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



L-functions and algebraic geometry
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](http://en.wikipedia.org/wiki/Langlands_dual) 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. 


