On the faithfulness of parabolic cohomology as a Hecke module over a finite field

Functional Hecke algebras and simple Bernstein blocks of a p-adic GL_n in non-defining characteristic

Let $G_{n}=\operatorname{GL}_{n}(F)$, where $F$ is a non-archimedean local field with residue characteristic $p$ and where $n=2k$ is even. In this article, we investigate a question occurring in the decomposition of the category of $\ell$-modular smooth representations of $G_n$ into Bernstein blocks (where $\ell\neq p$). The easiest block not investigated in \cite{guiraud} is the one defined by the standard parabolic subgroup with Levi factor $M=\GL_k(F) \times \GL_k(F)$ and by an $M$-representation of the form $π_0 \boxtimes π_0$ with $π_0$ a supercuspidal $\GL_k(F)$-representation. This bloc…

Hecke代数与非奇异深度零表示的局部朗兰兹对应

Hecke algebras and local Langlands correspondence for non-singular depth-zero representations

本文通过构造扭曲有理Hecke代数的范畴等价，建立了非奇异深度零表示的局部朗兰兹对应，并展示了其与抛物诱导的兼容性。

Parabolic eigenvarieties via overconvergent cohomology

Let $G'$ be a connected reductive group over $\mathbb{Q}$ such that $G = G'/\mathbb{Q}_p$ is quasi-split, and let $Q \subset G$ be a parabolic subgroup. We introduce parahoric overconvergent cohomology groups with respect to $Q$, and prove a classicality theorem showing that the small slope parts of these groups coincide with those of classical cohomology. This allows the use of overconvergent cohomology at parahoric, rather than Iwahoric, level, and provides flexible lifting theorems that appear to be particularly well-adapted to arithmetic applications. When $Q$ is the Borel, we recover the …

Parabolic induction in characteristic p

Let G be the group of rational points of a reductive connected group over a finite field (resp. nonarchimedean local field of characteristic p) and R a commutative ring. The unipotent (resp. pro-p Iwahori) invariant functor takes a smooth representation of G to a module over the unipotent (resp. pro-p Iwahori) Hecke R-algebra H of G. We prove that these functors for G and for a Levi subgroup of G commute with the parabolic induction functors, as well as with the right adjoints of the parabolic induction functors. However, they do not commute with the left adjoints of the parabolic induction fu…

Automorphic forms of higher order

In this paper a theory of Hecke operators for higher order modular forms is established. The definition of cusp forms and attached L-functions is extended beyond the realm of parabolic invariants. The role of representation theoretic methods is clarified and, motivated by higher order forms, new convolution products of L-functions are introduced.

$p$-adic 有限简约群的 Hecke 代数与伯恩斯坦块的局部朗兰兹对应关系

Hecke algebras for $p$-adic reductive groups and Local Langlands Correspondence for Bernstein blocks

本文研究了 $p$-adic 有限简约群的 Hecke 代数与伯恩斯坦块的局部朗兰兹对应关系，通过构建增强的 $L$-参数集并应用到各种群上，证明了某些猜想并简化了深度零情况。

On some properties of the functors ${\mathcal F}^G_P$ from Lie algebra to locally analytic representations

For a split reductive group $G$ over a finite extension $L$ of ${\mathbb Q}_p$, and a parabolic subgroup $P \subset G$ we examine functorial properties of the functors ${\mathcal F}^G_P$ introduced in \cite{OS2}. We discuss the aspects of faithfulness, projective and injective objects, Ext-groups and some kind of adjunction formulas.

The Steinberg Symbol and Special Values of L-functions

The main results of this article concern the definition of a compactly supported cohomology class for the congruence group $Γ_0(p^n)$ with values in the second Milnor $K$-group (modulo 2-torsion) of the ring of $p$-integers of the cyclotomic extension $\Q(μ_{p^n})$. We endow this cohomology group with a natural action of the standard Hecke operators and discuss the existence of special Hecke eigenclasses in its parabolic cohomology. Moreover, for $n=1$, assuming the non-degeneracy of a certain pairing on $p$-units induced by the Steinberg symbol when $(p,k)$ is an irregular pair, i.e. $p|\frac…

关于非分歧 Eisenstein 谱

On the unramified Eisenstein spectrum

本文研究整体域上分裂约化群的非分歧 Eisenstein 谱，借助 Langlands 对偶群 Springer 栈在导出商栈配边加法群中的分解，确定了最小抛物群对应球面 Hecke 代数的谱。

On pro-$p$-Iwahori invariants of $R$-representations of reductive $p$-adic groups

Let $F$ be locally compact field with residue characteristic $p$, and $\mathbf{G}$ a connected reductive $F$-group. Let $\mathcal{U}$ be a pro-$p$ Iwahori subgroup of $G = \mathbf{G}(F)$. Fix a commutative ring $R$. If $π$ is a smooth $R[G]$-representation, the space of invariants $π^{\mathcal{U}}$ is a right module over the Hecke algebra $\mathcal{H}$ of $\mathcal{U}$ in $G$. Let $P$ be a parabolic subgroup of $G$ with a Levi decomposition $P = MN$ adapted to $\mathcal{U}$. We complement previous investigation of Ollivier-Vignéras on the relation between taking $\mathcal{U}$-invariants and va…

虚二次域上GL(2)的自守符号与自守L-值

Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields

该研究将Kim–Sun的同调策略从经典模曲线推广到算术轨形，证明虚二次域上平行权2 Hecke本征形式经Coates–Wiles特征扭转的L-函数临界值，在正密度素数集合上模素非零的特征占比为正。

Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one, with appendices by Jean-Francois Mestre and Gabor Wiese

Two integral structures on the Q-vector space of modular forms of weight two on X_0(N) are compared at primes p exactly dividing N. When p=2 and N is divisible by a prime that is 3 mod 4, this comparison leads to an algorithm for computing the space of weight one forms mod 2 on X_0(N/2). For p arbitrary and N>4 prime to p, a way to compute the Hecke algebra of mod p modular forms of weight one on Gamma_1(N) is presented, using forms of weight p, and, for p=2, parabolic group cohomology with mod 2 coefficients. Appendix A is a letter from Mestre to Serre, of October 1987, where he reports on co…

Regularity of quotients of Drinfeld modular schemes

Let $A$ be the coordinate ring of a projective smooth curve over a finite field minus a closed point. For a nontrivial ideal $I \subset A$, Drinfeld defined the notion of structure of level $I$ on a Drinfeld module. We extend this to that of level $N$, where $N$ is a finitely generated torsion $A$-module. The case where $N=(I^{-1}/A)^d$, where $d$ is the rank of the Drinfeld module,coincides with the structure of level $I$. The moduli functor is representable by a regular affine scheme. The automorphism group $\mathrm{Aut}_{A}(N)$ acts on the moduli space. Our theorem gives a class of subgroup…

关于非阿基米德局部域上一般线性群内形式上的乘积函子

On the product functor on inner forms of the general linear group over a non-Archimedean local field

本文证明了在非阿基米德一般线性群内形式的光滑表示范畴的全子范畴上，抛物诱导函子是完全忠实的，并应用该结果证明由 Jacquet 函子产生的特定大导数具有不可约基座且重数为 1。