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Invariant Representations of $\mathrm {GSp}(2)$ under Tensor Product with a Quadratic Character

$Invariant Representations of $\mathrm {GSp}(2)$ under Tensor Product with a Quadratic Character PDF$ Author: Ping-Shun Chan
Publisher: American Mathematical Soc.
ISBN: 0821848224
Category : Mathematics
Languages : en
Pages : 185

Book Description
"Volume 204, number 957 (first of 5 numbers)."

Invariant Representations of $\mathrm {GSp}(2)$ under Tensor Product with a Quadratic Character

Book Description
"Volume 204, number 957 (first of 5 numbers)."

Invariant Representations of GSp(2)

Author: Ping-Shun Chan
Publisher:
ISBN:
Category : Automorphic forms
Languages : en
Pages : 255

Book Description
Abstract: Let F be a number field or a p-adic field. We introduce in Chapter 2 of this work two reductive rank one F-groups, H1, H2, which are twisted endoscopic groups of GSp(2) with respect to a fixed quadratic character [epsilon] of the idèle class group of F if F is global, F[superscript X] if F is local. If F is global, Langlands functoriality predicts that there exists a canonical lifting of the automorphic representations of H1, H2 to those of GSp(2). In Chapter 4, we establish this lifting in terms of the Satake parameters which parametrize the automorphic representations. By means of this lifting we provide a classification of the discrete spectrum automorphic representations of GSp(2) which are invariant under tensor product with [epsilon]. The techniques through which we arrive at our results are inspired by those of Kazhdan's in [K]. In particular, they involve comparing the spectral sides of the trace formulas for the groups under consideration. We make use of the twisted extension of Arthur's trace formula, and Kottwitz-Shelstad's stabilization of the elliptic component of the geometric side of the twisted trace formula. If F is local, in Chapter 5 we provide a classification of the irreducible admissible representations of GSp(2, F) which are invariant under tensor product with the quadratic character [epsilon] of F[superscript X]. Here, our techniques are also directly inspired by [K]. More precisely, we use the global results from Chapter 4 to express the twisted characters of these invariant representations in terms of the characters of the admissible representations of H[subscript i](F) (i = 1, 2). These (twisted) character identities provide candidates for the liftings predicted by the local component of the conjectural Langlands functoriality. The proofs rely on Sally-Tadić's classification of the irreducible admissible representations of GSp(2, F), and Flicker's results on the lifting from PGSp(2) to PGL(4).

Automorphic Forms

Author: Bernhard Heim
Publisher: Springer
ISBN: 3319113526
Category : Mathematics
Languages : en
Pages : 250

Book Description
This edited volume presents a collection of carefully refereed articles covering the latest advances in Automorphic Forms and Number Theory, that were primarily developed from presentations given at the 2012 “International Conference on Automorphic Forms and Number Theory,” held in Muscat, Sultanate of Oman. The present volume includes original research as well as some surveys and outlines of research altogether providing a contemporary snapshot on the latest activities in the field and covering the topics of: Borcherds products Congruences and Codes Jacobi forms Siegel and Hermitian modular forms Special values of L-series Recently, the Sultanate of Oman became a member of the International Mathematical Society. In view of this development, the conference provided the platform for scientific exchange and collaboration between scientists of different countries from all over the world. In particular, an opportunity was established for a close exchange between scientists and students of Germany, Oman, and Japan. The conference was hosted by the Sultan Qaboos University and the German University of Technology in Oman.

Modular Forms and Related Topics in Number Theory

Author: B. Ramakrishnan
Publisher: Springer Nature
ISBN: 9811587191
Category : Mathematics
Languages : en
Pages : 240

Book Description
This book collects the papers presented at the Conference on Number Theory, held at the Kerala School of Mathematics, Kozhikode, Kerala, India, from December 10–14, 2018. The conference aimed at bringing the active number theorists and researchers in automorphic forms and allied areas to demonstrate their current research works. This book benefits young research scholars, postdoctoral fellows, and young faculty members working in these areas of research.

Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors

Author: Jan H. Bruinier
Publisher: Springer
ISBN: 3540458727
Category : Mathematics
Languages : en
Pages : 159

Book Description
Around 1994 R. Borcherds discovered a new type of meromorphic modular form on the orthogonal group $O(2,n)$. These "Borcherds products" have infinite product expansions analogous to the Dedekind eta-function. They arise as multiplicative liftings of elliptic modular forms on $(SL)_2(R)$. The fact that the zeros and poles of Borcherds products are explicitly given in terms of Heegner divisors makes them interesting for geometric and arithmetic applications. In the present text the Borcherds' construction is extended to Maass wave forms and is used to study the Chern classes of Heegner divisors. A converse theorem for the lifting is proved.