Geometry and Topology without figures

In this post, I present explicit computations with elliptic curves of form $y^2=x^3+D$ (similar trick leads to curves $y^2 = x^3 + Dx$; the common point is they have complex multiplication). This was my TA for Ariane Mezard. at the Summer School on Galois Representations and Reciprocity last summer. The talk is based on the master thesis of Matteo Tamiozzo.  Reminder on zeta functions For a prime $p$ and let $f(x,y,z) \in \mathbb{F}_p[x,y,z]$ be a homogeneous polynomial so that $$C = V(f) =  \left \{[x:y:z] \in \mathbb{P}^2_{\mathbb{F}_p} \mid f(x,y,z) = 0 \right \}$$ is a smooth, projective curve. We define $$N_m = N_m(f) = \left \{P \ \text{has coordinates in} \ \mathbb{F}_{p^m} \right \}$$ to be the cardinality of solutions defined over the $\mathbb{F}_{p^m}$. We package these numbers $N_1,N_2,...,N_m,...$ into a zeta function $$Z_C(t) = \exp \left(\sum_{m=1}^{\infty} \frac{N_m}{m}t^m \right).$$ The celebrated result due to Weil and the school of Grothendieck gives us the (...

This serves as a starting point for my self-study of geometric Langlands. Currently, I am interested in geometric Satake equivalence, which establishes an equivalence between the category of representation of the Langland dual group (of a given reductive group) and the category of perverse sheaves on affine Grassmannians. Affine grassmannians are interesting objects, and appear naturally when studying stack of principal bundles. Let us now study them, and the story, of reductive groups, is often with $\mathrm{GL}_n$.  Affine Grassmannians of general linear groups via lattices Let $R$ be a commutative ring and denoteby $R[[t]]$ and $R((t))$ the formal power series and Laurent series, respectively.  Definition . A lattice $L \subset R((t))^n$ is a finite locally free $R[[t]]$-submodule of $R((t))^n$ such that the canonical morphism $L \otimes_{R[[t]]} R((t)) \longrightarrow R((t))^n$ is an isomorphism. Note that locally free means that a vector bundle at the level of schemes...

Last week, I participated in a summer school on arithmetic geometry and Langlands program. Although this is not my field of interests, I always give some respect those who work in those fields. I also did gave a motivational TA talk about couting points on elliptic curves. To be honest, I do not expect to learn much from this school but I got enough inspiration to read arithmetic stuff, which I am going to write. Since I am an algebraic geometer, I'd like to present things in a geometric way. Let $p$ be a prime number and denote $\mathbb{Q}_p$ the $p$-adic local field. Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $G_K = \mathrm{Gal}(\overline{K}/K)$ the absolute Galois group. A $\ell$- adic representation of $G_K$ is a continuous morphism (of groups) $$G_K \longrightarrow \mathrm{GL}_n(V),$$ where $V$ is a finite-dimensional $\mathbb{Q}_{\ell}$-vector space with $\ell \neq p$ a prime. We denote by $K^{unr}$ the maximal unramified extension and $I_K = \mathrm{Gal}(\overline...

In this post, I'd like to give a rapid introduction to the theory of $l$-adic Fourier transform developed by Laumon-Deligne-... My goal is not to how can we apply $l$-adic Fourier transform to prove the Weil conjectures but rather to see why their definitions are natural in comparison with the classical theory. My feeling is that it is easier to present Fourier transforms on finite fields than on measurable spaces (which require a lot of work and details) and the $l$-adic one is formally adapted from the one for finite fields. Fourier analysis on finite abelian groups Given a finite abelian group $G$, written additively, what we want to do here is to define a space $L^2(G)$ similar to the Hilbert space $L^2(X)$ of square integrable functions $X \longrightarrow \mathbb{C}$ (modulo equal almost everywhere relation) for $X$ being a measurable space. Then it is possible to develop a Fourier transform on $L^2(G)$. The finiteness seems to be a technical condition that you can see to be u...