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Quotients of $L$-functions: degrees $n$ and $n-2$

Ravi Raghunathan

Abstract

If $L(s,π)$ and $L(s,ρ)$ are the Dirichlet series attached to cuspidal automorphic representations $π$ and $ρ$ of ${\rm GL}_n({\mathbb A}_{\mathbb Q})$ and ${\rm GL}_{n-2}({\mathbb A}_{\mathbb Q})$ respectively, we show that $F_2(s)=L(s,π)/L(s,ρ)$ has infinitely many poles. We also establish analogous results for Artin $L$-functions and other $L$-functions not yet proven to be automorphic. Using the classification theorems of \cite{Ragh20} and \cite{BaRa20}, we show that cuspidal $L$-functions of ${\rm GL}_3({\mathbb A}_{\mathbb Q})$ are primitive in ${\mathfrak G}$, a monoid that contains both the Selberg class ${\mathcal{S}}$ and $L(s,σ)$ for all unitary cuspidal automorphic representations $σ$ of ${\rm GL}_n({\mathbb A}_{\mathbb Q})$.

Quotients of $L$-functions: degrees $n$ and $n-2$

Abstract

If and are the Dirichlet series attached to cuspidal automorphic representations and of and respectively, we show that has infinitely many poles. We also establish analogous results for Artin -functions and other -functions not yet proven to be automorphic. Using the classification theorems of \cite{Ragh20} and \cite{BaRa20}, we show that cuspidal -functions of are primitive in , a monoid that contains both the Selberg class and for all unitary cuspidal automorphic representations of .
Paper Structure (17 sections, 37 theorems, 98 equations)

This paper contains 17 sections, 37 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. The strategy of the proof and organisation of the paper
  3. Various axiomatically defined classes of Dirichlet series
  4. An analogue of Booker's theorem for the class $\mathfrak{G}$
  5. Local and global $L$-functions of automorphic representations
  6. Non-archimedean local factors
  7. Archimedean local factors
  8. The multiplicativity of $L$- and $\varepsilon$-factors
  9. The global theory
  10. A ${\rm GL}_2$-type functional equation for $F_2(s)$
  11. Eliminating the polynomial function from the functional equation of $F_2(s)$
  12. The epsilon factor associated to $F_2(s,\chi)$
  13. A ${\rm GL}_2$ converse theorem and the proof of Theorem \ref{['zerothm']}
  14. Applications to the Primitivity of cuspidal $L$-functions
  15. Comparing the zero sets of $L$-functions
  16. ...and 2 more sections

Key Result

Theorem 1.1

Suppose $\pi\in {\mathcal{A}}_n^{\circ}$ and $\rho\in\BOONDOX{T}_{n-2}$. The function $F_2(s)=L(s,\pi)/L(s,\rho)$ has infinitely many poles. Further, if $G_2(s)=L(s,\pi_{\infty})/L(s,\rho_{\infty})$ has a finite number of zeros, $F_2(s)$ has infinitely many poles in $0<\mathop{\mathrm{\rm{Re}}}\noli

Theorems & Definitions (71)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 3.1
  • Theorem 4.1
  • proof : Proof of Theorem \ref{['bookermodthm']}
  • Proposition 4.2
  • Corollary 4.3
  • ...and 61 more