27 votes
Accepted

What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

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 ...
user avatar
27 votes

nonabelian reciprocity law

Will Sawin's answer is perfectly correct, but I wanted to add some further perspective. What Peter Scholze has proven is a profound generalization of (one half) of Class Field Theory. The particular ...
user avatar
25 votes

What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

A brief update: in his amazing talk yesterday at MSRI (available here), Laurent Fargues explained (building on work of Peter Scholze, see his phenomenal talk two weeks ago here and his historic ...
user avatar
22 votes
Accepted

The modularity theorem as a special case of the Bloch-Kato conjecture

That is not what the link says. To quote (emphasis mine): ... in which this conjecture was reduced to a special instance of the Bloch-Kato conjecture for the symmetric square motive of an elliptic ...
user avatar
19 votes

Status of Fontaine-Mazur conjecture

It is true if $V$ is of dimension 1, essentially by class field theory (as you are considering only representations of $G_{\mathbb Q}$). Otherwise, it is still largely open for the following reasons. ...
user avatar
  • 9,500
18 votes
Accepted

When is the image of a 2-dim l-adic representation associated to a modular form open

This is more subtle than it looks. I asked exactly the same question some years back (see here); but I'm not going to flag this question as duplicate, because the answer that was given to my question ...
user avatar
17 votes

Is every 3-dim self-dual Galois representation a symmetic square of 2-dim representation?

The correct statement is: any 3-dimensional selfdual Galois representation is isomorphic to a quadratic twist of the adjoint of some 2-dimensional representation. (The quadratic twist is really ...
user avatar
17 votes
Accepted

Classify 2-dim p-adic galois representations

The classification of 1-dimensional representations, i.e. characters $G_K \to E^\times$, is a bit more complicated than you imply in your question. Any such character lands in $O_E^\times$ by ...
user avatar
16 votes
Accepted

Local-global principle for split extensions of Galois representations

Definitely no. To see why, consider just the case where $W=\mathbb Q_\ell$ is the trivial representation (this is not really a loss of generality, because extensions of $W$ by $V$ are "the same thing" ...
user avatar
  • 25k
16 votes
Accepted

n-th root of unity in n-th division field of abelian variety?

Actually, this is an exercise in Serre's Lectures on Mordell--Weil Theorem: $K(A[n])$ always contains $\mu_n$ if $char(K)$ does not divide $n$ and $A$ is an abelian variety of positive dimension over ...
user avatar
  • 4,730
15 votes
Accepted

Modularity theorem for abelian varieties

Abelian varieties over the rationals are modular if and only if they are of "$GL_2$"-type, which is a notion introduced by Ribet who proved that this statement is a consequence of Serre's conjecture ...
user avatar
15 votes
Accepted

Computing an eigencuspform in $S_2(\Gamma_0(1776))$

I did the computation in Sage, and there is no such form $f$. There are 21 Galois orbits of newforms of level $\Gamma_0(1776)$ and trivial character, of which the largest has size 3, and none of the ...
user avatar
15 votes
Accepted

Reference book for Galois Representations

Galois representations have to come from somewhere. If you are not interested in learning about modular forms and automorphic forms at this point, the other best source of representations are ...
user avatar
  • 17k
15 votes
Accepted

Is there a "Langlands philosophy" reason for the fact that the L-function of the Jacobi theta function is (almost) the Riemann zeta function?

Good question. I don't understand fully what's happening, but here is an idea. Let $f=\sum a_n q^n$ be a modular form of weight $k+1/2$, nebentypus $\chi$. Assuming $k \geq 1$, the Shimura ...
user avatar
  • 25k
14 votes
Accepted

Galois representations for the curve $y^{2} = x^{3} - x$

I don't know that this is written down anywhere, but it's possible. It is known in general that $GL(T_{\ell}(E))$ is contained in the normalizer of $R_{\ell}^{\times}$, where $R_{\ell} = \mathbb{Z}[i] ...
user avatar
  • 17.7k
13 votes
Accepted

Eichler-Shimura congruence

There's a little more geometry here that should be accounted for in characteristic $p$. Namely, the curve $X_0(Np)_{\mathbb{F}_p}$ is reducible -- its two components are isomorphic to $X_0(N)_{\...
user avatar
  • 2,361
13 votes

nonabelian reciprocity law

Peter Scholze explains the same example in more detail in this talk https://youtu.be/cFdm0B9KLcQ?t=48m and gives even more detail here https://youtu.be/gI3Ta73yVuo?t=23m55s. In brief, the ...
user avatar
  • 116k
13 votes

