It was mentioned in a lecture on Faltings’s proof of Mordell Conjecture that there’s some kind of correspondence between Galois representation (of cohomology, or some complex of constructible sheaves?) , but it hadn’t been explained explicitly. Also, I’m studying étale cohomology and Weil conjectures. The trace formula also seems suggest that correspondence: Frobenius elements form the Galois group of finite field, and the eigenvalues of them are the zeros and poles of Grothendieck’s L-function. But I can’t see the explicit ‘correspondence’(is there?) and I have no idea about the case of other base field.



L-functions and Galois representations: What’s the explicit relation?
The situation is:


One defines an $L$-function associated to a representation $V$ of the Galois group of a number field $K$ as $$L(V,s) = \prod\_{ \mathfrak p } \frac{ 1}{ \det( 1 -|\mathfrak p|^{-s} \operatorname{Frob}\_{\mathfrak p} , V^{I\_{\mathfrak p}} )}$$ with the product taken over the primes of $K$, where $V^{I\_{\mathfrak p}}$ are the inertia invariants of $p$ (which, for all reasonable Galois representations agree with $V$ away from finitely many primes).


So there is a correspondence between $L$-functions and Galois representations simply by definition. One can check that if two Galois representations over the rational numbers have the same $L$-function, then they have the same characteristic polynomial of Frobenius at each place, and if in addition they are semisimple, then they are isomorphic. So in some cases this is a one-to-one correspondence.


Furthermore one can define, associated to a scheme $X$ over $\mathcal O\_K$, the Hasse-Weil zeta function. This definition doesn't reference Galois representations at all, and just refers to the closed points of $X$. However, one can check using the Lefschetz fixed point formula (applied to the fiber of $X$ over each finite field) that the Hasse-Weil zeta function is


$$ \prod\_i L( H^i (X\_{\overline{K}}, \mathbb Q\_\ell), s)^{(-1)^i} $$ up to changing factors at finitely many primes (the primes where $X$ has singularities and the primes dividing $\ell$).


The zeta function appearing in the Weil conjectures is the Euler factor of the Hasse-Weil zeta function at a particular prime, and the proof of this is almost identical to the proof of the expression of the Weil zeta function in terms of cohomology in the proof of the Weil conjectures.


