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      Dates of Conference  
    04-07, September, 2017 

Harish-Chandra Research Institute

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Class groups of number fields and their cardinalities (i.e, class numbers) have been well studied since the time of Gauss. The study of class groups of number fields became the heart of algebraic number theory after the efforts of Kummer (towards Fermat's Last Theorem), Dedekind, Kronecker etc. In spite of long history of active research, class groups and their cardinalities remain one of the most mysterious object in algebraic number theory (save for “the finiteness of imaginary quadratic fields with class number one”).

There are two directions which are actively being explored in last 50 years or so. One being the study of annihilators of class groups (results of Iwasawa and Sinnot being corner stone), and, the other being Cohen-Lenstra heuristics. Annihilators of class groups give vital informations about class numbers (e.g. Theorems of Iwasawa and Sinnot) and Preda Mihailescu used them very cleverly to solve the longstanding conjecture of Catalan.

Though we are far from proving Cohen-Lenstra heuristics but there has been many small steps in this direction in last few decades. Infinitude of family of number fields of a given degree with class number divisible by a given number has been established by many mathematicians. Moreover, some significant results have been obtained on the density of quadratic number fields with class number a multiple of a given integer.

Another aspect which we shall highlight during the conference is the computation of class numbers of cyclotomic fields. Computing class number of cyclotomic fields is extremely tedious, and we have such computations available only for cyclotomic fields of prime conductor less than 151 (and up to 241 under GRH). In a recent work, Rene Schoof considers a subgroup of class group of maximal real subfield of p-th cyclotomic field whose cardinality can be computed easily. Schoof speculates that, most likely, this subgroup equals the class group of maximal real subfield. If the speculation of Schoof is proven right then it will make computation of class number of cyclotomic fields very easy.

The aim of this conference is to bring various experts on the subject at one place and provide young number theorists an oppurtunity to learn the techniques in the subject.

The conference will include talks in the following areas:
  • Ideal class groups of number fields.
  • Divisibility and indivisibility of class numbers of number fields.
  • Relative class numbers of number fields.
  • Class field Theory.
  • Diophantine equations.
  • Iwasawa Theory.