33 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 ...
Peter Scholze's 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 ...
Somerville Scholar's user avatar
24 votes
Accepted

A question on a paper of K. Murty

There is a flaw in Murty's argument. Once one corrects this flaw, the bound \eqref{1} weakens to $\ll x(\log\log x)^3/(\log x)^2$. Fortunately, Thorner and Zaman - A Chebotarev variant of the Brun–...
2734364041's user avatar
  • 5,059
22 votes
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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 ...
Jack Sprat's 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 ...
David Loeffler's 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 ...
David Loeffler's user avatar
16 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 ...
Myshkin's user avatar
  • 17.5k
15 votes
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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 ...
Joël's user avatar
  • 25.7k
14 votes
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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 $L$-...
Matt E's user avatar
  • 306
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 ...
Will Sawin's user avatar
  • 137k
13 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 ...
Peter Humphries's user avatar
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 ...
Stanley Yao Xiao's user avatar
13 votes

What is the smallest group not known to be a Galois group over $\mathbb{Q}$?

Doing an internet search I found the paper Groups of small order as Galois groups over $\mathbb{Q}$ by Jack Sonn, from 1989. Theorem 1 of that paper asserts that every group of order less than 672 is ...
Pace Nielsen's user avatar
  • 18.2k
12 votes
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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 ...
Joël's user avatar
  • 25.7k
12 votes
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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 ...
Jeremy Rouse's user avatar
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 $\...
Will Sawin's user avatar
  • 137k
11 votes

Effective weight-monodromy conjecture

You can bound the filtration length (assuming the WM conj) using the weight spectral sequence of Rapoport--Zink. This is a sp seq converging to $H^*(X_{\overline{k}})$; and if the WM conj holds, then ...
David Loeffler's user avatar
10 votes
Accepted

Artin reciprocity via Shimura varieties

If $K$ is 'everything 1 mod N' for some N, then the canonical model of $\mathbf{Q}^\times_+ \backslash \mathbf{A}^\times_{\mathrm{f}} / K$ is exactly $\mu_N / \mathbf{Q}$, the scheme of $N$-th roots ...
David Loeffler's user avatar
10 votes
Accepted

How does the cohomology of the Lubin-Tate/Drinfeld tower fit into categorical p-adic local Langlands?

Briefly (I will elaborate below): One expects that their fully faithful functor from (roughly) $p$-adic representations of $G(\mathbb Q_p)$ to (roughly) coherent sheaves on the Emerton--Gee stack ...
Peter Scholze's user avatar
10 votes
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Understanding absolute Galois group from its representations

The slogan "number theorists aim to understand $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$" is one that gets used a lot, but it's perhaps a tiny little bit misleading. ...
David Loeffler's user avatar
9 votes
Accepted

Selmer Group versus Selmer Variety

You should read the Bloch--Kato paper in the Grothendieck Festschrift. This was, I believe, the first paper to consider Selmer groups of Galois representations defined by local conditions coming from ...
David Loeffler's user avatar
9 votes
Accepted

Applications of anabelian geometry to Galois representations?

The standard theory of Galois representations is concerned with the actions of absolute Galois groups of number fields over abelian groups (in particular, vector spaces). Anabelian geometry is about ...
Joël's user avatar
  • 25.7k
9 votes
Accepted

Galois representation associated to CM-newforms

Let me abbreviate $\rho_{\lambda,f}$ as $\rho$, and $\psi_f$ as $\psi$. By definition, $L(s,\rho)=L(s,f)=L(s,\psi)$. The equality of the Euler factors of $L(s,\rho)$ and $L(s,\psi)$ at the split ...
GH from MO's user avatar
  • 99.2k
9 votes
Accepted

$B_{\mathrm{dR}}=B_{\mathrm{cris}}+{B_{\mathrm{dR}}^+}$?

Much more is true: the subring $B_{\mathrm{cris}}^{\varphi = 1}$ surjects onto $B_{\mathrm{dR}} / B_{\mathrm{dR}}^+$, so there is an exact sequence $$ 0 \to \mathbf{Q}_p \to B_{\mathrm{cris}}^{\varphi ...
David Loeffler's user avatar
9 votes

In practice, how explicitly can we describe a Galois representation?

(Too long for a comment, but maybe not what you are looking for.) I think the Galois representations that you are interested in are those coming from geometry with coefficients in a $p$-adic field. ...
Chris Wuthrich's user avatar
8 votes
Accepted

Weight filtration on certain Galois representations

No, life is not so easy I'm afraid. For instance, the group $\mathrm{Ext}^1_{G_{\mathbf{Q}}}(\mathbf{Z}_\ell, \mathbf{Z}_\ell(n)) = H^1(\mathbf{Q}, \mathbf{Z}_\ell(n))$ has positive rank for all odd ...
David Loeffler's user avatar
8 votes
Accepted

Frobenius at ramified primes

Are you looking for the following sort of answer? $a_\ell(E)=1$ if $E$ has split multiplicative reduction, $a_\ell(E)=-1$ if $E$ has non-split multiplicative reduction, and $a_\ell(E)=0$ if $E$ has ...
Joe Silverman's user avatar
8 votes

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

This is likely a very hard problem once $n \geq 7$. Such $E$ are parametrized by a twist of the modular curve ${\rm X}(n)$; this curve is rational for $n \leq 5$ (so for those $n$ one can use the ...
Noam D. Elkies's user avatar
8 votes

Bhargava's work on the BSD conjecture

Bhargava and Shankar have conjectured that the average size of the $n$-Selmer group $S_n(E)$ is the sum of the divisors of $n$. They proved this for $n \leq 5$. If you assume Equidistribution of ...
Ari Shnidman's user avatar
  • 2,481

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