# 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} }

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…

## 12 Citations

The functional equation and beyond endoscopy

- Mathematics
- 2012

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

- Mathematics
- 2010

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”

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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

- MathematicsJournal 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

- Mathematics
- 2013

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…

Beyond endoscopy via the trace formula: 1. Poisson summation and isolation of special representations

- MathematicsCompositio Mathematica
- 2015

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

- MathematicsInternational 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

- Mathematics
- 2020

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

- Mathematics
- 2015

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…

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