scholarly journals Fourier–Jacobi expansion of automorphic forms on Sp(1,q) generating quaternionic discrete series

2006 ◽  
Vol 239 (2) ◽  
pp. 638-682 ◽  
Author(s):  
Hiro-aki Narita
Keyword(s):  
Automorphic Forms ◽  
Discrete Series ◽  
Jacobi Expansion

SOME VECTOR-VALUED SINGULAR AUTOMORPHIC FORMS ON U(2, 2) AND THEIR RESTRICTION TO Sp(1, 1)

2012 ◽  
Vol 23 (10) ◽  
pp. 1250104
Author(s):  
ATSUO YAMAUCHI ◽  
HIRO-AKI NARITA
Keyword(s):  
Hyperbolic Space ◽  
Symplectic Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Discrete Series ◽  
Theta Series ◽  
Explicit Construction ◽  
Discrete Series Representations ◽  
Vector Valued ◽  
Real Symplectic Group

In this paper we provide a construction of theta series on the real symplectic group of signature (1,1) or the 4-dimensional hyperbolic space. We obtain these by considering the restriction of some vector-valued singular theta series on the unitary group of signature (2,2) to this indefinite symplectic group. Our (vector-valued) theta series are proved to have algebraic Fourier coefficients, and lead to a new explicit construction of automorphic forms generating quaternionic discrete series representations and automorphic functions on the hyperbolic space.

Algebraic Constructions of Minimal Compactifications

2013 ◽  
Author(s):  
Kai-Wen Lan
Keyword(s):  
Large Class ◽  
Automorphic Forms ◽  
Algebraic Construction ◽  
Local Structures ◽  
Jacobi Expansion ◽  
Integral Bases ◽  
Toroidal Compactifications ◽  
Jacobi Expansions ◽  
Algebraic Constructions ◽  
Moduli Problems

This chapter first studies the automorphic forms that are defined as global sections of certain invertible sheaves on the toroidal compactifications. The local structures of toroidal compactifications lead naturally to the theory of Fourier–Jacobi expansions and the Fourier–Jacobi expansion principle. The chapter also obtains the algebraic construction of arithmetic minimal compactifications (of the coarse moduli associated with moduli problems), which are projective normal schemes defined over the same integral bases as the moduli problems are. As a by-product of codimension counting, we obtain Koecher's principle for arithmetic automorphic forms (of naive parallel weights). Furthermore, this chapter shows the projectivity of a large class of arithmetic toroidal compactifications by realizing them as normalizations of blowups of the corresponding minimal compactifications.

Hard Objects in George Oppen's Discrete Series

2018 ◽  
Vol 13 (4) ◽  
pp. 496-517
Author(s):  
Ned Hercock
Keyword(s):  
Twentieth Century ◽  
Ludwig Wittgenstein ◽  
Discrete Series ◽  
Human Experience ◽  
John Ruskin ◽  
Gerard Manley Hopkins ◽  
Marcel Duchamp ◽  
The World ◽  
Production History ◽  
History Of

This essay examines the objects in George Oppen's Discrete Series (1934). It considers their primary property to be their hardness – many of them have distinctively uniform and impenetrable surfaces. This hardness and uniformity is contrasted with 19th century organicism (Gerard Manley Hopkins and John Ruskin). Taking my cue from Kirsten Blythe Painter I show how in their work with hard objects these poems participate within a wider cultural and philosophical turn towards hardness in the early twentieth century (Marcel Duchamp, Adolf Loos, Ludwig Wittgenstein and others). I describe the thinking these poems do with regard to industrialization and to human experience of a resolutely object world – I argue that the presentation of these objects bears witness to the production history of the type of objects which in this era are becoming preponderant in parts of the world. Finally, I suggest that the objects’ impenetrability offers a kind of anti-aesthetic relief: perception without conception. If ‘philosophy recognizes the Concept in everything’ it is still possible, these poems show, to experience resistance to this imperious process of conceptualization. Within thinking objects (poems) these are objects which do not think.

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