Related papers: Tori over number fields and special values at s=1

Let $G$ be a complex linear algebraic group, $\mathfrak{g}=\Lie(G)$ its Lie algebra and $e\in\mathfrak{g}$ a nilpotent element. Vust's theorem says that in case of $G=\GL(V)$, the algebra $\mbox{End}_{G_e}(V^{\otimes d})$, where $G_e\subset…

The notion of Weyl modules, both local and global, goes back to Chari and Pressley in the case of affine Lie algebras, and has been extensively studied for various Lie algebras graded by root systems. We extend that definition to a certain…

Let $\mathfrak{W}$ be the Lie algebra of vector fields on the line. Via computing extensions between all simple modules in the category $\mathcal{O}$, we give the block decomposition of $\mathcal{O}$, and show that the representation type…

The Weil reciprocity law asserts that given two meromorphic functions $f, g$ on a compact complex curve, the product of the values of $f$ over the roots and poles of $g$ is equal to the product of the values of $g$ over the roots and poles…

We study the constructible Witt theory of \'etale sheaves of $\Lambda$-modules on a scheme $X$ for coefficient rings $\Lambda$ having finite characteristic not equal to 2 and prime to the residue characteristics of the scheme $X$. Our…

Using work of Antieau and Bhatt-Morrow-Scholze, we define a filtration on topological Hochschild homology and its variants $TP$ and $TC^-$ of quasi-lci rings with bounded torsion, which recovers the BMS-filtration after $p$-adic completion.…

Inspired by Morse theory, we introduce a topological stack Broken, which we refer to as the moduli stack of broken lines. We show that Broken can be presented as a Lie groupoid with corners and provide a combinatorial description of sheaves…

We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on…

String diagrams turn algebraic equations into topological moves that have recurring shapes, involving the sliding of one diagram past another. We individuate, at the root of this fact, the dual nature of polygraphs as presentations of…

A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed. In one variable, Taylor varieties are given by rank constraints on…

We prove duality theorems for the {\'e}tale cohomology of logarithmic Hodge-Witt sheaves and split tori on smooth curves over a local field of positive characteristic. As an application, we obtain a description of the Brauer group of the…

This paper presents a $q$-analogue of an extension of the tensor algebra given by the same author. This new algebra naturally contains the ordinary tensor algebra and the Iwahori-Hecke algebra type $A$ of infinite degree. Namely this…

Using principal series Harish-Chandra modules, local cohomology with support in Schubert cells and twisting functors we construct certain modules parametrized by the Weyl group and a highest weight in the subcategory O of the category of…

An Ulrich sheaf on an embedded projective variety is a normalized arithmetically Cohen-Macaulay sheaf with the maximum possible number of independent sections. Ulrich sheaves are important in the theory of Chow forms, Boij-Soderberg theory,…

In the present article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight $2,4$ and 6. We define Hecke operators on them, find some analytic relations between these…

Duality for complete discrete valuation fields with perfect residue field with coefficients in (possibly p-torsion) finite flat group schemes was obtained by Begueri, Bester and Kato. In this paper, we give another formulation and proof of…

We define the notion of a Lie superalgebra over a field $k$ of characteristic $2$ which unifies the two pre-existing ones - $\mathbb{Z}/2$-graded Lie algebras with a squaring map and Lie algebras in the Verlinde category ${\rm Ver}_4^+(k)$,…

Turner's Conjecture describes all blocks of symmetric groups and Hecke algebras up to derived equivalence in terms of certain double algebras. With a view towards a proof of this conjecture, we develop a general theory of Turner doubles. In…

We extend the Euler's totient function (from arithmetic) to any irreducible subfactor planar algebra, using the Mobius function of its biprojection lattice, as Hall did for the finite groups. We prove that if it is nonzero then there is a…

We introduce a cohomology theory for a class of projective varieties over a finite field coming from the canonical trace on a C*-algebra attached to the variety. Using the cohomology, we prove the rationality, functional equation and the…