News & Updates Dates of Conference 0407, September, 2017

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
CohenLenstra 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 CohenLenstra 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 pth 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.
