Remarks on the symmetric powers of cusp forms on GL(2)

Abstract

In this paper we prove the following conditional result: Let F be a number field, and pi a cusp form on GL(2)/F which is not solvable polyhedral. Assume that all the symmetric powers sym^m(pi) are modular, i.e., define automorphic forms on GL(m+1)/F. If sym^6(pi) is cuspidal, then all the symmetric powers are cuspidal, for all m. Moreover, sym^6(pi) is Eisenteinian iff sym^5(pi) is an abelian twist of the functorial product of pi with the symmetric square of a cusp form pi' on GL(2)/F.

A sentence has been modified in the Introduction. It has nothing to do with the main result of the paper

References in corpus (1)

• Algebraic cycles on Hilbert modular fourfolds and poles of L-functions

Cited by in corpus (4)

• A conjecture on Whittaker-Fourier coefficients of cusp forms
• Cohomology of moduli spaces via a result of Chenevier and Lannes
• On the arithmetic of Shalika models and the critical values of $L$-functions for $\mathrm{GL}(2n)$
• Icosahedral Fibres of the Symmetric Cube and Algebraicity