# Tag Info

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 ...
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### 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 ...

### 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 ...
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### 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 ...
• 244

### 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. ...
• 9,500
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### 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 ...
• 31.5k

### 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 ...
• 31.5k
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### 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 ...
• 31.5k
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### 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" ...
• 25k
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### 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 ...
• 4,730
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### 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 ...
• 29.5k
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### 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 ...
• 31.5k
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### 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 ...
• 17k
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### 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 ...
• 25k
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• 2,361

### 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 ...
• 116k

### 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 ...
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### 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 ...
• 5,234
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### 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 ...
• 31.5k
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• 116k
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### 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 $L$-...
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### 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 ...