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How to build motivic realizations?

The theory of motives is a grand program envisioned by Grothendieck to encapsulate, within a single framework, the essential features shared by various cohomology theories developed by his school for smooth projective varieties over a field $k$, which are nowadays called Weil cohomology theories. Typical examples include $\ell$-adic cohomology, algebraic de Rham cohomology, and Betti cohomology. The notion of pure motives was introduced by Grothendieck, along with the expectation that there should exist a universal Weil cohomology theory reproducing all known properties of the existing ones. A natural candidate for the category of pure motives is the category of Chow motives introduced by Grothendieck. However, this approach immediately leads to the notorious standard conjectures, which remain unproven to this day. It is also natural to imagine that one can define motives for smooth but possibly non-projective varieties, thereby obtaining the notion of mixed motives $\mathrm{MM}(k)$, a...
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Spherical Hecke algebra and classical Satake isomorphism

The Satake isomorphism is a way to identify the spherical Hecke algebra of a reductive group $G$ with the invariant part (under the action of the Weyl group) of cocharacters (also called one-parameter subgroups) and both are isomorphic to the Grothendieck ring of the category of the Langlands dual group $G^{\vee}$. This fits perfectly with the so-called Langlands duality philosophy, suggesting that algebraic objects associated with $G^{\vee}$ should be captured by topological objects associated with $G$. More concretely, let $F$ be a non-archimedian local field, and $O$ the ring of integers, and let $G$ be a split reductive group over $\mathcal{O}$. Let $T \subset G$ be a maximal torus and $X_{\ast}(T) = \operatorname{Hom}(\mathbb{G}_m,T)$ the group of cocharacters. The group ring $\mathbb{Z}[X_{\ast}(T)]$ is endowed with an action of the Weyl group $W$. Let $q$ be the cardinality of the residue field of $K$, the Satake isomorphism reads $$\mathbb{Z}_c[G(\mathcal{O}) \setminus G(K)/G(\...
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Drinfeld curves and geometric Mackey Formula

This blog serves as my preparation for my talk at Bloch's seminar workshop on Deligne-Lusztig varieties. I will present a talk about $\ell$-adic cohomology of Drinfeld curves, which are easiest examples of Degline-Lusztig varieties. To begin with, the story dates back to representation theory of finite group of Lie type (by a finite group of Lie type, I mean $G^F$ with $G$ a reductive group over a finite field and $F$ the Frobenious endomorphism). To say why these groups are important, it is the classification of finite simple groups saying that any finite simplegroup belongs to four classes: cyclic, alternating, $26$ sporadic groups, and the others are all of Lie type.  The central object of this blog is $SL_2(\mathbb{F}_q)$ with $q=p^r$ a prime power. It is a finite group of Lie type whose cardinality is $q(q-1)(q+1)$ (it's a fun exercise to show this in case you do not know). Not only $SL_2(\mathbb{F}_q)$, but the representation theory of any finite group over complex numbe...
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Concrete counting points on elliptic curves of the form $y^2 = x^3 + D$

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 (...
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Affine Grassmannians of reductive groups

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...
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