scholarly journals -ADIC -FUNCTIONS FOR UNITARY GROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
ELLEN EISCHEN ◽  
MICHAEL HARRIS ◽  
JIANSHU LI ◽  
CHRISTOPHER SKINNER
Keyword(s):  
Differential Operators ◽  
Automorphic Forms ◽  
Unitary Groups ◽  
Shimura Varieties ◽  
Euler Product ◽  
Doubling Method ◽  
Hida Families

This paper completes the construction of $p$ -adic $L$ -functions for unitary groups. More precisely, in Harris, Li and Skinner [‘ $p$ -adic $L$ -functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math.Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such $p$ -adic $L$ -functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and $p$ -adic differential operators [Eischen, ‘A $p$ -adic Eisenstein measure for unitary groups’, J. Reine Angew. Math.699 (2015), 111–142; ‘ $p$ -adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble)62(1) (2012), 177–243], Part II of the present paper provides the calculations of local $\unicode[STIX]{x1D701}$ -integrals occurring in the Euler product (including at $p$ ). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method.

scholarly journals Vertical Distribution Relations for Special Cycles on Unitary Shimura Varieties

2018 ◽  
Vol 2020 (13) ◽  
pp. 3902-3926
Author(s):  
Réda Boumasmoud ◽  
Ernest Hunter Brooks ◽  
Dimitar P Jetchev
Keyword(s):  
Vertical Distribution ◽  
Three Dimensional ◽  
Complex Multiplication ◽  
Horizontal Distribution ◽  
Unitary Groups ◽  
Shimura Varieties ◽  
Heegner Points ◽  
Universal Norm

Abstract We consider cycles on three-dimensional Shimura varieties attached to unitary groups, defined over extensions of a complex multiplication (CM) field $E$, which appear in the context of the conjectures of Gan et al. [6]. We establish a vertical distribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of [8], and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $\Lambda $-module constructed from Heegner points.

AUTOMORPHIC LEFSCHETZ PROPERTIES FOR NONCOMPACT ARITHMETIC MANIFOLDS

2021 ◽  
pp. 1-48
Author(s):  
Arvind N. Nair ◽  
Ankit Rai
Keyword(s):  
Symmetric Spaces ◽  
Automorphic Forms ◽  
Hodge Theory ◽  
Hyperbolic Manifolds ◽  
Shimura Varieties ◽  
Restriction Maps ◽  
Locally Symmetric Spaces ◽  
Compact Congruence ◽  
Lefschetz Properties ◽  
Invent Math

Abstract We prove the injectivity of Oda-type restriction maps for the cohomology of noncompact congruence quotients of symmetric spaces. This includes results for restriction between (1) congruence real hyperbolic manifolds, (2) congruence complex hyperbolic manifolds, and (3) orthogonal Shimura varieties. These results generalize results for compact congruence quotients by Bergeron and Clozel [Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math.192 (2013), 505–532] and Venkataramana [Cohomology of compact locally symmetric spaces, Compos. Math.125 (2001), 221–253]. The proofs combine techniques of mixed Hodge theory and methods involving automorphic forms.

Shimura Varieties and the Selberg Trace Formula

1977 ◽  
Vol 29 (6) ◽  
pp. 1292-1299 ◽  
Author(s):  
R. P. Langlands
Keyword(s):  
Trace Formula ◽  
Automorphic Forms ◽  
Zeta Functions ◽  
Higher Dimensions ◽  
Shimura Varieties ◽  
Selberg Trace Formula ◽  
Work In Progress ◽  
Euler Products ◽  
Dimension One

This paper is a report on work in progress rather than a description of theorems which have attained their final form. The results I shall describe are part of an attempt to continue to higher dimensions the study of the relation between the Hasse-Weil zeta-functions of Shimura varieties and the Euler products associated to automorphic forms, which was initiated by Eichler, and extensively developed by Shimura for the varieties of dimension one bearing his name. The method used has its origins in an idea of Sato, which was exploited by Ihara for the Shimura varieties associated to GL(2).

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 Topological flatness of local models for ramified unitary groups. II. The even dimensional case

2013 ◽  
Vol 13 (2) ◽  
pp. 303-393 ◽  
Author(s):  
Brian D. Smithling
Keyword(s):  
Local Structure ◽  
Unitary Groups ◽  
Dimensional Case ◽  
Shimura Varieties ◽  
Local Models ◽  
Admissible Set ◽  
The Way ◽  
Moduli Problems

AbstractLocal models are schemes, defined in terms of linear-algebraic moduli problems, which are used to model the étale-local structure of integral models of certain$p$-adic PEL Shimura varieties defined by Rapoport and Zink. In the case of a unitary similitude group whose localization at${ \mathbb{Q} }_{p} $is ramified, quasi-split$G{U}_{n} $, Pappas has observed that the original local models are typically not flat, and he and Rapoport have introduced new conditions to the original moduli problem which they conjecture to yield a flat scheme. In a previous paper, we proved that their new local models are topologically flat when$n$is odd. In the present paper, we prove topological flatness when$n$is even. Along the way, we characterize the$\mu $-admissible set for certain cocharacters$\mu $in types$B$and$D$, and we show that for these cocharacters admissibility can be characterized in a vertexwise way, confirming a conjecture of Pappas and Rapoport.

scholarly journals RANKIN–COHEN TYPE DIFFERENTIAL OPERATORS FOR SIEGEL MODULAR FORMS

1998 ◽  
Vol 09 (04) ◽  
pp. 443-463 ◽  
Author(s):  
WOLFGANG EHOLZER ◽  
TOMOYOSHI IBUKIYAMA
Keyword(s):  
Half Space ◽  
Modular Forms ◽  
Automorphic Form ◽  
Differential Operators ◽  
Automorphic Forms ◽  
Siegel Modular Forms ◽  
Elliptic Case ◽  
Vector Valued

Let ℍn be the Siegel upper half space and let F and G be automorphic forms on ℍn of weights k and l, respectively. We give explicit examples of differential operators D acting on functions on ℍn × ℍn such that the restriction of [Formula: see text] to Z = Z1 = Z2 is again an automorphic form of weight k + l + v on ℍn. Since the elliptic case, i.e. n = 1, has already been studied some time ago by R. Rankin and H. Cohen we call such differential operators Rankin–Cohen type operators. We also discuss a generalisation of Rankin–Cohen type operators to vector valued differential operators.

Sign in / Sign up

Export Citation Format

Share Document