Guest Editorial

Abstract

This special issue of the Journal of IISc is devoted to the interrelated areas of number theory and representation theory, two evergreen and central subfields of mathematics that continue to see groundbreaking progress with each passing decade. This volume contains a collection of articles by experts, each surveying an important and active domain in the two subfields. We now briefly describe each of these works, starting with three articles on distinct themes in number theory that have seen extensive activity in recent times.The work of Wiles and Taylor on the Taniyama-Shimura conjecture catalysed a large amount of activity in the Langlands program. Roughly speaking, the Langlands program postulates a one-to-one correspondence between Galois representations on the one hand and certain automorphic forms on the other. Counterparts of some operations that are easily carried out for Galois representations turn out to be extremely difficult for automorphic forms (and vice versa). These form a part of Langlands's functoriality conjecture. One such operation is taking symmetric powers. The article "Modularity of Galois representations and Langlands functoriality" by James Newton gives a survey of recent results on modularity with a focus on his joint work with Thorne on proving Langlands functoriality of symmetric powers of a cuspidal Hecke eigenform. The article also provides an extensive list of references to expositions on various aspects and developments of modularity results that can be very helpful for someone to get initiated in the subject.Another very important theme in algebraic number theory is the relationship between analytic and arithmetic invariants. The subject has its origin in the works of Dirichlet and Kummer in the nineteenth century. Although there is a very satisfying conjectural picture due to the insights of many eminent mathematicians including Stark, Tate, Deligne, Beilinson, Bloch, and Kato, very few cases of these conjectures are proved. Perhaps the most studied case is the Birch and Swinnerton-Dyer conjecture (BSD).

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