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13 votes
3 answers
1k views

$\ell$- vs. $p$-adic and de Rham vs. Betti in the geometric Langlands correspondence

In Emerton–Gee–Hellmann’s IHÉS notes on the categorical $p$-adic local Langlands programme, one finds the following remark: The differences between the $\ell$-adic and $p$-adic settings are ...
1 vote
0 answers
98 views

Are there known effective bounds on the number of semisimple Galois representations?

In continuation to my question here, are there known effective bounds on the total number of semisimple $p$-adic Galois representations unramified outside a finite set of primes $S$, of dimension $d$, ...
7 votes
1 answer
613 views

Are all representations of the geometric étale fundamental group subquotients of representations of the arithmetic étale fundamental group?

Let $X$ be a variety over a field $k$. The étale fundamental group of $X$ fits into the exact sequence: $$1 \to \pi_1^{\text{geom}}(X) \to \pi_1^{\text{arith}}(X) \to \text{Gal}(\overline{k}/k) \to 1,$...
7 votes
1 answer
440 views

On a claim of Deligne about representations of Weil-Deligne groups

In Deligne's article "Les constantes des equations fonctionelles des fonctions L", we find the following claim: Proposition 8.9 (ibid.): Suppose $(V, \rho, N )$ and $(V', \rho', N')$ ...
15 votes
1 answer
910 views

A possible gap in Faltings note to prove the Tate conjecture for finitely generated field over $\mathbb{Q}$

Qeustion: Given a Lie algebra $\mathfrak{g}$ over $\mathbb{Q}_\ell$ with an ideal $\mathfrak{g}^O$ and a subalgebra $\mathfrak{h}$, such that $\mathfrak{g}=\mathfrak{g}^O+\mathfrak{h}$. Now given a ...
11 votes
0 answers
359 views

Representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$

Is there a nice reference for the finite dimensional (characteristic 0) representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$ and $\operatorname{PGL}_2(\mathbb Z/n\mathbb Z)$ for ...
3 votes
0 answers
174 views

On continuous seminorms on Fréchet-Stein algebras

Let $K$ be a discretely valued complete non-archimedean field and $U$ be a left Fréchet-Stein algebra as defined in Algebras of p-adic distributions and admissible representations, with a Fréchet-...
9 votes
1 answer
787 views

The cohomology of modular curves as a module over the Galois group

Consider the modular curve $\pi: X(N) \to X(1)$ where this map has Galois group $G = PSL_2(\mathbb Z/N\mathbb Z)$. In particular, $G$ acts on the singular cohomology $H^1(X(N),\mathbb Z)\otimes \...
5 votes
1 answer
435 views

Fourier coefficients of Siegel Eisenstein series

I am looking for reference about Fourier coefficients of Eisenstein series. Currently I am mainly interested Eisenstein series given by Siegel parabolic subgroup of $SP_{2n}$ and $U(n,n)$. Let's ...
2 votes
0 answers
549 views

Fontaine - Wintenberger field of norms and imperfect case

Let $K$ be a complete discrete valued field whose residue field $k_K$ has characteristic $p$ and has the property that $[k_K:k_K^p]=p^d$ for some $d$. Let $t_{\alpha}, 1 \leq \alpha \leq d$ be a set ...
2 votes
1 answer
881 views

How to prove the p-adic Galois representations atteched to the Tate module of an abelian variety is de Rham directly?

Recently I read a thesis p-adic Galois representations and elliptic curves. Using Tate's curve, the author proved the p-adic Galois representations atteched to the Tate module of an elliptic curve is ...
7 votes
2 answers
1k views

Classify 2-dim p-adic galois representations

Recently I have known how to classify 1-dim p adic Galois representations $\phi$. The p-adic Galois representations mean that a representation $G_K$ on a p-adic field $E$, where $K$ is also a p-adic ...
10 votes
1 answer
502 views

Arithmetic representation stability and Galois action

I am thinking about representation stability phenomena (as considered by Church, Farb, Ellenberg and others) in arithmetic settings where Galois action enters the picture, and am curious about their ...
2 votes
0 answers
165 views

Galois action on posets of number fields and $p$-adic fields

In the theory of group actions on posets one studies the action of a group $G$ on a poset $\mathcal{P}$ (via order-preserving permutations) partly through its derived action on the order complex of ...
6 votes
1 answer
397 views

A question on the Hecke L-function

For a Hecke L-function, if all of the local eigenvalues are roots of unity, is it an Artin L-function?
5 votes
2 answers
356 views

Is there a semisimple $\mathbf{Q}_\ell$-representation of $G_F$ ramified at an infinite set of places?

See http://math.uni.lu/~wiese/galois/Boeckle-Luxemburg-Notes.pdf, Theorem 1.4(a): Is there an example of a semisimple $\mathbf{Q}_\ell$-representation $V$ of $G_F$ ($F$ a global field) ramified at a ...
15 votes
3 answers
2k views

What is the intuition behind the definition of cuspidal representations?

Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the $\mathbb{G}(...
3 votes
1 answer
667 views

What is an automorphic representation of CM type ?

In a recent paper of BL-Gee-Geraghty: "Sato-Tate for Hilbert modular forms" (JAMS 2011), a theorem is proved for regular algebrai cuspidal automorphic representation of $GL_2(\mathbb A_F)$ with $F$ a ...
14 votes
2 answers
2k views

"Purely local" proof of local Langlands

As from this website http://math.uchicago.edu/~lxiao/workshop_site/ My question is: What does it mean by "purely local"? Also, I heard about this phrase "purely local" in other problems as well, ...
20 votes
3 answers
2k views

Geometric construction of depth zero local Langlands correspondence

Dear community, In light of the recent work of DeBacker/Reeder on the depth zero local Langlands correspondence, I was wondering if there is an attempt to "geometrize" the depth zero local Langlands ...
10 votes
0 answers
687 views

An analogue of Deligne-Lusztig theory for positive depth representations?

Deligne-Lusztig theory is an important tool in understanding the depth zero representations of $p$-adic groups. Is there an analogue of Deligne-Lusztig theory that helps in understanding positive ...
7 votes
1 answer
870 views

local to global Galois representation

Let $\rho_p : \mbox{Gal}(\overline{\mathbb{Q}}_p / {\mathbb{Q}_p}) \to \mbox{GL}_n(\mathbb{Q}_p)$ be a de Rham $p$-adic representation. Can one find a representation $\rho : \mbox{Gal}(\overline{\...
8 votes
1 answer
1k views

Geometric Intuition for Big Monodromy

In various contexts, I have come across results referred to as "big monodromy." A standard arithmetic example is the open image theorem for the image of Galois action on non-CM elliptic curves. A ...