International Conference on Class Groups of Number Fields and Related Topics

ag.algebraic-geometry nt.number-theory
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End Date
Harish-Chandra Research Institute 
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lass 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 FLT), 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 with exceptions like '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 P. 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 50 years. 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 due to efforts of a few mathematicians 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 70 (and up to 163 under GRH). In an article, R. 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 of the country a very needed thrust (it is after long time this topic is being highlighted so exclusively, even worldwide). Also we hope that this will kindle interest of upcoming generation in Algebraic Number Theory.


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