scholarly journals LOCAL BORCHERDS PRODUCTS FOR UNITARY GROUPS

2017 ◽  
Vol 234 ◽  
pp. 139-169
Author(s):  
ERIC HOFMANN
Keyword(s):  
Boundary Component ◽  
Unitary Group ◽  
Picard Group ◽  
Unitary Groups ◽  
Weil Representation ◽  
Cusp Forms ◽  
Modular Variety ◽  
Arithmetic Subgroup ◽  
Torsion Element ◽  
Vector Valued

For the modular variety attached to an arithmetic subgroup of an indefinite unitary group of signature $(1,n+1)$, with $n\geqslant 1$, we study Heegner divisors in the local Picard group over a boundary component of a compactification. For this purpose, we introduce local Borcherds products. We obtain a precise criterion for local Heegner divisors to be torsion elements in the Picard group, and further, as an application, we show that the obstructions to a local Heegner divisor being a torsion element can be described by certain spaces of vector-valued elliptic cusp forms, transforming under a Weil representation.

scholarly journals Generators for modules of vector-valued Picard modular forms

2013 ◽  
Vol 212 ◽  
pp. 19-57 ◽  
Author(s):  
Fabien Cléry ◽  
Gerard Van Der Geer
Keyword(s):  
Modular Forms ◽  
Unitary Group ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Picard Modular Forms ◽  
Vector Valued

AbstractWe construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2, 1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.

scholarly journals Generators for modules of vector-valued Picard modular forms

2013 ◽  
Vol 212 ◽  
pp. 19-57
Author(s):  
Fabien Cléry ◽  
Gerard Van Der Geer
Keyword(s):  
Modular Forms ◽  
Unitary Group ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Picard Modular Forms ◽  
Vector Valued

AbstractWe construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2, 1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.

scholarly journals UNITARY GROUPS OVER LOCAL RINGS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350093 ◽  
Author(s):  
J. CRUICKSHANK ◽  
A. HERMAN ◽  
R. QUINLAN ◽  
F. SZECHTMAN
Keyword(s):  
Local Ring ◽  
Unitary Group ◽  
Quadratic Extension ◽  
Unitary Groups ◽  
Weil Representation ◽  
Local Rings ◽  
Hermitian Forms ◽  
Finite Local Ring

We study hermitian forms and unitary groups defined over a local ring, not necessarily commutative, equipped with an involution. When the ring is finite we obtain formulae for the order of the unitary groups as well as their point stabilizers, and use these to compute the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring.

A TRACE FORMULA FOR HECKE OPERATORS ON VECTOR-VALUED MODULAR FORMS

2016 ◽  
Vol 59 (1) ◽  
pp. 143-165
Author(s):  
NICOLE RAULF ◽  
OLIVER STEIN
Keyword(s):  
Trace Formula ◽  
Modular Forms ◽  
Hecke Operators ◽  
Weil Representation ◽  
Cusp Forms ◽  
Dimension Formula ◽  
Integral Weight ◽  
Vector Valued

AbstractWe present a ready to compute trace formula for Hecke operators on vector-valued modular forms of integral weight for SL2(ℤ) transforming under the Weil representation. As a corollary, we obtain a ready to compute dimension formula for the corresponding space of vector-valued cusp forms, which is more general than the dimension formulae previously published in the vector-valued setting.

scholarly journals An Integral Representation of Standard AutomorphicLFunctions for Unitary Groups

2007 ◽  
Vol 2007 ◽  
pp. 1-22
Author(s):  
Yujun Qin
Keyword(s):  
Integral Representation ◽  
Eisenstein Series ◽  
Number Field ◽  
Unitary Group ◽  
Theta Series ◽  
Automorphic Representation ◽  
Unitary Groups ◽  
Cusp Forms ◽  
Cuspidal Automorphic Representation ◽  
Type Integral

LetFbe a number field,Ga quasi-split unitary group of rankn. We show that given an irreducible cuspidal automorphic representationπofG(A), its (partial)LfunctionLS(s,π,σ)can be represented by a Rankin-Selberg-type integral involving cusp forms ofπ, Eisenstein series, and theta series.

scholarly journals -ADIC EISENSTEIN SERIES AND -FUNCTIONS OF CERTAIN CUSP FORMS ON DEFINITE UNITARY GROUPS

2014 ◽  
Vol 15 (3) ◽  
pp. 471-510 ◽  
Author(s):  
Ellen Eischen ◽  
Xin Wan
Keyword(s):  
Positive Integer ◽  
Eisenstein Series ◽  
Constant Term ◽  
Unitary Groups ◽  
Cusp Forms ◽  
Imaginary Field

We construct$p$-adic families of Klingen–Eisenstein series and$L$-functions for cusp forms (not necessarily ordinary) unramified at an odd prime$p$on definite unitary groups of signature$(r,0)$(for any positive integer$r$) for a quadratic imaginary field${\mathcal{K}}$split at$p$. When$r=2$, we show that the constant term of the Klingen–Eisenstein family is divisible by a certain$p$-adic$L$-function.

scholarly journals Fast Constructive Recognition of Black-Box Unitary Groups

2003 ◽  
Vol 6 ◽  
pp. 162-197 ◽  
Author(s):  
Peter A. Brooksbank
Keyword(s):  
Homomorphic Image ◽  
Polynomial Time ◽  
Unitary Group ◽  
Black Box ◽  
Unitary Groups ◽  
Discrete Logarithms ◽  
Cyclic Groups

AbstractIn this paper, the author presents a new algorithm to recognise, constructively, when a given black-box group is a homomorphic image of the unitary group SU(d, q) for known d and q. The algorithm runs in polynomial time, assuming the existence of oracles for handling SL(2, q) subgroups, and for computing discrete logarithms in cyclic groups of order q ± 1.

scholarly journals Commuting elements in central products of special unitary groups

2012 ◽  
Vol 56 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Alejandro Adem ◽  
F. R. Cohen ◽  
José Manuel Gómez
Keyword(s):  
Prime Number ◽  
Moduli Space ◽  
Unitary Group ◽  
Unitary Groups ◽  
Connected Components ◽  
Flat Connections ◽  
Isomorphism Classes ◽  
Central Product ◽  
Central Products ◽  
Path Connected

AbstractWe study the space of commuting elements in the central product Gm,p of m copies of the special unitary group SU(p), where p is a prime number. In particular, a computation for the number of path-connected components of these spaces is given and the geometry of the moduli space Rep(ℤn, Gm,p) of isomorphism classes of flat connections on principal Gm,p-bundles over the n-torus is completely described for all values of n, m and p.

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