L Functions associated to galois representations

We review how to construct an L-function from a representation \(\rho\colon Gal(\bar{K}/K) \to GL(V)\)

Assume that the representation factors through some \(Gal(\bar{K}/K)\to Gal(L/K)\to GL(V)\),

Assume this representation factors through some \(Gal(L/K)\). Let \(\mathfrak{q}\) be a prime of \(L\) lying over \(\mathfrak{p}\) a prime of \(K\). Recall that we have the subgroup of \(Gal(L/K)\), \(G_{\mathfrak{q}}=\{\sigma \in G \colon \sigma \mathfrak{q}= \mathfrak{q}\}\) called the decomposition group. One can then prove we have a map \(G_{\mathfrak{q}}\to Gal(\mathcal{O}_L/\mathfrak{q} / \mathcal{O}_K/\mathfrak{p})\) with kernel exactly given by \(I_{\mathfrak{q}}= \{\sigma \in G_{\mathfrak{q}} \colon \sigma x = x \pmod(\mathfrak{q}) \forall x \in \mathcal{O}_K \}\). The galois group \(Gal(\mathcal{O}_L/\mathfrak{q} / \mathcal{O}_K/\mathfrak{p})\) is cyclic and generated by an element we call \(Frob_{\mathfrak{q}}\). We take a representative of this element in \(G_{\mathfrak{q}}\) which we shall use the same notation for.

Now we have enough to define \(L(s,\rho)=\prod_{\mathfrak{p}\subset \mathcal{O}_K}det(1-N(\mathfrak{q})^{-s}\rho(Frob_{\mathfrak{q}}) | V^{I_{\mathfrak{q}}} )^{-1}\) where \(\mathfrak{q}\) is an arbitrary prime above \(\mathfrak{p}\).

We will prove that this definition is well defined up to choice of lift, then choice of prime \(\mathfrak{q}\). First note that if \(I_{\mathfrak{p}}\neq \{1\}\) then for two representatives \(g\) and \(gh\) with \(h\in I_{\mathfrak{p}}\) then \(gh\dot v =g\dot v\) since \(v\in V^{I_{\mathfrak{p}}}\) so the action of any two representatives of \(Frob_{\mathfrak{p}}\) are equal.

Last we must check our definition is well defined for primes \(\mathfrak{q}\) over \(\mathfrak{p}\). The idea is that since all primes above \(\mathfrak{p}\) are conjugate, then we have that the intertia groups should be conjugate, thus our lifts can be taken to be conjugate which means that they shall have the same characteristic polynomial.

So we have a well defined L function which converges for \(Re(s)>1\). I will note because it came up in my research last week that you would get the same L function if instead of taking your representation you instead took \(\chi_{\rho}(g)=Tr(\rho(g))\), the character of the representation and built the L function for this representation.

Artins Conjecture: If \(\rho\) is an irreducible representation then \(L(s,\rho)\) is an entire function. This is known if \(dim \rho =1\) and in some other cases. The dimension 1 case was apparently known to Artin and Hecke, but I am confused on the connection between this result and tate’s thesis.