their early stages and some were difficult or impossible to understand for their contemporaries. Three main generalisations of CFT and their further extensions will be presented and some of their key fundamental features will be discussed. This text proposes eight fundamental problems. We start with Kummer theory, a purely algebraic exercise, whose highly non-trivial arithmetic analogues over arithmetic fields are supplied by CFT. Kummer theory is an algebraic predecessor of CFT including its existence theorem. Then we discuss the fundamental split of (one-dimensional) CFT into special CFT (SCFT) and general CFT (GCFT). This split has enormously affected many developments in number theory. Section 3 delves into four fundamental parts of CFT including the reciprocity map, existence theorem, explicit formulas for the Hilbert symbol and its generalisations, and interaction with ramification theory. Section 4 briefly touches on higher Kummer theory using Milnor K-groups, i.e. the norm residue isomorphism property. Three generalisations of CFT: Langlands correspondences (LC), higher CFT, and anabelian geometry are discussed in section 5. We note that the split of CFT into SCFT and GCFT is currently somehow reproduced at the level of generalisations of CFT: LC over number fields does not yet have any development parallel to GCFT, while higher CFT is parallel to GCFT and it does not have substantial developments similar to SCFT. In the last section we specialise to elliptic curves over global fields, as an illustration. There we consider two further developments: Mochizuki's inter-universal Teichmuller theory (IUT) which is pivoted on anabelian geometry and two-dimensional adelic analysis and geometry which uses structures of two-dimensional CFT. We also consider the fundamental role of zeta integrals which may unite different generalisations of CFT. Similarly to the situation with LC, the current studies of special values of zeta-and L-functions of elliptic curves over number fields, except two-dimensional adelic analysis and geometry, use special structures and are not of general type. There is no attempt to mention all the main results in CFT and all of its generalisations or all of their parts, and the text does not include all of bibliographical references. 1 [64] 2 in Foreword to [13].