Zeta and L-functions have been classically associated, first to schemes of finite type over Z, and then to the l-adic cohomology of smooth projective varieties over a global field. In the latter definition, independence from l and actual existence remain partly conjectural in characteristic 0. In these lectures, I will explain how to associate unconditionally an L-function to objects in triangulated categories of motives. For the motive of a smooth projective variety this definition differs from the one above in general, but only up to a finite number of Euler factors. This might be useful to tackle the Beilinson conjectures.

We prove the geometric Breuil--M\'ezard conjecture for two-dimensional semi-stable representations of $G_{\mathbf{Q}_{p^2}}$ with parallel Hodge--Tate weights $(0,r)$, where $r$ is odd and lies in the Fontaine--Laffaille range. Our strategy is to prove the numerical Breuil--M\'ezard conjecture at a maximally non-split mod-$p$ representation on each irreducible component of the special fiber of the Emerton--Gee stack. Building on previous work on strongly divisible modules, we obtain explicit descriptions of the relevant semi-stable deformation rings at these points and use them to establish the numerical Breuil--M\'ezard conjecture. We then explain how these local results lead to the geometric Breuil--M\'ezard conjecture in this setting. This is joint work with Seongjae Han and Hyeonmin Kim.

Zeta and L-functions have been classically associated, first to schemes of finite type over Z, and then to the l-adic cohomology of smooth projective varieties over a global field. In the latter definition, independence from l and actual existence remain partly conjectural in characteristic 0. In these lectures, I will explain how to associate unconditionally an L-function to objects in triangulated categories of motives. For the motive of a smooth projective variety this definition differs from the one above in general, but only up to a finite number of Euler factors. This might be useful to tackle the Beilinson conjectures.