This paper is devoted to explaining what symmetric power \(L\)-functions attached to cusp forms on \(GL(2)\) are and how they can be applied to some important problems in number theory. A main result is that if \(\pi\) and \(\pi'\) are two cusp forms on \(GL(2)\), an \(L\)-function attached to \(\text{Ad}^3 (\pi) \times \pi'\) extends to a meromorphic function on \(C\) satisfying a standard functional equation. Here \(\text{Ad}^3 (\pi)i\) is the adjoint cube of \(\pi\). This result gives us further evidence for the existence of symmetric cube of a cusp form on \(GL(2)\).
Another main result is that the fourth symmetric power \(L\)-function attached to an arbitrary non-monomial cusp form \(GL(2)\) over any number field with non-trivial central character is holomorphic and non-zero on the half plane \(\text{Re} (s)\geq 1\).
As an appendix the paper contains a letter of January 24, 1992 from Jean-Pierre Serre to the author in which results on asymptotic distribution of Hecke eigenvalues of modular cusp forms are proved. For example, Serre shows that there are infinitely many \(p\)’s with positive upper density such that \(|\tau(p)/ p^{11/2} |< 0.8165\), where \(\tau(p)\) is the \(p\)th Fourier coefficient of the Ramanujan cusp form of weight 12. Serre’s proof is based on properties of the fifth symmetric power \(L\)-functions.
For the entire collection see [Zbl 0788.00052].