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 \(Gal(\bar{K}/K)\to Gal(L/K)\to GL(V)\), then we can 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. 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

#We get conjugate inertia grops with the conjugate element giving an isomorphism, #which won’t change the characteristic polynomial.