scholarly journals ON THE COHOMOLOGY OF SOME SIMPLE SHIMURA VARIETIES WITH BAD REDUCTION

2016 ◽  
Vol 18 (1) ◽  
pp. 1-24
Author(s):  
Xu Shen
Keyword(s):  
Galois Representations ◽  
Shimura Varieties

We determine the Galois representations inside the$\ell$-adic cohomology of some quaternionic and related unitary Shimura varieties at ramified places. The main results generalize the previous works of Reimann and Kottwitz in this setting to arbitrary levels at$p$, and confirm the expected description of the cohomology due to Langlands and Kottwitz.

scholarly journals CATEGORICITY OF MODULAR AND SHIMURA CURVES

2015 ◽  
Vol 16 (5) ◽  
pp. 1075-1101 ◽  
Author(s):  
Christopher Daw ◽  
Adam Harris
Keyword(s):  
Model Theory ◽  
Galois Representations ◽  
Arithmetic Geometry ◽  
Shimura Varieties ◽  
Shimura Curves ◽  
Shimura Variety ◽  
Unique Model ◽  
Infinite Cardinality ◽  
Special Points ◽  
Generic Points

We describe a model-theoretic setting for the study of Shimura varieties, and study the interaction between model theory and arithmetic geometry in this setting. In particular, we show that the model-theoretic statement of a certain${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$-sentence having a unique model of cardinality$\aleph _{1}$is equivalent to a condition regarding certain Galois representations associated with Hodge-generic points. We then show that for modular and Shimura curves this${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$-sentence has a unique model in every infinite cardinality. In the process, we prove a new characterisation of the special points on any Shimura variety.

A stable trace formula for Igusa varieties

2010 ◽  
Vol 9 (4) ◽  
pp. 847-895 ◽  
Author(s):  
Sug Woo Shin
Keyword(s):  
Trace Formula ◽  
Positive Characteristic ◽  
Galois Representations ◽  
Type A ◽  
Shimura Varieties ◽  
Explicit Method ◽  
Good Reduction ◽  
Orbital Integrals ◽  
Fundamental Lemma ◽  
Stable Trace Formula

AbstractIgusa varieties are smooth varieties in positive characteristic p which are closely related to Shimura varieties and Rapoport–Zink spaces. One motivation for studying Igusa varieties is to analyse the representations in the cohomology of Shimura varieties which may be ramified at p. The main purpose of this work is to stabilize the trace formula for the cohomology of Igusa varieties arising from a PEL datum of type (A) or (C). Our proof is unconditional thanks to the recent proof of the fundamental lemma by Ngô, Waldspurger and many others.An earlier work of Kottwitz, which inspired our work and proves the stable trace formula for the special fibres of PEL Shimura varieties with good reduction, provides an explicit way to stabilize terms at ∞. Stabilization away from p and ∞ is carried out by the usual Langlands–Shelstad transfer as in work of Kottwitz. The key point of our work is to develop an explicit method to handle the orbital integrals at p. Our approach has the technical advantage that we do not need to deal with twisted orbital integrals or the twisted fundamental lemma.One application of our formula, among others, is the computation of the arithmetic cohomology of some compact PEL-type Shimura varieties of type (A) with non-trivial endoscopy. This is worked out in a preprint of the author's entitled ‘Galois representations arising from some compact Shimura varieties’.

scholarly journals Shimura varieties at level and Galois representations

2020 ◽  
Vol 156 (6) ◽  
pp. 1152-1230 ◽  
Author(s):  
Ana Caraiani ◽  
Daniel R. Gulotta ◽  
Chi-Yun Hsu ◽  
Christian Johansson ◽  
Lucia Mocz ◽  
...  
Keyword(s):  
Symmetric Spaces ◽  
Galois Representations ◽  
Hecke Algebras ◽  
Derived Category ◽  
Shimura Varieties ◽  
Nilpotent Ideal ◽  
Locally Symmetric Spaces ◽  
Compactly Supported ◽  
Symmetric Varieties

We show that the compactly supported cohomology of certain $\text{U}(n,n)$- or $\text{Sp}(2n)$-Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$-level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for $\text{GL}_{n}/F$. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze [On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), 945–1066; MR 3418533] and Newton–Thorne [Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma 4 (2016), e21; MR 3528275].

scholarly journals ON THE -ADIC COHOMOLOGY OF SOME -ADICALLY UNIFORMIZED SHIMURA VARIETIES

2016 ◽  
Vol 17 (5) ◽  
pp. 1197-1226
Author(s):  
Xu Shen
Keyword(s):  
Galois Representations ◽  
Shimura Varieties ◽  
Finite Products

We determine the Galois representations inside the$\ell$-adic cohomology of some unitary Shimura varieties at split places where they admit uniformization by finite products of Drinfeld upper half spaces. Our main results confirm Langlands–Kottwitz’s description of the cohomology of Shimura varieties in new cases.

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