L-functions and algebraic geometry

Question

Robert Langlands commented in a letter to Deligne that perhaps some of the deepest problems of algebraic geometry lie in L-functions. I want to understand the general philosophy and the connection between L-functions and algebraic geometry beyond the well-know Weil conjectures. L-functions encode something about counting points on varieties in weil conjectures. What similiar things are done by the other kinds ie. a general motivic L-function

Answer 1

In the general philosophy I don't know much about the connection between an $L$-function and algebro-geometric questions, but there is a big relation lying behind theory of automorphic $L$-function and $L$-functions arising from problems of Arithmetic geometry; namely Langlands functoriality which was started by Langlands to establish non-abelian class field theory and reciprocity. The conjecture can be described roughly as follows:

To every $L$-homomorphism $\phi:H^L\to G^L$ between to L-groups of quasi-split reductive groups H and G, there exists a natural lifting or transfer of automorphic representations of H to those of G.

Another conjecture: Let $G$ be a connected, quasi-split reductive group over a number field $F$. Let $\pi=\otimes'_{v \ places}\pi_v$ be a cuspidal automorphic representation of $G(\mathbb{A}_F)$ ($\mathbb{A}_F$ adele ring of $F$ and $\otimes'$ is restricted tensor product). Consider the canonical homomorphism $\xi_v:G(F_v)\to G(F)$ and for a finite dimensional representation $r$ of $G^L$ define $r_v=r\circ\xi_v$. Now the local Langlands $L$ function is defined by$$L_v(s, \pi_v,r_v)=\det(I-r_vc(\pi_v)q_v^{-s})^{-1},$$ where $q_v$ is the order of the residue field $\mathcal{O}_v(F)/\mathcal{p}_v$, $\mathcal{p}_v$ is the maximal ideal. Now if $S$ is a finite set (carefully chosen to avoid ramification) of places of $F$ Langlands conjecture says:

$$L_S(s,\pi,r)=\prod_{v\notin S}L_v(s,\pi_v,r_v)$$ has a meromorphic continuation to the whole complex plane and satisfies a standard functional equation.

Artin $L$-function is a motivic $L$-function you can think of. An Artin $L$-function $L(\rho,s)$ of some Galois representation $\rho$ is conjecturally (Artin conjecture) analytic over whole complex plane, for every non-trivial irreducible representation $\rho$. Langlands showed that, an automorphic representation of $GL_n(\mathbb{A})$ can be associated to every $n$-dimensional representation of a Galois group, also the associated automorphic representation will be cuspidal if the Galois representation is irreducible. Langlands attached automorphic L-functions to these automorphic representations, and conjectured (reciprocity conjecture) that every Artin $L$-function arising from a finite-dimensional representation of the Galois group of a number field is equal to one arising from an automorphic cuspidal representation.

Answer 2

The book "Introduction to the Langlands program" (edited by Bernstein and Gelbart, 2003) contains articles by Bump, Cogdell, Gaitsgory, Kowalski, Kudla and Shalit on various topics pertaining this very subject.