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An evident corollary arising from Newton-Thorne

Shenghao Hua

TL;DR

Problem: understand Langlands functoriality for automorphic lifts from tensor products and symmetric powers of GL2 forms. Approach: decompose compositions as isobaric sums using Schur-polynomial identities and leverage Newton-Thorne automorphy plus Friedlander–Iwaniec to derive L-function factorizations and coefficient asymptotics. Contributions: constructing automorphic lifts Π on GL_n for compositions of symmetric powers, tensor products, and parabolic inductions, with complete L-functions at level 1; establishing explicit L-factor decompositions and power-saving Fourier coefficient asymptotics. Significance: advances Langlands functoriality for higher-rank lifts and provides tools for modularity lifting and arithmetic information encoded in L-functions.

Abstract

We present a special class of examples of automorphic lifts of multiple tensor products of automorphic representations, motivated by combinatorial identities for Schur polynomials and a celebrated result of Newton and Thorne.

An evident corollary arising from Newton-Thorne

TL;DR

Problem: understand Langlands functoriality for automorphic lifts from tensor products and symmetric powers of GL2 forms. Approach: decompose compositions as isobaric sums using Schur-polynomial identities and leverage Newton-Thorne automorphy plus Friedlander–Iwaniec to derive L-function factorizations and coefficient asymptotics. Contributions: constructing automorphic lifts Π on GL_n for compositions of symmetric powers, tensor products, and parabolic inductions, with complete L-functions at level 1; establishing explicit L-factor decompositions and power-saving Fourier coefficient asymptotics. Significance: advances Langlands functoriality for higher-rank lifts and provides tools for modularity lifting and arithmetic information encoded in L-functions.

Abstract

We present a special class of examples of automorphic lifts of multiple tensor products of automorphic representations, motivated by combinatorial identities for Schur polynomials and a celebrated result of Newton and Thorne.

Paper Structure

This paper contains 3 sections, 2 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Proof

Key Result

Theorem 1.1

Let $\pi$ be the automorphic representation of $\operatorname{GL}_2(\mathbb{A}_\mathbb{Q})$ associated with a non-CM regular algebraic cuspidal Hecke eigenform $f$. Then $\operatorname{Sym}^n \pi$ exists and is cuspidal for all $n \geq 1$.

Theorems & Definitions (3)

  • Theorem 1.1: Newton--Thorne NewtonThorne2021aNewtonThorne2021b
  • Theorem 2.1
  • proof