scholarly journals On the Ramanujan conjecture for automorphic forms over function fields I. Geometry

2020 ◽  
pp. 1
Author(s):  
Will Sawin ◽  
Nicolas Templier
Keyword(s):  
Automorphic Forms ◽  
Function Fields ◽  
Ramanujan Conjecture

scholarly journals Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

2009 ◽  
Vol 146 (1) ◽  
pp. 21-57 ◽  
Author(s):  
Harald Grobner
Keyword(s):  
Algebraic Group ◽  
Eisenstein Series ◽  
Symplectic Group ◽  
Automorphic Forms ◽  
General Theorem ◽  
Simple Algebraic Group ◽  
Ramanujan Conjecture ◽  
Eisenstein Cohomology ◽  
Derivatives Of

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

scholarly journals Automorphic Forms and Sums of Squares over Function Fields

1999 ◽  
Vol 79 (2) ◽  
pp. 301-329 ◽  
Author(s):  
Jeffrey Hoffstein ◽  
Kathy D. Merrill ◽  
Lynne H. Walling
Keyword(s):  
Automorphic Forms ◽  
Function Fields ◽  
Sums Of Squares

scholarly journals BEYOND ENDOSCOPY VIA THE TRACE FORMULA – III THE STANDARD REPRESENTATION

2018 ◽  
Vol 19 (4) ◽  
pp. 1349-1387 ◽  
Author(s):  
S. Ali Altuğ
Keyword(s):  
Number Theory ◽  
Asymptotic Expansion ◽  
Trace Formula ◽  
Automorphic Forms ◽  
Fourier Transforms ◽  
Cusp Forms ◽  
Selberg Trace Formula ◽  
Johns Hopkins University ◽  
Ramanujan Conjecture ◽  
Beyond Endoscopy

We finalize the analysis of the trace formula initiated in S. A. Altuğ [Beyond endoscopy via the trace formula-I: Poisson summation and isolation of special representations, Compos. Math.151(10) (2015), 1791–1820] and developed in S. A. Altuğ [Beyond endoscopy via the trace formula-II: asymptotic expansions of Fourier transforms and bounds toward the Ramanujan conjecture. Submitted, preprint, 2015, Available at: arXiv:1506.08911.pdf], and calculate the asymptotic expansion of the beyond endoscopic averages for the standard $L$-functions attached to weight $k\geqslant 3$ cusp forms on $\mathit{GL}(2)$ (cf. Theorem 1.1). This, in particular, constitutes the first example of beyond endoscopy executed via the Arthur–Selberg trace formula, as originally proposed in R. P. Langlands [Beyond endoscopy, in Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 611–698 (The Johns Hopkins University Press, Baltimore, MD, 2004), chapter 22]. As an application we also give a new proof of the analytic continuation of the $L$-function attached to Ramanujan’s $\unicode[STIX]{x1D6E5}$-function.

scholarly journals Toroidal automorphic forms for function fields

2012 ◽  
Vol 194 (2) ◽  
pp. 555-596 ◽  
Author(s):  
Oliver Lorscheid
Keyword(s):  
Automorphic Forms ◽  
Function Fields

scholarly journals Toroidal automorphic forms for some function fields

2009 ◽  
Vol 129 (6) ◽  
pp. 1456-1463 ◽  
Author(s):  
Gunther Cornelissen ◽  
Oliver Lorscheid
Keyword(s):  
Automorphic Forms ◽  
Function Fields
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