# Galois classes of L-functions

Around one month ago, I posted on math.stackexchange a draft I wrote in which I define the notion of Galois class of L-functions: see http://math.stackexchange.com/questions/280876/definition-of-a-peculiar-quotient-group-of-isometries

Edit: As Marc Palm asked for it, I copy and paste this draft here:

"Main ideas towards a proof of GRH

Definition 1

Let $A$ be be a subclass of the Selberg class containing 1, closed under products, and such that every element of $A$ can be factored in a unique fashion in a product of primitive elements of $\mathcal{S}$, these primitive elements belonging to $A$. Such a subclass of $\mathcal{S}$ will be called a Galois class of L-functions.

Definition 2

Let $A$ be a Galois class of L-functions. $T$ is an automorphism of $A$ iff the following properties simultaneously hold true:

1) $T$ is a bijective map from A to itself
2) $T$ maps a primitive element of $A$ to a primitive element of $A$
3) for all $F$ in $A$, the degree of $F$ and the degree of $T(F)$ are the same
4) for all $F$, $G$ in $A$, $T(F.G)=T(F).T(G)$

Definition 3

$\tilde{\phi}:F\mapsto\phi\circ F\circ \phi^{-1}$ is said to be an isometric automorphism of $A$ if $\tilde{\phi}$ is an automorphism of $A$ and $\phi$ is an isometry of the complex plane preserving any rectangle centered in 1/2.

Proposition 1

$\tilde{\phi}(F)=F$ implies $\phi(0)=0$.

Definition 4

A group of equivalence classes of isometries is said to be nice iff each equivalence class contains exactly one representant $\phi$ such that $\phi(0)=0$.

Let $F\in A$ and let $G_F$ be the group of all equivalence classes of isometries preserving the set of non trivial zeros of $F$, where the equivalence relation $R_F$ is defined by : $f R_F g$ iff for every non-trivial zero $s$ of $F$, $f(s)=g(s)$.

Let $G′_F$ be the group of isometric automorphisms of $A$ preserving $F$.

Proposition 2

$G_F$ and $G′_F$ are isomorphic iff GF is nice.

Definition 5

Let $A$ and $B$ two Galois classes of L-functions such that $A$ is a sub Galois class of L-functions of $B$. Let's define the L-type (Isometric) Structure group of $B$ over $A$ as the group of (isometric) automorphisms of $B$ preserving $A$ pointwise. The L-type structure group of $B$ over $A$ will be denoted as $LStr(B/A)$ and the L-type Isometric Structure Group of $B$ over $A$ $LIStr(B/A)$.

Definition 6

Let $GZ(A)$ be the set $\bigcup_{F\in A}\{Z(F):=\{s,F(s)=0, 0<\Re(s)<1\}\}$. The group of all isometries preserving $GZ(A)$ will be denoted as $PG_A$. Now let's consider the relation $R'_A$ defined by $f R'_A g,$, where $f$, $g$ are elements of $PG_A$, iff for all $F$, $H$ in $A$, $f R_F g$ iff $f R_H g$. The quotient group $PG_A/R'_A$ will be denoted as $SG_A$. Let's now consider the group of elements of $SG_B$ preserving $GZ(A)$ pointwise. This last group will be called "Z-type Isometric Structure Goup of $B$ over $A$" and denoted as ZIStr(B/A).

Conjecture 1

Let $A$ be a Galois class of L-functions, $F$ an element of $A$. $G_F$ is nice iff for all $H$ in $A$, $G_H$ is nice.

Conjecture 2 (Main Conjecture)

Let $A$ and $B$ be two Galois classes of L-functions such that $A$ is a sub-Galois class of L-functions of $B$. Then $LIStr(B/A)$ and $ZIStr(B/A)$ are isomorphic.

Idea of proof: prove that $LIStr(B/A)$ and $ZIStr(B/A)$ are both isomorphic to $Gal(F_B/F_A)$, where $F_B$ is the field generated by the union of all images of $R−{1}$ by the elements of $B$, and $F_A$ is defined in a similar way. If $B$ is the maximal Galois class of L-functions and $A=\{\zeta^{n},n\geq 0\}$, one can expect to have a proof of the Riemann Hypothesis."

My question is: can one hope to establish the isomorphy of the groups $ZIStr(B/A)$ and $LIStr(B/A)$ through Hadamard's factorization theorem (and maybe some other results), or must one necessarily consider the fields $F_A$ and $F_B$?
I somehow think that one can rephrase RH for an L-function $G$ as: $ZIStr(T/\langle G\rangle)$ is isomorphic to $Gal(\mathbb{C}/F_{\langle G\rangle})$, where $T$ is the maximal Galois class of L-functions and $\langle G\rangle$ is the Galois class of L-functions generated by $G$. Is this true? If so, one could try to prove the two following statements: 1) $LIStr(T/\langle G\rangle)$ is isomorphic to $Gal(\mathbb{C}/F_{\langle G\rangle})$ 2) $LIStr(T/\langle G\rangle)$ is isomorphic to $ZIStr(T/\langle G\rangle)$. – Sylvain JULIEN Feb 17 '13 at 20:23
If RH is false for $G$, then $ZIStr(T/\langle G \rangle)$ contains two distinct elements (the identity map and the map $s\mapsto \overline{1-s}$) if $F_{\langle G\rangle}=\mathbb{C}$, plus the complex conjugation and the map $s\mapsto 1-s$ if $F_{\langle G\rangle}=\mathbb{R}$. So if $ZIStr(T/\langle G\rangle)$ is isomorphic to $Gal(\mathbb{C}/F_{\langle G\rangle}$, then RH holds true for $G$. – Sylvain JULIEN Feb 18 '13 at 20:57
Moreover, if RH is true for $G$, then for any zero $s_{0}$ of $G$ $s_{0}=\overline{1-s_{0}}$, thus the identity map and the map $s\mapsto \overline{1-s}$ can't be distinguished considering their action on $s_{0}$, thus they correspond to a single element of $ZIStr(T/\langle G \rangle)$. Same reasoning with the complex conjugation and the map $s\mapsto 1-s$ if $F_{\langle G\rangle}=\mathbb{R}$. Hence, if RH is true for $G$, then $ZIStr(T/\langle G \rangle)$ is the trivial group if $F_{\langle G\rangle}=\mathbb{C}$, and is a group of order 2 if $F_{\langle G\rangle}=\mathbb{R}$. – Sylvain JULIEN Feb 18 '13 at 21:09