Let $F$ be a totally real field, $\mathfrak{p}$ an unramified place of $F$ dividing $p$ and $\overline{r}: \mathrm{Gal}(\overline{F}/F)\rightarrow\mathrm{GL}_2(\overline{\mathbb{F}}_p)$ a continuous irreducible modular representation. The work of Buzzard, Diamond and Jarvis associates to $\overline{r}$ an admissible smooth representation of $\mathrm{GL}_2(F_\mathfrak{p})$ on the mod $p$ cohomology of Shimura curves attached to indefinite division algebras which split at $\mathfrak{p}$. When $\overline{r}|_{\mathrm{Gal}(\overline{F_\mathfrak{p}}/F_\mathfrak{p})}$ is tamely ramified and generic (and under some technical assumptions), we determine the subspace of invariants of this representation under the principal congruence subgroup of level $\mathfrak{p}$. In particular, it depends only on $\overline{r}|_{\mathrm{Gal}(\overline{F_\mathfrak{p}}/F_\mathfrak{p})}$ and verifies a multiplicity one property.
In this paper, we investigate a general method to establish tame and norm relations for special cycles in Shimura varieties, using unitary cycles in odd orthogonal Shimura varieties as a guiding example.
Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum of Hodge type. Let $p$ be an odd prime such that $\mathsf{G}_{\mathbb{Q}_p}$ splits after a tamely ramified extension and $p\nmid |π_1(\mathsf{G}^{\rm der})|$. Under some mild additional assumptions that are satisfied if the associated Shimura variety is proper and $\mathsf{G}_{\mathbb{Q}_p}$ is either unramified or residually split, we prove the generalisation of Mantovan's formula for the $l$-adic cohomology of the associated Shimura variety. On the way we derive some new results about the geometry of the Newton stratification of the reduction…
我们证明了Breuil和Breuil-Dembele（C. Breuil，“Sur un probleme de compatibilite local-global modulo p pour GL（2）”），包括从主列到尖点情形的概括，服从一个温和的全局假设，我们为了应用某些R=T定理。更准确地说，我们证明了一个多重性的结果为模p上同调的Shimura曲线在Iwahori水平，我们表明，某些显然是全球定义的格上同调的Shimura曲线是由相应的本地p-adic伽罗瓦表示。我们还指出了Buzzard-Diamond-Jarvis猜想在一般情况下的一个新的证明。我们的主要工具是几何Breuil-Mezard哲学的两位作者开发的，和一个新的和更多的功能的泰勒-怀尔斯-Kisin修补方法的角度。一路上，我们确定了驯服潜在的Barsotti-Tate变形环的通用二维模p表示，推广的结果Breuil-Mezard在主列的情况下。
We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, as well as their resolutions by moduli stacks of two-dimensional Breuil-Kisin modules with tame descent data. We study the local geometry of these moduli stacks by comparing them with local models of Shimura varieties at hyperspecial and Iwahori level.
Xue proved an equational refinement of the unitary Shimura curve case of the arithmetic Gan-Gross-Prasad conjecture via the Gross-Zagier formula for quaternionic Shimura curves. On the other hand, Rapoport, Smithling and Zhang posed a variant of the conjecture, using modified PEL type Shimura varieties, which we call RSZ Shimura varieties. We reinterpret the result of Xue in terms of the modified Shimura curves. We then use the reinterpretation to prove a case of the variant of the conjecture. Our result combined with the work of Xue establishes a connection between the variant and the Gross-Z…
Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular Galois representation which satisfies the Taylor-Wiles hypotheses and is generic at a place $v$ above $p$. Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. Then the $\mathfrak{m}$-torsion in the mod $p$ cohomology of Shimura curves with full congruence level at $v$ coincides with the $\mathrm{GL}_2(k_v)$-representation $D_0(\overline{r}|_{G_{F_v}})$ constructed by Breuil and Paškūnas. In particular, it depends only on the local representa…
We show that the mod $p$ cohomology of a simple Shimura variety treated in Harris-Taylor's book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we show the vanishing outside the middle degree.
