scholarly journals Distinction of the Steinberg representation III: the tamely ramified case

2017 ◽  
Vol 67 (4) ◽  
pp. 1521-1607 ◽  
Author(s):  
François Courtès
Keyword(s):  
Steinberg Representation

scholarly journals ON TENSORING WITH THE STEINBERG REPRESENTATION

2019 ◽  
Vol 25 (4) ◽  
pp. 981-1008 ◽  
Author(s):  
C. P. BENDEL ◽  
D. K. NAKANO ◽  
C. PILLEN ◽  
P. SOBAJE
Keyword(s):  
Steinberg Representation

scholarly journals Level-raising for Saito–Kurokawa forms

2009 ◽  
Vol 145 (4) ◽  
pp. 915-953
Author(s):  
Claus M. Sorensen
Keyword(s):  
Trace Formula ◽  
Modular Forms ◽  
Automorphic Forms ◽  
Automorphic Representation ◽  
Weil Representation ◽  
Local Component ◽  
Fundamental Lemma ◽  
Error Terms ◽  
Stable Trace Formula ◽  
Steinberg Representation

AbstractThis paper provides congruences between unstable and stable automorphic forms for the symplectic similitude group GSp(4). More precisely, we raise the level of certain CAP representations Π arising from classical modular forms. We first transfer Π to π on a suitable inner form G; this is achieved by θ-lifting. For π, we prove a precise level-raising result that is inspired by the work of Bellaiche and Clozel and which relies on computations of Schmidt. We thus obtain a $\tilde {\pi }$ congruent to π, with a local component that is irreducibly induced from an unramified twist of the Steinberg representation of the Klingen parabolic. To transfer $\tilde {\pi }$ back to GSp(4), we use Arthur’s stable trace formula. Since $\tilde {\pi }$ has a local component of the above type, all endoscopic error terms vanish. Indeed, by results due to Weissauer, we only need to show that such a component does not participate in the θ-correspondence with any GO(4); this is an exercise in using Kudla’s filtration of the Jacquet modules of the Weil representation. We therefore obtain a cuspidal automorphic representation $\tilde {\Pi }$ of GSp(4), congruent to Π, which is neither CAP nor endoscopic. It is crucial for our application that we can arrange for $\tilde {\Pi }$ to have vectors fixed by the non-special maximal compact subgroups at all primes dividing N. Since G is necessarily ramified at some prime r, we have to show a non-special analogue of the fundamental lemma at r. Finally, we give an application of our main result to the Bloch–Kato conjecture, assuming a conjecture of Skinner and Urban on the rank of the monodromy operators at the primes dividing N.

Spherical buildings and the character of the Steinberg representation

1980 ◽  
Vol 58 (3) ◽  
pp. 201-210 ◽  
Author(s):  
C. W. Curtis ◽  
G. I. Lehrer ◽  
J. Tits
Keyword(s):  
Steinberg Representation

scholarly journals A QUOTIENT OF THE LUBIN–TATE TOWER

2017 ◽  
Vol 5 ◽  
Author(s):  
JUDITH LUDWIG
Keyword(s):  
Borel Subgroup ◽  
Series Representation ◽  
Principal Series ◽  
Principal Series Representation ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Steinberg Representation

In this article we show that the quotient${\mathcal{M}}_{\infty }/B(\mathbb{Q}_{p})$of the Lubin–Tate space at infinite level${\mathcal{M}}_{\infty }$by the Borel subgroup of upper triangular matrices$B(\mathbb{Q}_{p})\subset \operatorname{GL}_{2}(\mathbb{Q}_{p})$exists as a perfectoid space. As an application we show that Scholze’s functor$H_{\acute{\text{e}}\text{t}}^{i}(\mathbb{P}_{\mathbb{C}_{p}}^{1},{\mathcal{F}}_{\unicode[STIX]{x1D70B}})$is concentrated in degree one whenever$\unicode[STIX]{x1D70B}$is an irreducible principal series representation or a twist of the Steinberg representation of$\operatorname{GL}_{2}(\mathbb{Q}_{p})$.

scholarly journals Distinction of the Steinberg Representation

2013 ◽  
Vol 2014 (11) ◽  
pp. 3140-3157 ◽  
Author(s):  
Paul Broussous ◽  
François Courtès
Keyword(s):  
Steinberg Representation
Sign in / Sign up

Export Citation Format

Share Document