# L-functions and algebraic geometry

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

• Did you have a look at this? mathoverflow.net/questions/119326/… – Tim Dokchitser Jan 5 '14 at 10:33
• yes, i had but how does it help in algebraic geometric problems – Koushik Jan 5 '14 at 12:38
• There are lots of articles and books and such on the "general philosophy" of $L$-functions from various points of view. You might want to ask something a little more pointed. For example, are you perhaps asking about what algebreo-geometric information can be gleaned from $L$-functions? The Weil conjectures you mention, for example, say that $L$-functions encode something about counting points on varieties. – Ramsey Jan 5 '14 at 15:43
• yes that's i am asking – Koushik Jan 5 '14 at 15:56

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.

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.