On supersingular representations of $GL_2(D)$ with a division algebra $D$ over a $p$-adic field

A classification of irreducible admissible mod p representations of p-adic reductive groups

Let F be a locally compact non-archimedean field, p its residue characteristic, and G a connected reductive group over F. Let C an algebraically closed field of characteristic p. We give a complete classification of irreducible admissible C-representations of G = G(F), in terms of supercuspidal C-representations of the Levi subgroups of G, and parabolic induction. Thus we push to their natural conclusion the ideas of the third-named author, who treated the case G = GL_m, as further expanded by the first-named author, who treated split groups G. As in the split case, we first get a classificati…

On the reduction modulo p of representations of a quaternion division algebra over a p-adic field

Let F be a non-Archimedean local field of residue characteristic p. In this paper, we first compute the reduction modulo p of irreducible smooth representations of a quaternion division algebra over F and of two-dimensional irreducible smooth representations of the Weil group of F. It turns out that a natural correspondence of mod p representations of the two groups and the composite of the local Langlands and Jacquet-Langlands correspondence are not compatible with the reduction, except in the cases considered by Vigneras.

Algebres de Hecke affines generiques

Let $H$ be a generic affine Hecke algebra (Iwahori-Matsumoto definition) over a polynomial algebra with a finite number of indeterminates over the ring of integers. We prove the existence of an integral Bernstein-Lusztig basis related to the Iwahori-Matsumoto basis by a strictly upper triangular matrix, from which we deduce that the center $Z$ of $H$ is finitely generated and that $H$ is a finite type $Z$-module (this was proved after inversion of the parameters by Bernstein-Lusztig), and we give some applications to the theory of $H$-modules where the parameters act by 0. These results are re…

论在 $\mathbb{Z}_p^2$-扩张上超奇异阿贝尔簇的 Mordell-Weil 秩

On the Mordell-Weil Ranks of supersingular abelian varieties over $\mathbb{Z}_p^2$-extensions

本文研究在分裂素数 $p$ 处超奇异的阿贝尔簇沿虚二次域 $\mathbb{Z}_p^2$-扩张的 Mordell–Weil 秩增长，并讨论相关多符号 Selmer 群与 GSp(4) 的 $p$-进 $L$-函数。

Compatibility between Satake and Bernstein-type isomorphisms in characteristic p

We study the center of the pro-p Iwahori-Hecke ring H of a connected split p-adic reductive group G. For k an algebraically closed field with characteristic p, we prove that the center of the k-algebra H_k:= H\otimes_Z k contains an affine semigroup algebra which is naturally isomorphic to the Hecke algebra attached to any irreducible smooth k-representation of a given hyperspecial maximal compact subgroup of G. This isomorphism is obtained using the inverse Satake isomorphism constructed in arXiv:1207.5557. We apply this to classify the simple supersingular H_k-modules, study the supersingula…

Iwasawa theory of automorphic representations of $\mathrm{GL}_{2n}$ at non-ordinary primes

Let $Π$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $Π$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed possibly unbounded $p$-adic $L$-functions interpolating complex $L$-values of $Π$ in the non-ordinary case. Under certain assumptions, we construct two \textit{bounded} $p$-adic $L$-functions for $Π$, thus extending an earlier work of Rockwood by relaxing the Pollack condition. Using Langlands local-global compatibility, we define signed Selmer groups over the $p$-adic cyclotomic extension of $\…

Deligne-Lusztig Constructions for Division Algebras and the Local Langlands Correspondence

Let $K$ be a local non-Archimedean field of positive characteristic and let $L$ be the degree-$n$ unramified extension of $K$. Via the local Langlands and Jacquet-Langlands correspondences, to each sufficiently generic multiplicative character of $L$, one can associate an irreducible representation of the multiplicative group of the central division algebra $D$ of invariant $1/n$ over $K$. In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive $p$-adic groups analogous to Deligne-Lusztig theory for finite reductive groups. In this paper we prove th…

Restriction of $p$-adic representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ to parahoric subgroups

Without using the $p$-adic Langlands correspondence, we prove that for many finite length smooth representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ on $p$-torsion modules the $\mathrm{GL}_2(\mathbf{Q}_p)$-linear morphisms coincide with the morphisms that are linear for the normalizer of a parahoric subgroup. We identify this subgroup to be the Iwahori subgroup in the supersingular case, and $\mathrm{GL}_2(\mathbf{Z}_p)$ in the principal series case. As an application, we relate the action of parahoric subgroups to the action of the inertia group of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q…

