Related papers: Weight conjectures for $\ell$-compact groups and s…

In this paper, we present a unifying approach to the general theory of abelian Stark conjectures. To do so we define natural notions of `zeta element', of `Weil-\'etale cohomology complexes' and of `integral Selmer groups' for the…

We formulate an analogue of the Breuil-M\'ezard conjecture for the group of units of a central division algebra over a $p$-adic local field, and we prove that it follows from the conjecture for $\mathrm{GL}_n$. To do so we construct a…

It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring of a finite group is conjugate to a group element within the rational group algebra. The object of this note is the computational aspect of a method developed…

Let $G$ be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic $p>0$. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of…

Condensed mathematics, developed by Clausen and Scholze over the last few years, is a new way of studying the interplay between algebra and geometry. It replaces the concept of a topological space by a more sophisticated but better-behaved…

Let $H$ be a complex reductive group, with finite-dimensional representations $W$ and $U$. The module of covariants for $W$ of type $U$ is the space of all $H$-equivariant polynomial maps $\varphi: W \longrightarrow U$. In this paper, we…

The classical Serre-Swan's theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these…

We prove Carlos Simpson's "semi-strictification" (or "weak unit") conjecture in the case of infinity-groupoids. More precisely, we introduce two precise versions of the conjecture, the "general" and the "regular" conjecture, involving two…

The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of…

We prove a version of the weight part of Serre's conjecture for mod $p$ Galois representations attached to automorphic forms on rank 2 unitary groups which are non-split at $p$. More precisely, let $F/F^+$ denote a CM extension of a totally…

We prove the Farrell-Jones Conjecture for (non-connective) $A$-theory with coefficients and finite wreath products for hyperbolic groups, CAT(0)-groups, cocompact lattices in almost connected Lie groups and fundamental groups of manifolds…

In this short note we study the cohomology algebra of saturated fusion systems using finite groups which realize saturated fusion systems and Hochschild cohomology of group algebras. A similar result to a theorem of Alperin is proved for…

From a certain induced representation $\mathcal{P}_\ell$ of a double affine Weyl group, we construct a ring $\mathcal{F}_\ell$ that is isomorphic to the fusion ring, or Verlinde algebra, associated to affine Lie algebras at fixed positive…

H.J. Zassenhaus conjectured that any unit of finite order and augmentation $1$ in the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra $\mathbb{Q}G$ to an element of $G$. We prove the…

We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian $p$-groups with an abelian subgroup of index $p$. In particular, this gives many new examples illustrating the enormous variety…

We determine the numerical invariants of blocks with defect group D_{2^n}\times C_{2^m}, where D_{2^n} denotes a dihedral group of order 2^n and C_{2^m} denotes a cyclic group of order 2^m. This generalizes Brauer's results for m=0. As a…

We formulate and prove the analogue of the Ramanujan Conjectures for modular forms of half-integral weight subject to some ramification restriction in the setting of a polynomial ring over a finite field. This is applied to give an…

We investigate a stronger formulation of Webb's conjecture on the contractibilty of the orbit space of the p-subgroup complexes in terms of finite topological spaces. The original conjecture, which was first proved by Symonds and, more…

The theta-block conjecture proposed by Gritsenko--Poor--Yuen in 2013 characterizes Siegel paramodular forms which are simultaneously Borcherds products and additive Jacobi lifts. In this paper, we prove this conjecture for two new infinite…

We prove the weight elimination direction of the Serre weight conjectures as formulated by Herzig for forms of $U(n)$ which are compact at infinity and split at places dividing $p$ in generic situations. That is, we show that all modular…