Quadratic base change and the analytic continuation of the Asai L-function: A new trace formula approach

@article{Herman2010QuadraticBC,
  title={Quadratic base change and the analytic continuation of the Asai L-function: A new trace formula approach},
  author={P. Edward Herman},
  journal={American Journal of Mathematics},
  year={2010},
  volume={138},
  pages={1669 - 1729}
}
  • P. E. Herman
  • Published 23 August 2010
  • Mathematics
  • American Journal of Mathematics
Abstract:Using Langlands's beyond endoscopy idea, we study the Asai $L$-function associated to a real quadratic field $\Bbb{K}/\Bbb{Q}$. We prove that the Asai $L$-function associated to a cuspidal automorphic representation over $\Bbb{K}$ has analytic continuation to the complex plane with at most a simple pole at $s=1$. We then show if the $L$-function has a pole then the representation is a base change from $\Bbb{Q}$. While this result is known using integral representations from the work of… 
View PDF on arXiv
12 Citations
Background Citations
2

12 Citations

The functional equation and beyond endoscopy
In his paper “Beyond endoscopy,” Langlands tries to understand functoriality via poles of L-functions. This paper further investigates the analytic continuation of an L-function associated to a GL2…
BEYOND ENDOSCOPY: ITS CONNECTION TO L-FUNCTIONS AND THE FUNDAMENTAL LEMMA
One tool a number theorist often uses is an L-function. L-functions arise from many arithmetic objects of interestsuch as elliptic curves, Galois groups, and Dirichlet characters. The L-functions…
Rankin–Selberg L-functions and “beyond endoscopy”
Let f and g be two holomorphic cuspidal Hecke eigenforms on the full modular group $$ \text {SL}_{2}({\mathbb {Z}}). $$ We show that the Rankin–Selberg L-function $$L(s, f \times g)$$ has no pole at…
BEYOND ENDOSCOPY FOR THE RELATIVE TRACE FORMULA II: GLOBAL THEORY
  • Y. Sakellaridis
  • Mathematics
    Journal of the Institute of Mathematics of Jussieu
  • 2017
For the group $G=\operatorname{PGL}_{2}$ we perform a comparison between two relative trace formulas: on the one hand, the relative trace formula of Jacquet for the quotient $T\backslash G/T$ , where…
A nonabelian trace formula
AbstractLet E/F be an everywhere unramified extension of number fields with Gal(E/F) simple and nonabelian. In a recent paper, the first named author suggested an approach to nonsolvable base change…
The base change L-function for modular forms and Beyond Endoscopy
Beyond endoscopy via the trace formula: 1. Poisson summation and isolation of special representations
With analytic applications in mind, in particular beyond endoscopy, we initiate the study of the elliptic part of the trace formula. Incorporating the approximate functional equation into the…
On the Fourier Transform of Regularized Bessel Functions on Complex Numbers and Beyond Endoscopy Over Number Fields
  • Zhi Qi
  • Mathematics
    International Mathematics Research Notices
  • 2019
In this article, we prove certain Weber–Schafheitlin-type integral formulae for Bessel functions over complex numbers. A special case is a formula for the Fourier transform of regularized Bessel…
Hankel transform, Langlands functoriality and functional equation of automorphic L-functions
This is a survey on recent works of Langlands’s work on functoriality conjectures and related works including the works of Braverman and Kazhdan on the functional equation of automorphic L…
Isolating Rankin-Selberg lifts
Let F be a number field and let π be a cuspidal unitary automorphic representation of GLmn(AF ) where m and n are integers greater than one. We propose a conjecturally necessary condition for π to be…
...
...

References

SHOWING 1-10 OF 29 REFERENCES
The functional equation and beyond endoscopy
In his paper “Beyond endoscopy,” Langlands tries to understand functoriality via poles of L-functions. This paper further investigates the analytic continuation of an L-function associated to a GL2…
Formule des Traces et Fonctorialit\'e: le D\'ebut d'un Programme
We outline an approach to proving functoriality of automorphic representations using trace formula. More specifically, we construct a family of integral operators on the space of automorphic forms…
A relative trace formula proof of the Petersson trace formula
The Petersson trace formula relates spectral data coming from cusp forms to Kloosterman sums and Bessel functions. It was discovered in 1932 [Pe] long before Selberg’s trace formula and can be…
Kuznetsov's Trace Formula and the Hecke Eigenvalues of Maass Forms
Introduction Preliminaries Bi-$K_\infty$-invariant functions on $\operatorname{GL}_2(\mathbf{R})$ Maass cusp forms Eisenstein series The kernel of $R(f)$ A Fourier trace formula for…
BASE CHANGE FOR GL(2)
R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, ?) attached to a cuspdidal automorphic…
Contributions to automorphic forms, geometry, and number theory
In Contributions to Automorphic Forms, Geometry, and Number Theory, Haruzo Hida, Dinakar Ramakrishnan, and Freydoon Shahidi bring together a distinguished group of experts to explore automorphic…
Beyond endoscopy for the Rankin-Selberg L-function
Modular forms associated to real quadratic fields
The purpose of this paper is to construct modular forms, both for SL27Z (and certain of its congruence subgroups) and for the Hilbert modular group of a real quadratic field. In w 1 we fix a real…
Beyond Endoscopy and special forms on GL(2)
Abstract We carry out (with technical modifications) some cases of a procedure proposed by R. Langlands in Beyond Endoscopy. This gives a new proof of the classification of “dihedral forms” on GL(2),…
Distinguished representations, base change, and reducibility for unitary groups
We show the equality of the local Asai L-functions defined via the Rankin-Selberg method and the Langlands-Shahidi method for a square-integrable representation of GL n (E). As a consequence, we…
...
...