41 research outputs found
Mixed moments and the joint distribution of random groups
We study the joint distribution of random abelian and non-abelian groups. In
the abelian case, we prove several universality results for the joint
distribution of the multiple cokernels for random -adic matrices. In the
non-abelian case, we compute the joint distribution of random groups given by
the quotients of the free profinite group by random relations. In both cases,
we generalize the known results on the distribution of the cokernels of random
-adic matrices and random groups. Our proofs are based on the observation
that mixed moments determine the joint distribution of random groups, which
extends the works of Wood for abelian groups and Sawin for non-abelian groups.Comment: 27 page
Counting -dimensional algebraic tori over
In this paper we count the number of -dimensional
algebraic tori over whose Artin conductor is bounded by . We
prove that , and this upper bound can be improved to
under the Cohen-Lenstra
heuristics for . We also prove that for out of conjugacy classes
of finite nontrivial subgroups of , Malle's
conjecture for tori over holds up to a bounded power of
under the Cohen-Lenstra heuristics for and Malle's conjecture for quartic
-fields.Comment: 33 pages, to appear in J. Number Theor
Joint distribution of the cokernels of random -adic matrices
In this paper, we study the joint distribution of the cokernels of random
-adic matrices. Let be a prime and be monic polynomials whose reductions modulo in
are distinct and irreducible. We determine the limit of the
joint distribution of the cokernels for a random matrix over
with respect to Haar measure as . By applying the
linearization of a random matrix model, we also provide a conjecture which
generalizes this result. Finally, we provide a sufficient condition that the
cokernels and become independent as , where is a fixed matrix over
for each and is a random matrix over
.Comment: 18 page
On the lower bound of the number of abelian varieties over
In this paper, we prove that the number of isomorphism classes of
abelian varieties over a prime field of dimension has a
lower bound as . This is
the first nontrivial result on the lower bound of . We also improve the
upper bound of given by
Lipnowski and Tsimerman (Duke Math. J. 167:3403-3453, 2018) to .Comment: 20 pages, to appear in IMR