In this article, we generalize the work of H.Hida and V.Pilloni to construct $p$-adic families of $μ$-ordinary modular forms on Shimura varieties of Hodge type $Sh(G,X)$ associated to a Shimura datum $(G,X)$ where $G$ is a connected reductive group over $\mathbb{Q}$ and is unramified at $p$, such that the adjoint quotient $G^\mathrm{ad}$ has no simple factors isomorphic to $\mathrm{PGL}_2$.
For a new class of Shimura varieties of orthogonal type over a totally real number field, we construct special cycles and show the the modularity of Kudla's generating series in the cohomology group.
We study the motivic cohomology of the special fiber of quaternionic Shimura varieties at a prime of good reduction. We exhibit classes in these motivic cohomology groups and use this to give an explicit geometric realization of level raising between Hilbert modular forms. The main ingredient for our construction is a form of Ihara's Lemma for compact quaternionic Shimura surfaces which we prove by generalizing a method of Diamond-Taylor. Along the way we also verify the Hecke orbit conjecture for these quaternionic Shimura varieties which is a key input for our proof of Ihara's Lemma.
Let $F$ be a totally real field in which a prime number $p>2$ is inert. We continue the study of the (generalized) Goren--Oort strata on quaternionic Shimura varieties over finite extensions of $\mathbb F_p$. We prove that, when the dimension of the quaternionic Shimura variety is even, the Tate conjecture for the special fiber of the quaternionic Shimura variety holds for the cuspidal $π$-isotypical component, as long as the two unramified Satake parameters at $p$ are not differed by a root of unity.
We give a new proof of Faltings's p-adic Eichler-Shimura decomposition of the modular curves via BGG methods and the Hodge-Tate period map. The key property is the relation between the Tate module and the Faltings extension, which was already used in the original proof. Then, we construct overconvergent Eichler-Shimura (ES) maps for the modular curves providing ''the second half'' of the overconvergent ES map of Andreatta-Iovita-Stevens. We use higher Coleman theory on the modular curve developed by Boxer-Pilloni to show that the small slope part of the ES maps interpolates the classical p-adi…
We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain $p$-adic estimates of Satake parameters of certain non-regular self dual automorphic representations of $\mathrm{GL}_n$.
We generalize the local-global compatibility result in arXiv:1506.04022 to higher dimensional cases, by examining the relation between Scholze's functor and cohomology of Kottwitz-Harris-Taylor type Shimura varieties. Along the way we prove a cuspidality criterion from type theory. We also deal with compatibility for torsion classes in the case of semisimple mod $p$ Galois representations with distinct irreducible components under certain flatness hypotheses.
Let F be a totally real number field. Using a recent geometric approach developed by Andreatta and Iovita we construct several variables p-adic families of finite slope quaternionic automorphic forms over F. It is achieved by interpolating the modular sheaves defined over some auxiliary unitary Shimura curves. Secondly, we attach p-adic L-functions to triples of ordinary p-adic families of quaternionic automorphic eigenforms. This is done by relating trilinear periods to some trilinear products over unitary Shimura curves which can be interpolated adapting the work of Liu-Zhang-Zhang to our fa…
We compute the arithmetic volumes of integral models of unitary Shimura curves. This establishes the base case of an inductive argument to compute the arithmetic volumes of unitary Shimura varieties of higher dimension, to appear in subsequent work of Bruinier and the author.
By considering the intersections of Shimura curves and Humbert surfaces on the Siegel modular threefold, we obtain new class number relations. The result is a higher-dimensional analogue of the classical Hurwitz-Kronecker class number relation.
We study the special fiber of the integral models for Shimura varieties of Hodge type with parahoric level structure constructed by Kisin and Pappas in [KP]. We show that when the group is residually split, the points in the mod $p$ isogeny classes have the form predicted by the Langlands Rapoport conjecture in [LR]. We also verify most of the He-Rapoport axioms for these integral models without the residually split assumption. This allows us to prove that all Newton strata are non-empty for these models.