Bhargava's work on the BSD conjecture

For a prime $p$ and an elliptic curve $E/\mathbb{Q}$, we have the exact sequence $$\displaystyle 0 \rightarrow E(\mathbb{Q})/p E(\mathbb{Q}) \rightarrow S_p(E) \rightarrow \text{Sha}_E[p]\rightarrow ...
user avatar
12 votes

What are the possible motivic Galois groups over $\mathbb Q$?

I realise that I'm a bit late to this party, but maybe it is still interesting to you: Under the Hodge conjecture the motivic Galois group coincides with the Mumford–Tate group. These have been ...
user avatar
  • 5,234
12 votes
Accepted

Example of a variety over a number field with non-semisimple Galois representation on $\ell$-adic cohomology

Here's an example, if I'm not mistaken. Let $E / K$ be an elliptic curve and $x \in E$ a non-torsion $K$-point. Then the image of the divisor $\{x\} - \{\infty\}$ under the etale cycle class map is a ...
user avatar
12 votes
Accepted

Explicit Chebotarev and Langlands - irreducibility of X^5-X-1 mod primes

The object (conjecturally) associated to an Artin representation by Langlands is not a classical modular form except in a very limited number of situations (odd two dimensional representations). Let $...
user avatar
12 votes

To what extent are modular parametrizations expected to generalize?

A natural generalization of the geometric modularity conjecture which is compatible with your formulation Do you expect some form of modularity to correspond to the existence of a map from some ...
user avatar
  • 9,500
12 votes
Accepted

A question on the Hecke L-function

Whatever your answer to my question in comment, the answer to the title question is yes. Take the case of the $L$-function attached to a modular form $f$, which we assume an eigenform for almost all ...
user avatar
  • 25k
12 votes
Accepted

Is every $GL_2(\mathbb{Z}/n\mathbb{Z})$-extension contained in some elliptic curve's torsion field?

I have two things to add to this discussion. $\bullet$ For $n = 2$ and $n = 3$, every Galois extension of $\mathbb{Q}$ with Galois group ${\rm GL}_{2}(\mathbb{Z}/n\mathbb{Z})$ does arise from an ...
user avatar
  • 17.7k
12 votes
Accepted

Etale cohomology of singular varieties?

Just to answer the last question, the key statement is that for a proper variety over $\mathbb F_q$, the number of $\mathbb F_{q^n}$ points is equal to the alternating sum of the traces of $\...
user avatar
  • 116k
12 votes
Accepted

Cohomology of Shimura varieties and coherent sheaves on the stack of Langlands parameters

The anticipated analogue is as follows: There is a map $f: \mathcal X \to \prod_{v \in S} \mathcal X_v,$ where $\mathcal X$ is the stack of $p$-adic representations of $G_{E,S}$ into ${}^LG$ (the $L$-...
user avatar
  • 286
11 votes
Accepted

Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?

It follows from results of Ribet in "On l-adic representations attached to modular forms" (Invent. Math. 28 (1975), 245–275) that the Mumford-Tate conjecture holds for the motives attached to modular ...
11 votes
Accepted

Abelian $\ell$-adic representations in $\widehat{\mathrm{SL}(2,\mathbb{Z})}$

What Grothendieck is doing to get all the Galois representations is: 1) Take a finite index subgroup of $\pi_1(\mathcal M_{1,1})$ that is stable by the $\operatorname{Gal}(\overline{\mathbb Q}|\...
user avatar
  • 116k
11 votes
Accepted

What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

I've seen Conjecture 1 and Conjecture 1' stated in the literature in many places. I don't believe I have seen Conjecture 1'' so stated. I'd also like to point out that (EDIT: a weaker version of) ...
user avatar
  • 17.7k
11 votes
Accepted

"Extra Euler factors" in one definition of the L-function of a twist of a modular form

Let me translate this into a problem purely about automorphic forms: Take a newform $f \in \mathcal{S}_k^{\ast}(q,\chi)$, and a primitive Dirichlet character $\psi$ modulo $q'$. Then there exists a ...
user avatar

Only top scored, non community-wiki answers of a minimum length are eligible