# Structure groups and a special class of L-functions

Hello,

Let $X$ and $Y$ be two mathematical objects such that there exists a canonical embedding $f:X\hookrightarrow Y$. I define the structure group of $Y$ relatively to $X$, denoted $Str(Y/X)$, as the group of automorphisms of $Y$ such that their restriction to $X$ is the identity map, with composition as group law. For example, whenever $K$ is a subfield of a field $L$, $Str(L/K)$ is simply the Galois group $Gal(L/K)$.

Let's now consider the set $\mathcal{A}$ of automorphic L-functions, and the Selberg class $\mathcal{S}$. Whenever $\mathcal{F}$ is a family of L-functions such that the following properties P1 to P3 simultaneously hold, I say that $\mathcal{F}$ is a French family of L-functions:
P1: $\mathcal{F}\subset \mathcal{A}\cap\mathcal{S}$
P2: the constant function which maps a complex number to $1$ belongs to $\mathcal{F}$
P3: whenever $F,G\in\mathcal{F}$, $F.G\in\mathcal{F}$

Let's denote $\tilde{\mathcal{S}}$ the maximal (under inclusion) French family of L-functions, and $\tilde{\mathcal{S}}_{\mathbb{R}}$ the sub-French family of L-functions of $\tilde{\mathcal{S}}$ such that $\forall F\in\tilde{\mathcal{S}}_{\mathbb{R}}$, $F(s)\in\mathbb{R}$ whenever $s$ is a real number such that $F(s)$ is well-defined. There exists a canonical embedding $g:\tilde{\mathcal{S}}_{\mathbb{R}}\hookrightarrow\tilde{\mathcal{S}}$, so that $Str(\tilde{\mathcal{S}}/\tilde{\mathcal{S}}_{\mathbb{R}})$ is well defined.

Let's now consider, given a non trivial French family of L-functions $\mathcal{F}$, the set $Z_{\mathcal{D}}(\mathcal{F})$ defined as the set of elements $w$ of $\mathcal{D}$ such that there exists $F\in\mathcal{F}$ such that $F(w)=0$, and let denote $CS$ (for Critical Strip) the set of complex numbers $z$ such that $0<\Re(z)<1$.

My question is: can it be proven (maybe from Weierstrass factorization theorem and the set of axioms defining the Selberg class) that $Str(\tilde{\mathcal{S}}/\tilde{\mathcal{S}}_{\mathbb{R}})$ is isomorphic to $Str(Z_{CS}(\tilde{\mathcal{S}})/Z_{CS}(\tilde{\mathcal{S}}_{\mathbb{R}}))$, where automorphisms of $\tilde{\mathcal{S}}$ are defined as in Automorphisms of the Selberg class without assuming Selberg's orthonormality conjecture (since the condition P1 above entails that elements of $\tilde{\mathcal{S}}$ can be factored in a product of primitive elements in a unique fashion), and automorphisms of $Z_{CS}(\tilde{\mathcal{S}})$ are defined as affine isometries that preserve both $CS$ and $Z_{CS}(\tilde{\mathcal{S}})$?

An automorphism of $Y$ doesn't necessarily restrict to an automorphism of $X$. –  Qiaochu Yuan Nov 28 '12 at 3:12
Thanks for your comment. But if I only consider a special kind of automorphisms of $Y$, namely those which restrict to identity on $X$, I guess $Str(Y/X)$ is well defined (as it contains at least the identity map on $Y$), isn't it? To say it differently, let's say that $\phi\in Str(Y/X)\Longrightarrow\phi$ is an automorphism of $Y$ which restricts to $Id_{X}:X\to X, \ \ x\mapsto x$. –  Sylvain JULIEN Nov 28 '12 at 17:38
It appears that I can't type braces in LaTeX, I don't know why. I will replace them by left and right angles in the following to give a better definition of $Z_{\mathcal{D}}(\mathcal{F})$: $Z_{\mathcal{D}}(\mathcal{F}):=\cup_{F\in\mathcal{F}}\langle Z_{\mathcal{D}}(F)\rangle$, where $Z_{\mathcal{D}}(F)$ is $\langle s\in\mathcal{D}, \ \ F(s)=0\rangle$. –  Sylvain JULIEN Nov 28 '12 at 18:11