13 problems

Let ${\mathcal O}\subseteq M_n(K)$ be a classical order over a discrete valuation ring $D$ with quotient field $K$, and let $\pi$ be a uniformizing parameter of $D$. The integer…

Let an arithmetic order be a subring of a finite-dimensional semisimple algebra over a number field, and let a lattice over that order be a finitely generated torsion-free module.…

Equal-elasticity conjecture. If

Let $p\in\mathbb N$ be prime, let $K=\mathbb Q[\sqrt{p}]$, and let $R_p$ be the order of index $p$ in $K$. An order is locally associated when it has the local association property…

Kainrath's conjecture. Every order $\mathcal{O}'$ between a half-factorial order and $\mathcal{O}_K$ is itself half-factorial.

Graduated-order conjecture. The order $H_{n,\lambda}$, represented as a matrix order using the standard basis of $S_\lambda(V)$, is a graduated order: there exists a matrix…

Lattice-power invertibility conjecture. The lattice $\mathfrak c^{n-1}$ and every higher power $\mathfrak c^m$ with $m\geq n-1$ are invertible in their common endomorphism ring.

Let $K$ be a number field, let $\mathcal{O}_K$ be its ring of integers, and let $\alpha_K$ be the exponent in the asymptotic formula … where $N_K(n)$ counts orders in…

Let $A$ be an order of ${\mathbb Z}$-rank $n \ge 1$, and let $\ell$ be a prime number with $\ell > n^2$. Theorem asserts that there exists a maximal ideal ${\mathfrak{p}}$ of $A$ t…

Let ${\mathcal O}$ be a tiled classical order over a discrete valuation ring, with uniformizing parameter $\pi$, and set $\Lambda={\mathcal O}/\pi{\mathcal O}$. For a $\Lambda$-mod…

Local half-factoriality conjecture. Every half-factorial order is locally half-factorial.

Let $C$ be the canonically embedded complete-intersection curve of degree $8$ and genus $5$ arising as the base locus of a net of quadrics in $\mathbb{P}^4$. Fix ramification data…

Let $Q$ be a conic bundle and let $\operatorname{Cl}_0(Q)$ be its even Clifford algebra. Consider orders Morita equivalent to $\operatorname{Cl}_0(Q)$ and having the same rank and…