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Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 3: contribution of the elliptic part

Yuhao Cheng

Abstract

We continue to work on \emph{Beyond Endoscopy} for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification at $S = \{\infty, q_1, \dots, q_r\}$ (where $2 \in S$), generalizing the final step of Altuğ's work in the unramified setting. We derive an explicit asymptotic formula for the elliptic part when summing over $n<X$ with arbitrary smooth test functions at places in $S$ for the standard representation. As a consequence, we obtain the desired limit for the simple trace formula which only occurs in the ramified case. Moreover, we prove an asymptotic formula for the traces of Hecke operators on cusp forms with arbitrary level and weight $>2$, directly generalizing Altuğ's final result. Our approach differs entirely from Altuğ's: We apply a second Poisson summation with respect to the determinant, obtaining a formula on the Hitchin-Steinberg base $\mathfrak{g}/\!/ \mathsf{G}$. By changing variables from $(T, N)$ to $(T, Δ)$ on $\mathfrak{g}/\!/ \mathsf{G}$, we perform analysis in the new coordinates.

Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 3: contribution of the elliptic part

Abstract

We continue to work on \emph{Beyond Endoscopy} for over with ramification at (where ), generalizing the final step of Altuğ's work in the unramified setting. We derive an explicit asymptotic formula for the elliptic part when summing over with arbitrary smooth test functions at places in for the standard representation. As a consequence, we obtain the desired limit for the simple trace formula which only occurs in the ramified case. Moreover, we prove an asymptotic formula for the traces of Hecke operators on cusp forms with arbitrary level and weight , directly generalizing Altuğ's final result. Our approach differs entirely from Altuğ's: We apply a second Poisson summation with respect to the determinant, obtaining a formula on the Hitchin-Steinberg base . By changing variables from to on , we perform analysis in the new coordinates.

Paper Structure

This paper contains 37 sections, 80 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. A new way to detect functority
  3. The main results
  4. An overview of the proof
  5. Notations
  6. Preliminaries
  7. The elliptic part of the trace formula and measure normalizations
  8. The modified $p$-adic norm
  9. Singularities of the orbital integrals
  10. Nonarchimedean case
  11. Archimedean case
  12. Global notations
  13. The functions $F$ and $V$
  14. The generalized Kloosterman sum
  15. Norm with respect to the Mellin transform
  16. ...and 22 more sections

Key Result

Theorem 1.1

Suppose that $\varrho<1/6$ is a bound towards the Ramanujan conjecture for $\mathsf{GL}_2$ over $\mathbb{Q}$, namely for any $\varepsilon>0$. Then for any $\varepsilon>0$, we have where $\mathfrak{A}$ is defined in thm:contributenontempered, $\mathfrak{B},\mathfrak{C},\mathfrak{D}$ are defined in thm:contributealpha0, $\upgamma_S$ is defined in eq:defgammas. The implied constant only depends on

Theorems & Definitions (83)

  • Theorem 1.1: Contribution of the elliptic part
  • Theorem 1.2: Contribution of degree $1$ term of the elliptic part
  • Remark 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Corollary 2.7
  • Lemma 2.8
  • ...and 73 more