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
[Math] L-functions and algebraic geometry
ag.algebraic-geometryl-functionsnt.number-theorysoft-question
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Now that our paper Geometrization of the local Langlands correspondence with Fargues is finally out (ooufff!!), it may be worth giving an update to Ben-Zvi's answer above. In brief: we give a formulation of Local Langlands over a $p$-adic field $F$ so that it is finally
- an actual conjecture, in the sense that it asks for properties of a given construction, not for a construction;
- of a form as in geometric Langlands, in particular about an equivalence of categories, not merely a bijection of irreducibles.
First, I should say that in the notation of the OP, we construct a canonical map $\Pi(G)\to \Phi(G)$, and prove some properties about it. However, we are not able to say anything yet about its fibres (not even finiteness).
Moreover, we give a formulation of local Langlands as an equivalence of categories, and (essentially) construct a functor in one direction that one expects to realize the equivalence. In particular, this nails down what the local Langlands correspondence should be, it "merely" remains to establish all the desired properties of it.
Let me briefly state the main result here. Let $\mathrm{Bun}_G$ be the stack of $G$-bundles on the Fargues--Fontaine curve. We define an ($\infty$-)category $\mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$ of $\ell$-adic sheaves on $\mathrm{Bun}_G$. The stack $\mathrm{Bun}_G$ is stratified into countably many strata enumerated by $b\in B(G)$, and on each stratum, the category $\mathcal D(\mathrm{Bun}_G^b,\overline{\mathbb Q}_\ell)$ is the derived ($\infty$-)category of smooth representations of the group $G_b(F)$. In particular, for $b=1$, one gets smooth representations of $G(F)$.
Moreover, there is an Artin stack $Z^1(W_F,\hat{G})/\hat{G}$ of $L$-parameters over $\overline{\mathbb Q}_\ell$.
Our main result is the construction of the "spectral action":
There is a canonical action of the $\infty$-category of perfect complexes on $Z^1(W_F,\hat{G})/\hat{G}$ on $\mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$.
The main conjecture is basically that this makes $\mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)^\omega$ a "free module of rank $1$ over $\mathrm{Perf}(Z^1(W_F,\hat{G})/\hat{G})$", at least if $G$ is quasisplit (or more generally, has connected center).
More precisely, assume that $G$ is quasisplit and fix a Borel $B\subset G$ and a generic character $\psi$ of $U(F)$, where $U\subset B$ is the unipotent radical, giving the Whittaker representation $c\text-\mathrm{Ind}_{U(F)}^{G(F)}\psi$, thus a sheaf on $[\ast/G(F)]$, which is the open substack of $\mathrm{Bun}_G$ of geometrically fibrewise trivial $G$-bundles; extending by $0$ thus gives a sheaf $\mathcal W_\psi\in \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$, called the Whittaker sheaf.
Conjecture. The functor $$ \mathrm{Perf}(Z^1(W_F,\hat{G})/\hat{G})\to \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$$ given by acting on $\mathcal W_\psi$ is fully faithful, and extends to an equivalence $$\mathcal D^{b,\mathrm{qc}}_{\mathrm{coh}}(Z^1(W_F,\hat{G})/\hat{G})\cong \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)^{\omega}.$$
Here the superscript $\mathrm{qc}$ means quasicompact support, and $\omega$ means compact objects. As $Z^1(W_F,\hat{G})$ is not smooth (merely a local complete intersection), there is a difference between perfect complexes and $\mathcal D^b_{\mathrm{coh}}$, and there is still a minor ambiguity about how to extend from perfect complexes to all complexes of coherent sheaves. Generically over the stack of $L$-parameters, there is however no difference.
It takes a little bit of unraveling to see how this implies more classical forms of the correspondence, like the expected internal parametrization of $L$-packets; in the case of elliptic $L$-parameters, everything is very clean, see Section X.2 of our paper.
(There are related conjectures and results by Ben-Zvi--Chen--Helm--Nadler, Hellmann and Zhu; see also the work of Genestier--Lafforgue in the function field case. And this work is heavily inspired by previous work in geometric Langlands, notably the conjectures of Arinkin--Gaitsgory, and the work of Nadler--Yun and Gaitsgory--Kazhdan--Rozenblyum--Varshavsky on spectral actions.)
PS: It may be worth pointing out that this conjecture is, at least a priori, of a quite different nature than Vogan's conjecture, mentioned in the other answers, which is based on perverse sheaves on the stack of $L$-parameters; here, we use coherent sheaves.
An algebraic analog of Chern-Weil theory (explicitly taking symmetric polynomials of curvature) is given by the Atiyah class. Given a vector bundle $E$ on a smooth variety we can consider the short exact sequence $$ 0\to End(E) \to A(E) \to T_X\to 0$$ where $T_X$ is the tangent sheaf and $A(E)$ is the "Atiyah algebroid" --- differential operators of order at most one acting on sections of $E$, whose symbol is a scalar first order diffop (hence the map to the tangent sheaf). A (holomorphic or algebraic) connection is precisely a splitting of this sequence, and a flat connection is a Lie algebra splitting. Now algebraically such splittings will often not exist (having a holomorphic connection forces your characteristic classes to have type $(p,0)$ rather than the $(p,p)$ you want..) but nonetheless we can define the extension class, which is the Atiyah class $$a_E\in H^1(X, End(E)\otimes \Omega^1_X).$$ This is the analog of the curvature form in the Riemannian world -- we now can take symmetric polynomials in the $End(E)$ factor to get the characteristic classes of $E$ in $H^p(X,\Omega_X^p)$ as desired.
This answer and Mariano's agree of course in the sense that Atiyah classes can be interpreted via Hochschild and cyclic (co)homology and generalized to arbitrary coherent sheaves (or complexes) on varieties (or stacks) (let me stick to characteristic zero to be safe). Namely the Atiyah class of the tangent sheaf can be used to define a Lie algebra structure (or more precisely $L_\infty$) on the shifted tangent sheaf $T_X[-1]$, and Hochschild cohomology is its enveloping algebra. This Lie algebra acts as endomorphisms of any coherent sheaf (which is another way to say Hochschild cohomology is endomorphisms of the identity functor on the derived category), and one can take characters for these modules, recovering the characteristic classes defined concretely above.
(In fact the notion of characters is insanely general... for example an object of any category - with reasonable finiteness - defines a class (or "Chern character") in the Hochschild homology of that category, which is cyclic and so descends to cyclic homology. An example of this is the category of representations of a finite group, whose HH is class functions, recovering usual characters, or coherent sheaves on a variety, recovering usual Chern character. or one can go more general.)
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Best Answer
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.