中心除代数上的 Breuil-Mézard 猜想

Breuil-Mézard conjectures for central division algebras

本文将 Breuil-Mézard 猜想推广到中心除代数的单位群，利用 Deligne-Lusztig 理论构造惯性类型与 Serre 权重的转移，并借助惯性 Jacquet-Langlands 对应和显式特征标公式证明其可由 GL_n 情形推出。

$D$ 为 $p$-进域上四元数可除代数时 $\rm{SL}_n(D)$ 的判别表示

Distinguished Representations for $\rm{SL}_n(D)$ where $D$ is a quaternion division algebra over a $p$-adic field

本文研究 $p$-进域上四元数可除代数情形中 $\rm{SL}_n(D)$ 被 $\rm{SL}_n^*(E)$ 判别的不可约容许表示，并给出相应 Hom 空间维数的重数公式。

Deligne--Lusztig constructions for division algebras and the local Langlands correspondence, II

In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive $p$-adic groups, analogous to Deligne-Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig's program. Precisely, let $X$ be the $p$-adic Deligne-Lusztig ind-scheme associated to a division algebra $D$ of invariant k/n over a non-Archimedean local field $K$. We study its homology groups by establishing a Deligne-Lusztig theory for families of finite unipotent groups that arise as subquotients of $D^\times$. The homology of $X$ induces a natural corresp…

$p$-进族中的傅里叶理论

$p$-adic Fourier theory in families

本研究构造了有限高度$p$-可除刚性解析群上函数与分布和对偶范畴中$\mathbb{Z}_p$-局部系统对象的傅里叶变换，推广了前人构造，还构造了相容积分傅里叶变换，并应用于构造整体艾森斯坦测度及新四元数模形式族。

Représentations modulo $p$ de $\mathrm{GL}(m,D)$, $D$ algèbre à division sur un corps local

This Ph.D. thesis belongs to the realm of mod $p$ representation theory of $p$-adic groups. The main object of study is the inner form of the general linear group $\mathrm{GL}(m,D)$ where $D$ is a division algebra over a non-Archimedean local field. The first part focuses on the rank $1$ case ($m=2$): it develops in detail the irreducibility criterion for parabolically induced representations and a classification of irreducible admissible smooth representations "à la Barthel-Livné". Then one turns to the generalized Steinberg representations: building on ideas of Grosse-Klönne and Herzig, this…

On the p-adic cohomology of the Lubin-Tate tower

We prove a finiteness result for the p-adic cohomology of the Lubin-Tate tower. For any n>=1 and p-adic field F, this provides a canonical functor from admissible p-adic representations of GL_n(F) towards admissible p-adic representations of Gal_F x D^*, where Gal_F is the absolute Galois group of F, and D/F is the central division algebra of invariant 1/n. Moreover, we verify a local-global-compatibility statement for this correspondence, and compatibility with the patching construction of Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin.

On special representations of $p$-adic reductive groups

Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(*)\quad\quad\quad\quad C^{\infty}(G/Q,L)/\sum_{Q'\supsetneq Q}C^{\infty}(G/Q',L).$$Let $I\subset G$ denote an Iwahori subgroup. We define a certain free finite rank $L$-module ${\mathfrak M}$ (depending on $Q$; if $Q$ is a Borel subgroup then $(*)$ is the Steinberg representation and ${\mathfrak M}$ is of rank one) and construct an $I$-equivariant embedding of $(*)$ into $C^{\infty}…

Iwahori-Hecke model for mod p representations of GL(2,F)

For a $p$-adic field $F$, the space of pro-$p$-Iwahori invariants of a universal supersingular mod $p$ representation $τ$ of ${\rm GL}_2(F)$ is determined in the works of Breuil, Schein, and Hendel. The representation $τ$ is introduced by Barthel and Livné and this is defined in terms of the spherical Hecke operator. In earlier work of Anandavardhanan-Borisagar, an Iwahori-Hecke approach was introduced to study these universal supersingular representations in which they can be characterized via the Iwahori-Hecke operators. In this paper, we construct a certain quotient $π$ of $τ$, making use o…

Remark on representation theory of general linear groups over a non-archimedean local division algebra

