Symmetric power functoriality for holomorphic modular forms

Abstract

Let $f$ f be a cuspidal Hecke eigenform of level 1. We prove the automorphy of the symmetric power lifting $\operatorname{Sym}^{n} f$ Sym n f for every $n \geq 1$ n ≥ 1 .We establish the same result for a more general class of cuspidal Hecke eigenforms, including all those associated to semistable elliptic curves over $\mathbf{Q}$ Q .

Symmetric power functoriality for holomorphic modular forms, II

“…-If k ≥ 0, let (H k ) denote the hypothesis that the conclusion of the proposition holds when |sc(π )| ≤ k, and let (H k ) denote the hypothesis that the conclusion of the proposition holds when |sc(π )| ≤ k and π is seasoned with respect to some tuple (q, t, r). As remarked above, (H 0 ) follows from the results of [NT21]. It therefore suffices to prove the implications (H k ) ⇒ (H k+1 ) and (H k ) ⇒ (H k ).…”

“…We fix n, which we can assume to be ≥ 3. The proof of Theorem 3.1 will be roughly by induction on the cardinality of sc(π ), the set of primes p such that π p is supercuspidal; the case where sc(π ) is empty is exactly the main result of [NT21].…”

“…For a more detailed discussion of the context surrounding this problem, including known results by other authors, we refer the reader to the introduction of [NT21], of which this paper is a continuation. In that paper we studied the problem of symmetric power functoriality in the case that F = Q and π is regular algebraic (in which case π corresponds to a twist of a cuspidal Hecke eigenform f of weight k ≥ 2, cf.…”

“…Here we associate to any regular algebraic, cuspidal automorphic representation π of GL 2 (A Q ) the set sc(π ) of primes p such that π p is supercuspidal. Fixing n ≥ 1, we prove the existence of Sym n π by induction on the cardinality of the set sc(π ), the case |sc(π )| = 0 being exactly the main result of [NT21].…”

Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

“…Recently, Nice(π, r) was proved for all r ∈ N, all π ∈ Π * cus (l, q) and all non-zero prime ideals q if we restrict our case to elliptic modular forms (F = Q). Indeed, Sym r (π) is an irreducible C-algebraic cuspidal automorphic representation of GL r+1 (A Q ) by Newton and Thorne [34] which is applied to the case where the conductor of π is square-free. They also treated in [35] the case of general levels of elliptic modular forms.…”