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.

scholarly journals GALOIS CONJUGATES OF SPECIAL POINTS AND SPECIAL SUBVARIETIES IN SHIMURA VARIETIES

2019 ◽  
pp. 1-15
Author(s):  
Martin Orr
Keyword(s):  
Special Point ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Canonical Models ◽  
Special Points

Let $S$ be a Shimura variety with reflex field $E$ . We prove that the action of $\text{Gal}(\overline{\mathbb{Q}}/E)$ on $S$ maps special points to special points and special subvarieties to special subvarieties. Furthermore, the Galois conjugates of a special point all have the same complexity (as defined in the theory of unlikely intersections). These results follow from Milne and Shih’s construction of canonical models of Shimura varieties, based on a conjecture of Langlands which was proved by Borovoi and Milne.

scholarly journals Towards the Andre–Oort conjecture for mixed Shimura varieties: The Ax–Lindemann theorem and lower bounds for Galois orbits of special points

2017 ◽  
Vol 2017 (732) ◽  
pp. 85-146 ◽  
Author(s):  
Ziyang Gao
Keyword(s):  
Lower Bounds ◽  
Riemann Hypothesis ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Weierstrass Theorem ◽  
Generalized Riemann Hypothesis ◽  
Galois Orbits ◽  
Special Points

Abstract We prove in this paper the Ax–Lindemann–Weierstraß theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular, we reprove a result of Silverberg [57] in a different approach. Then combining these results we prove the André–Oort conjecture unconditionally for any mixed Shimura variety whose pure part is a subvariety of {\mathcal{A}_{6}^{n}} and under the Generalized Riemann Hypothesis for all mixed Shimura varieties of abelian type.

scholarly journals On the Bloch–Kato conjecture for Hilbert modular forms

2021 ◽  
Author(s):  
Matteo Tamiozzo
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Euler System ◽  
Central Point ◽  
Shimura Curves ◽  
Hilbert Modular Forms ◽  
Reciprocity Laws ◽  
Hilbert Modular ◽  
Special Points

AbstractThe aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

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 On Goren–Oort stratification for quaternionic Shimura varieties

2016 ◽  
Vol 152 (10) ◽  
pp. 2134-2220 ◽  
Author(s):  
Yichao Tian ◽  
Liang Xiao
Keyword(s):  
Line Bundle ◽  
Real Field ◽  
Necessary Condition ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Maximal Level ◽  
Totally Real ◽  
Totally Real Field

Let $F$ be a totally real field in which a prime $p$ is unramified. We define the Goren–Oort stratification of the characteristic-$p$ fiber of a quaternionic Shimura variety of maximal level at $p$. We show that each stratum is a $(\mathbb{P}^{1})^{r}$-bundle over other quaternionic Shimura varieties (for an appropriate integer $r$). As an application, we give a necessary condition for the ampleness of a modular line bundle on a quaternionic Shimura variety in characteristic $p$.

scholarly journals Arithmetic gauge theory: A brief introduction

2018 ◽  
Vol 33 (29) ◽  
pp. 1830012 ◽  
Author(s):  
Minhyong Kim
Keyword(s):  
Moduli Spaces ◽  
Automorphic Forms ◽  
Galois Representations ◽  
Arithmetic Geometry ◽  
Quantum Field ◽  
Principal Bundles ◽  
Langlands Program ◽  
Effective Version ◽  
Diophantine Geometry ◽  
Higher Genus

Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles holds the key to an effective version of Faltings’ theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of Galois representations, the structures linking motives to automorphic forms according to the Langlands program. In this paper, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory.

scholarly journals Local models in the ramified case. III Unitary groups

2009 ◽  
Vol 8 (3) ◽  
pp. 507-564 ◽  
Author(s):  
G. Pappas ◽  
M. Rapoport
Keyword(s):  
Local Structure ◽  
Level Structure ◽  
Quadratic Extension ◽  
Unitary Groups ◽  
Shimura Varieties ◽  
Flag Varieties ◽  
Shimura Variety ◽  
Local Models ◽  
Affine Flag

AbstractWe continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primespat which the group defining the Shimura variety ramifies. We describe ‘good’p-adic integral models of these Shimura varieties and study their étale local structure. In the present paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties that we developed in a previous paper.

scholarly journals The μ-ordinary Hasse invariant of\break unitary Shimura varieties

2017 ◽  
Vol 2017 (728) ◽  
Author(s):  
Wushi Goldring ◽  
Marc-Hubert Nicole
Keyword(s):  
Shimura Varieties ◽  
Shimura Variety

AbstractWe construct a generalization of the Hasse invariant for any Shimura variety of PEL-type

scholarly journals Unitary cycles on Shimura curves and the Shimura lift II

2014 ◽  
Vol 150 (12) ◽  
pp. 1963-2002
Author(s):  
Siddarth Sankaran
Keyword(s):  
Modular Form ◽  
Recent Work ◽  
Chow Groups ◽  
Shimura Varieties ◽  
Shimura Curves ◽  
Shimura Lift ◽  
Generating Series

AbstractWe consider two families of arithmetic divisors defined on integral models of Shimura curves. The first was studied by Kudla, Rapoport and Yang, who proved that if one assembles these divisors in a formal generating series, one obtains the$q$-expansion of a modular form of weight 3/2. The present work concerns the Shimura lift of this modular form: we identify the Shimura lift with a generating series comprising divisors arising in the recent work of Kudla and Rapoport regarding cycles on Shimura varieties of unitary type. In the prequel to this paper, the author considered the geometry of the two families of cycles. These results are combined with the Archimedean calculations found in this work in order to establish the theorem. In particular, we obtain new examples of modular generating series whose coefficients lie in arithmetic Chow groups of Shimura varieties.

scholarly journals Foliations on unitary Shimura varieties in positive characteristic

2018 ◽  
Vol 154 (11) ◽  
pp. 2267-2304 ◽  
Author(s):  
Ehud de Shalit ◽  
Eyal Z. Goren
Keyword(s):  
Tangent Bundle ◽  
Positive Characteristic ◽  
Blow Up ◽  
Level Structure ◽  
Horizontal Component ◽  
Zariski Closure ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Integral Manifolds ◽  
Imaginary Field

When$p$is inert in the quadratic imaginary field$E$and$m<n$, unitary Shimura varieties of signature$(n,m)$and a hyperspecial level subgroup at$p$, carry a naturalfoliationof height 1 and rank$m^{2}$in the tangent bundle of their special fiber$S$. We study this foliation and show that it acquires singularities at deep Ekedahl–Oort strata, but that these singularities are resolved if we pass to a natural smooth moduli problem$S^{\sharp }$, a successive blow-up of$S$. Over the ($\unicode[STIX]{x1D707}$-)ordinary locus we relate the foliation to Moonen’s generalized Serre–Tate coordinates. We study the quotient of$S^{\sharp }$by the foliation, and identify it as the Zariski closure of the ordinary-étale locus in the special fiber$S_{0}(p)$of a certain Shimura variety with parahoric level structure at$p$. As a result, we get that this ‘horizontal component’ of$S_{0}(p)$, as well as its multiplicative counterpart, are non-singular (formerly they were only known to be normal and Cohen–Macaulay). We study two kinds of integral manifolds of the foliation: unitary Shimura subvarieties of signature$(m,m)$, and a certain Ekedahl–Oort stratum that we denote$S_{\text{fol}}$. We conjecture that these are the only integral submanifolds.

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