arXiv subjects

Eric Brussel

Publications and source records attributed to Eric Brussel.

10 recordsLinked to original sources

arXiv2024

The Stack of Similarity Classes of Triangles

We construct the smooth, compact moduli space of similarity classes of labeled, oriented triangles. The space, denoted $\mathfrak D$, is a connected sum of three projective planes, and projects via blowdown to two shape spaces that have appeared in the literature: the well-known (Riemann) sphere (\cite{Kend84}, \cite{Beh}, \cite{Montgomery}, \cite{ES15}), and the less-well-known 2-torus (\cite{BG23}). A natural action by the dihedral group $D_6$ defines the quotient stack $[\mathfrak D/D_6]$ of absolute (unlabeled, unoriented) classes.

math.AG↗

arXiv2023

The Torus of Triangles

We prove the 2-torus $\mathbb T$, an abelian linear algebraic group, is a fine moduli space of labeled, oriented, possibly-degenerate inscribable similarity classes of triangles, where a triangle is {\it inscribable} if it can be inscribed in a circle. A natural action by the dihedral group $D_6$ defines a quotient stack $[\mathbb T/D_6]$, which is the stack of absolute (unlabeled, unoriented) possibly-degenerate inscribable classes. We show the main triangle types form distinguished algebraic substructures: subgroups, cosets, and elements of small order, and we apply the natural metric on $\mathbb T$ to compare them.

math.MG↗

arXiv2021

Hasse Invariant for the Tame Brauer Group of a Higher Local Field

We generalize the Hasse invariant of local class field theory to the tame Brauer group of a higher dimensional local field, and use it to study the arithmetic of central simple algebras over such fields, which are given {\it a priori} as tensor products of standard cyclic algebras. We also compute the tame Brauer dimension (or {\it period-index bound}) and the cyclic length of a general henselian-valued field of finite rank and finite residue field.

math.NT↗

arXiv2019

Noncyclic Division Algebras over Fields of Brauer Dimension One

Let $K$ be a complete discretely valued field of rank one, with residue field $\Q_p$. It is well known that period equals index in $\Br(K)$. We prove that when $p=2$ there exist noncyclic $K$-division algebras of every $2$-power degree divisible by four. Otherwise, every $K$-division algebra is cyclic.

math.RA↗

arXiv2014

Division Algebra Cyclicity in Prime Degree over a p-Adic Curve

We reprove two results of Saltman, Theorem 5.1 and Corollary 5.2 of [Sa07]: If F is the function field of a smooth p-adic curve and D is an F-division algebra of prime degree l\neq p, then D is Z/l-cyclic, and that if D is an F-division algebra of prime period l\neq p then D has index l if and only if its ramification locus on a suitable 2-dimensional model forF has no "hot points".

math.RA↗

arXiv2013

Cyclic Length in the Tame Brauer Group of the Function Field of a p-Adic Curve

Let $F$ be the function field of a smooth curve over the $p$-adic number field $\Q_p$. We show that for each prime-to-$p$ number $n$ the $n$-torsion subgroup $\H^2(F,μ_n)={}_n\Br(F)$ is generated by $\Z/n$-cyclic classes; in fact the $\Z/n$-length is equal to two. It follows that the Brauer dimension of $F$ is two (first proved in \cite{Sa97}), and any $F$-division algebra of period $n$ and index $n^2$ is decomposable.

math.RA↗

arXiv2011

Tame Covers and Cohomology of Relative Curves over Complete Discrete Valuation Rings, with Applications to the Brauer Group

We prove the existence of noncrossed product and indecomposable division algebras over the function field of a smooth p-adic curve, especially when the curve does not admit a smooth model over Z_p. Thus we generalize arXiv 0907.0670. To make our constructions, we investigate the lifting of cohomology classes from the total fraction ring of the closed fiber to the function field of the curve, over an arbitrary discrete valuation ring of mixed characteristic.

math.NT↗

arXiv2011

Fixed Points of the q-Bracket on the p-Adic Unit Disk

We study the fixed points of the q-bracket on the complex unit disk, and prove the following. The set of (nontrivial) pairs (x,q) such that [x]_q=x form a manifold whose standard projections both have degree p-2. There is an analytic function Q(X) taking x to q for which [x]_q=x, which is a (bijective) contraction unless the multiplicity of the residue of x in the fiber over q is two. The restriction of the theory to Z_p is trivial unless p=3.

math.NT↗

arXiv2010

Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves

We construct indecomposable and noncrossed product division algebras over function fields of smooth curves X over Z_p. This is done by defining an index preserving morphism s:Br(\hat K(X))' -> Br(K(X))' which splits res:Br(K(X)) -> Br(\hat K(X)), where \hat K(X) is the completion of K(X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over \hat K(X).

math.RA↗

arXiv2010

Open Problems on Central Simple Algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.

math.RA↗