3,098 research outputs found
The Cohen-Lenstra Heuristic: Methodology and Results
In number theory, great efforts have been undertaken to study the
Cohen-Lenstra probability measure on the set of all finite abelian -groups.
On the other hand, group theorists have studied a probability measure on the
set of all partitions induced by the probability that a randomly chosen
-matrix over \FF_p is contained in a conjucagy class associated
with this partitions, for .
This paper shows that both probability measures are identical. As a
consequence, a multitide of results can be transferred from each theory to the
other one. The paper contains a survey about the known methods to study the
probability measure and about the results that have been obtained so far, from
both communities
A weighted M\"obius function
For a fixed odd prime , we define a variant of the classical M\"{o}bius
function on the poset of isomorphism classes of finite abelian -groups,
then we prove an analog of Hall's theorem on the vanishing of the M\"{o}bius
function.Comment: 7 pages, new introduction, updated reference
Counting Co-Cyclic Lattices
There is a well-known asymptotic formula, due to W. M. Schmidt (1968) for the
number of full-rank integer lattices of index at most in .
This set of lattices can naturally be partitioned with respect to the
factor group . Accordingly, we count the number of full-rank
integer lattices such that is
cyclic and of order at most , and deduce that these co-cyclic lattices are
dominant among all integer lattices: their natural density is . The problem is motivated by
complexity theory, namely worst-case to average-case reductions for lattice
problems
Gauss composition over an arbitrary base
The classical theorems relating integral binary quadratic forms and ideal
classes of quadratic orders have been of tremendous importance in mathematics,
and many authors have given extensions of these theorems to rings other than
the integers. However, such extensions have always included hypotheses on the
rings, and the theorems involve only binary quadratic forms satisfying further
hypotheses. We give a complete statement of the relationship between binary
quadratic forms and modules for quadratic algebras over any base ring, or in
fact base scheme. The result includes all binary quadratic forms, and commutes
with base change. We give global geometric as well as local explicit
descriptions of the relationship between forms and modules.Comment: submitte
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