# Does the strong multiplicity one theorem imply the isomorphy of these two automorphism groups?

Defining the notion of "Galois class of L-functions" as a set of automorphic L-functions belonging to the Selberg class closed under the usual product and the Rankin-Selberg convolution and containing the constant map equal to $$1$$ and the Riemann zeta function, say $$\mathcal{G}(F)$$ is the Galois class of L-functions generated by $$F$$ if it is the smallest Galois class of L-functions containing $$F$$.

Do(es) the so far proven multiplicity one theorem(s) imply that an element of a Galois class of L-functions is entirely determined by its set of Satake parameters at all but finitely many places? If so, is the group of automorphisms of $$\mathcal{G}(F)$$ for a given $$F$$ (hence bijective maps from $$\mathcal{G}(F)$$ to itself preserving the usual product and RS convolution and sending $$s\mapsto 1$$ to itself and $$\zeta$$ to itself) naturally isomorphic to the Galois group $$Gal(K_{F}/Q)$$ where $$K_{F}$$ is the compositum of the number fields $$K_{p}(F)$$ obtained by adjoining to $$Q$$ the Satake parameters of $$F$$ at $$p$$ for all but finitely many $$p$$?

Would this imply that the group $$Aut(\mathcal{M})$$ where $$\mathcal{M}$$ is the maximal Galois class of L-functions is naturally isomorphic to the absolute Galois group of the rationals?

• I assume that by "automorphic $L$-function" you mean "$L$-function of an automorphic form on $\mathrm{GL}_n$ over $\mathbb{Q}$ for some $n$". If $F$ is not a product of shifted Dirichlet $L$-functions, then I don't think it is known that $\mathcal{G}(F)$ exists. Assuming Langlands functoriality, $\mathcal{G}(F)$ exists of course, hence the multiplicity one theorem applies to its elements (as does to any principal automorphic $L$-function). Also, I don't think that $\mathcal{G}(F)$ has much to do with Galois groups of number fields (cf. my response to mathoverflow.net/questions/50581). – GH from MO Jul 11 at 22:52
• But aren't Satake parameters algebraic numbers? – Sylvain JULIEN Jul 12 at 5:03
• Also, if I'm not mistaken, for any pair $(k,m)$ of non-negative integers, the Rankin-Selberg convolution $\zeta^{k}\otimes\zeta^{m}$ exists and equals $\zeta^{km}$. So that there exists at least one Galois class of L-functions unconditionally, namely $\mathcal{G}(\zeta)$. – Sylvain JULIEN Jul 12 at 6:00
• First, Satake parameters are not algebraic numbers in general. This is what I told you 8.5 years ago. Second, if $F$ is a product of shifted Dirichlet $L$-functions (e.g. when $F=\zeta$), then of course $\mathcal{G}(F)$ exists, but otherwise not, because Rankin-Selberg $L$-functions are only known to be automorphic for small degrees (cf. known cases of Langlands functoriality). – GH from MO Jul 12 at 7:41
• Ok. Some Satake parameters are transcendental, so $Aut(\mathcal{M})$ can't be isomorphic to $G_{\mathbb{Q}}$. But don't the formal properties of the two operations and the fact that $\mathcal{M}$ is the maximal Galois class of L-functions imply that its automorphism group is isomorphic to the absolute Galois group of some field $\mathbb{K}$ such as $\mathbb{R}$? – Sylvain JULIEN Jul 14 at 10:02