Related papers: Local-Global Principle for Unitary Groups Over Fun…

For a given l-adic sheaf F on a commutative algebraic group over a finite field k and an integer r we define the r-th local norm L-function of F at a point t in G(k) and prove its rationality. This function gives information on the sum of…

Let $F$ be a $p$-adic field ($p\neq 2$), let $E$ be a quadratic Galois extension of $F$, and let $n \geq 2$. We construct representations in the discrete spectrum of the $p$-adic symmetric space $H \backslash G$, where $G =…

We study universal groups for right-angled buildings. Inspired by Simon Smith's work on universal groups for trees, we explicitly allow local groups that are not necessarily finite nor transitive. We discuss various topological and…

Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…

We define the local fundamental group scheme, defined by the F-trivial vector bundles, and give necessary and sufficient conditions for it to base change.

Let $K$ be an algebraically closed field. Let $G$ be a non-trivial connected unipotent group, which acts effectively on an affine variety $X.$ Then every non-empty component $R$ of the set of fixed points of $G$ is a $K$-uniruled variety,…

Let $G$ be a connected reductive group over a $p$-adic field $F$ of characteristic 0 and let $M$ be an $F$-Levi subgroup of $G.$ Given a discrete series representation $\sigma$ of $M(F),$ we prove that there exists a locally constant and…

Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve with positive self-intersection. We prove that if there exists a non-constant meromorphic function on $F$, then the…

In this paper we construct classifying localic categories and groupoids for various bundles equipped with logical structure. When these bundles are local homeomorphisms, we recover the localic groupoids that classify geometric theories,…

Fix a non-negative integer g and a positive integer I dividing 2g-2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we construct a curve C over K of genus g and index I.…

Let $G$ be a simple algebraic group over an algebraically closed field $k$. Let $\Gamma$ be a finite group acting on $G$. We classify and compute the local types of $(\Gamma, G)$-bundles on a smooth projective $\Gamma$-curve in terms of the…

Let F be a finite field and l a prime not equal to the characteristic of F. Let K be the function field of a surface over F. Assume that K contains a primitive lth root of unity. In the paper we prove a certain local-global principle for…

We initiate the study of the $p$-local commensurability graph of a group, where $p$ is a prime. This graph has vertices consisting of all finite-index subgroups of a group, where an edge is drawn between $A$ and $B$ if $[A : A\cap B]$ and…

We prove the existence of global minimal models for rational morphisms $\phi:{\mathbb P}^N\rightarrow{\mathbb P}^N$ of projective space defined over the field of fractions of a principal ideal domain.

For a reductive group $G$, we prove that complex irreducible rigid $G$-local systems with quasi-unipotent monodromies and finite order abelianization on a smooth curve are motivic, generalizing a theorem of Katz for $GL_n$. We do so by…

This paper studies the jumping coefficients of principal ideals of regular local rings. Recently M. Blickle, M. Mustata and K. Smith showed that, when $R$ is of essentially finite type over a field and $F$-finite, bounded intervals contain…

Let $F$ be a non-archimedean local field of characteristic different from $2$ and $G$ be either an odd special orthogonal group ${\rm SO}_{2r+1}(F)$ or a symplectic group ${\rm Sp}_{2r}(F)$. In this paper, we establish the local converse…

For any smooth connected linear algebraic group G over an algebraically closed field k, we describe the Picard group of the universal moduli stack of principal G-bundles over pointed smooth k-projective curves.

We prove the following extension of Tits' simplicity theorem. Let $k$ be an infinite field, $G$ an algebraic group defined and quasi-simple over $k,$ and $G(k)$ the group of $k$-rational points of $G.$ Let $G(k)^+$ be the subgroup of $G(k)$…

This paper examines fields of rationality in families of cuspidal automorphic representations of unitary groups. Specifically, for a fixed $A$ and a sufficiently large family $\mathcal{F}$, a small proportion of representations $\pi\in…