In this paper we give a simple (local) proof of two principal results about irreducible tempered representations of general linear groups over a non-archimedean local division algebra. We give a proof of the parameterization of the irreducible square integrable representations of these groups by segments of cuspidal representations, and a proof of the irreducibility of the tempered parabolic induction. Our proofs are based on Jacquet modules (and the Geometric Lemma, incorporated in the structure of a Hopf algebra). We use only some very basic general facts of the representation theory of redu…

Mod p Hecke algebras and dual equivariant cohomology I: the case of $GL_2$

Let $F$ be a p-adic local field and $G=GL_2(F)$. Let $\mathcal{H}^{(1)}$ be the pro-p Iwahori-Hecke algebra of $G$ with coefficients in an algebraic closure of $\mathbb{F}_p$. We show that the supersingular irreducible $\mathcal{H}^{(1)}$-modules of dimension 2 can be realized through the equivariant cohomology of the flag variety of the mod p Langlands dual group of $G$.

Nilpotent orbits of orthogonal groups over $p$-adic fields, and the DeBacker parametrization

For local non-archimedean fields $k$ of sufficiently large residual characteristic, we explicitly parametrize and count the rational nilpotent adjoint orbits in each algebraic orbit of orthogonal and special orthogonal groups. We separately give an explicit algorithmic construction for representatives of each orbit. We then, in the general setting of groups $\mathrm{GL}_n(D)$, $\mathrm{SL}_n(D)$ (where $D$ is a central division algebra over $k$) or classical groups, give a new characterisation of the "building set" (defined by DeBacker) of an $\mathfrak{sl}_2(k)$-triple in terms of the buildin…

Field Embeddings which are conjugate under a unit of a p-adic classical Group

Let (V,h) be a Hermitian space over a division algebra D which is of index at most two over a non-Archimedean local field k of residue characteristic not 2. Let G be the unitary group defined by h and let σbe the adjoint involution. Suppose we are given two σ-invariant but not σ-fixed field extensions E1 and E2 of k in End_D(V) which are isomorphic under conjugation by an element g of G and suppose that there is a point x in the Bruhat-Tits building of G which is fixed by the action of E1\{0} and E2\{0} on the reduced building of Aut_D(V). Then E1 is conjugate to E2 under an element of the sta…

Geometric approach to the local Jacquet-Langlands correspondence

In this paper, we give a purely geometric approach to the local Jacquet-Langlands correspondence for GL(n) over a p-adic field, under the assumption that the invariant of the division algebra is 1/n. We use the l-adic etale cohomology of the Drinfeld tower to construct the correspondence at the level of the Grothendieck groups with rational coefficients. Moreover, assuming that n is prime, we prove that this correspondence preserves irreducible representations. This gives a purely local proof of the local Jacquet-Langlands correspondence in this case. We need neither a global automorphic techn…

Rankin-Selberg local factors modulo $\ell$

After extending the theory of Rankin-Selberg local factors to pairs of $\ell$-modular representations of Whittaker type, of general linear groups over a non-archimedean local field, we study the reduction modulo $\ell$ of $\ell$-adic local factors and their relation to these $\ell$-modular local factors. While the $\ell$-modular local $γ$-factor we associate to such a pair turns out to always coincide with the reduction modulo $\ell$ of the $\ell$-adic $γ$-factor of any Whittaker lifts of this pair, the local $L$-factor exhibits a more interesting behaviour; always dividing the reduction modul…

Block decomposition of the category of l-modular smooth representations of finite length of GL(m,D)

Let F be a non-Archimedean locally compact field of residue characteristic p, let G be an inner form of GL(n,F) with n>0, and let l be a prime number different from p. We describe the block decomposition of the category of finite length smooth representations of G with coefficients in an algebraically closed field of characteristic l. Unlike the case of complex representations of an arbitrary p-adic reductive group and that of l-modular representations of GL(n,F), several non-isomorphic supercuspidal supports may correspond to the same block. We describe the (finitely many) supercuspidal suppo…

关于某些有限环上 $\mathrm{GL}_2$ 的模 $p$ 主级数表示的结构

On the structure of modular principal series representations of $\mathrm{GL}_2$ over some finite rings

本文研究 $p$-adic 域整数环模幂商环上 $\mathrm{GL}_2$ 的模 $p$ 主级数表示结构，确定其 Jordan-Hölder 合成因子，并在 $n=2$ 时按分歧类型给出子模结构、基座与根滤过等精细结果，深化了对 Breuil 等人相关构造的理解。