Levels of cancellativity in commutative monoids M, determined by stable-rank values in $\mathbb {Z}_{> 0} \cup \{\infty \}$ for elements of M, are investigated. The behavior of the stable ranks of multiples $ka$, for $k \in \mathbb {Z}_{> 0}$ and $a \in M$, is determined. In the case of a refinement monoid M, the possible stable-rank values in archimedean components of M are pinned down. Finally, stable rank in monoids built from isomorphism or other equivalence classes of modules over a ring is discussed.

Let A be an abelian variety defined over a global function field F and let p be a prime distinct from the characteristic of F. Let $F_\infty $ be a p-adic Lie extension of F that contains the cyclotomic $\mathbb {Z}_p$-extension $F^{\mathrm {cyc}}$ of F. In this paper, we investigate the structure of the p-primary Selmer group $\mathrm {Sel}(A/F_\infty )$ of A over $F_\infty $. We prove the $\mathfrak {M}_H(G)$-conjecture for $A/F_\infty $. Furthermore, we show that both the $\mu $-invariant of the Pontryagin dual of the Selmer group $\mathrm {Sel}(A/F^{\mathrm {cyc}})$ and the generalized $\mu $-invariant of the Pontryagin dual of the Selmer group $\mathrm {Sel}(A/F_\infty )$ are zero, thereby proving Mazur’s conjecture for $A/F$. We then relate the order of vanishing of the characteristic elements, evaluated at Artin representations, to the corank of the Selmer group of the corresponding twist of A over the base field F. Assuming the finiteness of the Tate–Shafarevich group, we establish that this corank equals the order of vanishing of the L-function of $A/F$ at $s=1$. Finally, we extend a theorem of Sechi—originally proved for elliptic curves without complex multiplication—to abelian varieties over global function fields. This is achieved by adapting the notion of generalized Euler characteristic, introduced by Zerbes for elliptic curves over number fields. This new invariant allows us, via Akashi series, to relate the generalized Euler characteristic of $\mathrm {Sel}(A/F_\infty )$ to the Euler characteristic of $\mathrm {Sel}(A/F^{\mathrm {cyc}})$.

A ${\mathcal Z}$-subalgebra $U_{\mathcal Z}^\jmath {(n)}$ (${\mathcal Z}=\mathbb Z[\upsilon ,\upsilon ^{-1}]$) for the i-quantum group ${\mathbf {U}}^{\jmath }(n)$ over the field $\mathbb Q(\upsilon )$ is constructed by two of the authors [‘A new realisation of the i-quantum group $U^\jmath {(n)}$’, J. Pure Appl. Algebra 226(1) (2022), Paper no. 106793, 27 pages, Theorem 6.5], using a Beilinson–Lusztig–MacPherson (BLM) type realisation. In this paper, we construct bases for $U_{\mathcal Z}^\jmath {(n)}$, including the monomial basis conjectured in [‘A new realisation of the i-quantum group $U^\jmath {(n)}$’, J. Pure Appl. Algebra 226(1) (2022), Paper no. 106793, 27 pages, Remark 6.6(4)]. This proves that the ${\mathcal Z}$-algebra $U_{\mathcal Z}^\jmath {(n)}$ is a free ${\mathcal Z}$-module. Hence, $U_{\mathcal Z}^\jmath {(n)}$ is in fact an integral form of Lusztig type. This construction is further extended to the i-quantum hyperalgebra over a field of any characteristic. By specialising $\upsilon $ to an l th primitive root $\varepsilon $ of $1$ with l odd, a realisation of the quotient of modulo the ideal generated by $d_i^l-1$, for all $1\leqslant i\leqslant n+1$, is also given as a by-product.

We introduce a generating function approach to the affine Brauer and Kauffman categories, and show how it allows one to efficiently recover important sets of relations in these categories. We use this formalism to deduce restrictions on possible categorical actions and show how this recovers admissibility results that have appeared in the literature on cyclotomic Birman–Murakami–Wenzl (BMW) algebras and their degenerate versions, also known as cyclotomic Nazarov–Wenzl algebras or VW algebras.

Let ${\mathrm {U}}_n({\mathbb {F}}_q)$ be the unitriangular group and ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_q)$ the four-block unipotent radical of the standard parabolic subgroup of $\mathrm {GL}_{n}$, where $a+b+c+d=n$. In this paper, we study the class of all pattern subgroups of ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_{q})$. We establish character-number formulae of degree $q^e$ for all these pattern groups. For pattern subgroups $G_{{\mathcal {D}}_m}({\mathbb {F}}_q)$ in this class, we provide an algebraic geometric approach to their polynomial properties, which verifies an analogue of Lehrer’s conjecture for these pattern groups.