II型塞雷猜想伪简约群

Tamagawa Numbers and Other Invariants of Pseudo-reductive Groups Over Global Function Fields

We study Tamagawa numbers and other invariants (especially Tate-Shafarevich sets) attached to commutative and pseudo-reductive groups over global function fields. In particular, we prove a simple formula for Tamagawa numbers of commutative groups and pseudo-reductive groups. We also show that the Tamagawa numbers and Tate-Shafarevich sets of such groups are invariant under inner twist, as well as proving a result on the cohomology of such groups which extends part of classical Tate duality from commutative groups to all pseudo-reductive groups. Finally, we apply this last result to show that f…

常群下的泛平凡主丛

Generically trivial torsors under constant groups

本文解决了任意基域上的 Grothendieck-Serre 问题，通过发展伪完全和拟约化群下 torsor 的纯性与扩展定理，结合群概型结构理论，证明了泛平凡 torsor 的半局部平凡性。

代数群的Hom概形

Hom schemes for algebraic groups

本文拓展了SGA3中约化群同态函子可表的经典结论，覆盖G非约化的情形且证明更简洁，结论适用于任意基抛物子群，还关联了Serre完全可约性理论、解答Furter-Kraft问题并给出示例。

分布与正规性定理

Distributions and normality theorems

本文推导特征$p$下环路群分布代数的Serre表示，并据此为无$p$-挠基本群的约化群的抛物仿射格拉斯曼流形中Schubert簇正规性提供新证明。

The weight part of Serre's conjecture for GL(2)

Let p > 2 be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call "pseudo-Barsotti-Tate representations", over arbitrary finite extensions of the p-adics. As a consequence, we establish (under the usual Taylor-Wiles hypothesis) the weight part of Serre's conjecture for GL(2) over arbitrary totally real fields.

Cohomology of GL(4,Z) with Non-trivial Coefficients

In this paper we compute the cohomology groups of GL(4,Z) with coefficients in symmetric powers of the standard representation twisted by the determinant. This problem arises in Goncharov's approach to the study of motivic multiple zeta values of depth 4. The techniques that we use include Kostant's formula for cohomology groups of nilpotent Lie subalgebras of a reductive Lie algebra, Borel-Serre compactification, a result of Harder on Eisenstein cohomology. Finally, we need to show that the ghost class, which is present in the cohomology of the boundary of the Borel-Serre compactification, di…

A Satake isomorphism in characteristic p

Suppose that G is a connected reductive group over a p-adic field F, that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported, K-biequivariant functions f: G(F) \to End V. These Hecke algebras were first considered by Barthel-Livne for GL_2. They play a role in the recent mod p and p-adic Langlands correspondences for GL_2(Q_p), in generalisations of Serre's conjecture on the modularity of mod p …

Compactifications of S-arithmetic quotients for the projective general linear group

Let F be a global field, and let S be a finite set of places of F containing all archimedean places. Consider the product X of the symmetric spaces and Bruhat-Tits buildings for PGL_d of the completions of F at archimedean and non-archimedean places in S, respectively. We construct compactifications of the quotient of X by S-arithmetic subgroups of PGL_d(F). The constructions make delicate use of reductive Borel-Serre spaces for archimedean places and polyhedral and seminorm compactifications at nonarchimedean places. We also briefly discuss a few potential applications of our compacifications…

Pseudo-abelian varieties

Chevalley's theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian variety over an arbitrary field k to be a smooth connected k-group in which every smooth connected affine normal k-subgroup is trivial. This gives a new point of view on the classification of algebraic groups: every smooth connected group over a field is an extension of a pseudo-abelian variety by a smooth connected affine group, in a unique way. We work out…

Independence of ell of Monodromy Groups

Let X be a smooth curve over a finite field of characteristic p, let E be a number field, and consider an E-compatible system of lisse sheaves on the curve X. For each place lambda of E not lying over p, the lambda-component of the system is a lisse E_lambda-sheaf on X, whose associated arithmetic monodromy group is an algebraic group over the local field E_lambda. We use Serre's theory of Frobenius tori and Lafforgue's proof of Deligne's conjecture to show that when the E-compatible system is semisimple and pure of some integer weight, the isomorphism type of the identity component of these m…

Théorie de Bruhat-Tits pour les groupes quasi-réductifs

We extend (scheme-theoretic) Bruhat-Tits theory to quasi-reductive groups i.e. with trivial split unipotent radical over discretely valued henselian non-archimedean fields $K$, whose ring of integers is excellent and residue field is perfect, building on Bruhat-Tits I et II and the classification of pseudo-reductive groups by Conrad-Gabber-Prasad. The buildings had been constructed previously by Solleveld in a different manner, but they ultimately coincide with ours. We were motivated by the geometry of affine Grassmannians and give some applications at the end (compare also with previously su…

Generalized pseudo-coefficients of discrete series of $p$-adic groups

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 $σ$ of $M(F),$ we prove that there exists a locally constant and compactly supported function on $M(F),$ which generalizes a pseudo-coefficient of $σ.$ This function satisfies similar properties to the pseudo-coefficient, and its lifting to $G(F)$ is applied to the Plancherel formula.

通过μ_p-等变性对晶体伽罗瓦表示进行模p约化

Reduction modulo p of crystalline Galois representations via μ_p-equivariance

研究\(\mathbb{Q}_p\)绝对伽罗瓦群晶体表示模p约化的惯性权重，通过棱柱技术使布雷伊尔 - 基辛模模p约化获\(\mu_p\)-等变结构，结合仿射格拉斯曼流形结果，明确塞尔权重猜想并证其消除方向。