How to construct the symmetric power function from a modular form?

Question

I want to understand how we construct from a modular form $f$ its symmetric power function $Sym^rf.$ I read that there is a particular representation that does this but I am not familiar with this notion so can you explain to me or give me a reference that helps me to understand clearly this application ? Best wishes, Khadija

Answer 1

The standard casual usage of this terminology is itself confusing. To be a little formal, for a modular/automorphic form $f$ on $GL_2$, the $L$-function (or at least the good-prime factors) looks like $L(s,f)=\prod_p 1/\det(1_2-p^{-s}A_p)$ where $A_p$ is a two-by-two matrix (or conjugacy class thereof). This gives the quadratic-in-$p^{-s}$ denominator we expect. The symmetric $r$-th power $L$-functions is directly described as a Dirichlet series with Euler product as $$ L(s,f,{\rm Sym}^r) \;=\; \prod_p {1\over \det(1_r-p^{-s}{\rm Sym}^r A_p)} $$ That is, the symmetric power is that of the two-by-two matrices giving local-at-$p$ information.

Yes, it is conjectured (Langlands functoriality) that any such multilinear algebra operations on the local data $A_p$ produces the standard $L$-function attached to a cuspform on $GL_n$ of the right size to have the (good-prime) $L$-factors be of the proper degree in $p^{-s}$. This is not known in many cases...

That is, on one hand, the symmetric power $L$-function is set up rather trivially and definitionally, via the symmetric power of the two-dimensional repn of $GL_2$, in which the local data $A_p$ sits, for every $p$. On the other hand, while we imagine/hope that there really is an automorphic form whose standard $L$-function is that "defined" one, we do not know that this is so. That is, we do not generally know that there is an automorphic representation producing that $L$-function.