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Register a new userOn supercharacter theoretic generalizations of monomial groups and Artins conjecture
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We extend the notions of quasi-monomial groups and almost monomial groups, in the framework of supercharacter theories, and we study their connection with Artins conjecture regarding the holomorphy of Artin $L$-functions.
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In this paper we consider the density of maximal order elements in $mathrm{GL}_n(q)$. Fixing any of the rank $n$ of the group, the characteristic $p$ or the degree $r$ of the extension of the underlying field $mathbb{F}_q$ of size $q=p^r$, we compute the expected value of the said density and establish that it follows a distribution law.
In the 1990s, J.H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the topograph, Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pells equation. It appears that the crux of his method is the coincidence between the arithmetic group $PGL_2({mathbb Z})$ and the Coxeter group of type $(3,infty)$. There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conways topograph, and generalizations to other arithmetic Coxeter groups. This includes a study of arithmetic flags and variants of binary quadratic forms.
As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${rm L}_n(q)$ is prime. We present heuristic arguments and computational evidence to support a conjecture that for each prime $nge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$.
Work on generalizations of the Cohen-Lenstra and Cohen-Martinet heuristics has drawn attention to probability measures on the space of isomorphism classes of profinite groups. As is common in probability theory, it would be desirable to know that these measures are determined by their moments, which in this context are the expected number of surjections to a fixed finite group. We show a wide class of measures, including those appearing in a recent paper of Liu, Wood, and Zurieck-Brown, have this property. The method is to work locally with groups that are extensions of a fixed group by a product of finite simple groups. This eventually reduces the problem to the case of powers of a fixed finite simple group, which can be handled by a simple explicit calculation. We can also prove a similar theorem for random modules over an algebra.
We give a new proof of the Baum--Connes conjecture with coefficients for any second countable, locally compact topological group that acts properly and cocompactly on a finite-dimensional CAT(0)-cubical space with bounded geometry. The proof uses the Julg-Valette complex of a CAT(0)-cubical space introduced by the first three authors, and the direct splitting method in Kasparov theory developed by the last author.
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