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Results tagged with langlands-conjectures
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user 582
Higher reciprocity laws
26
votes
Accepted
Kapranov's analogies
There are notes from more recent (2000) lectures of Kapranov on the subject on my webpage.
Despite this, as far as I know, there isn't any convincing argument at this point that there should be a gen …
- 24k
10
votes
What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$...
To elaborate on Marty's comment, the simple moral one learns from both the Kazhdan-Lusztig classification of tamely ramified representations and the real local Langlands classification is that L-packe …
- 24k
4
votes
What is a tamely-ramified Weil-Deligne representation?
This is more of an extended comment. If I'm understanding the question correctly, I don't believe it's correct to say that tamely ramified Galois representations with unipotent monodromy correspond pr …
- 24k
26
votes
What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$...
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 Berkel …
- 24k
5
votes
Why is Langlands functoriality usually related with period integral in a third group?
I'm not close to familiar enough with the references you cite or examples you ask about to address them, but here's a picture coming out of Sakellaridis and Venkatesh [SV]. Let us label a period not b …
- 24k
27
votes
New Geometric Methods in Number Theory and Automorphic Forms
To complement Joel's wonderful and (as far as I understand) very much on point answer, let me quote from the proposal for the parallel program on Geometric Representation Theory, which touches on seve …
- 24k
3
votes
Accepted
Categorical-geometric Langlands for tori
In case $G=T$ is a torus and $G^\vee=T^\vee$ is the dual torus, the geometric Langlands conjecture — or “categorical geometric class field theory” (for a smooth projective curve $C$ over $\mathbb C$) …
- 24k
13
votes
Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh
The (fairly poetic and ill-formed) idea in this story is that the Kapustin-Witten story and the Langlands program are about the SAME four-dimensional TQFTs, but evaluated on different "manifolds" - i. …
- 24k
15
votes
How can I see the relation between shtukas and the Langlands conjecture?
I was hoping someone arithmetically qualified would take this on, but here are some comments from a geometer. One nice perspective I learned from Wei Zhang's ICM address - namely, over function fields …
- 24k
19
votes
Accepted
Implications and consequences of the recent proof of the geometric Langlands conjecture
To avoid writing a long essay I'll be very telegraphic.
Here are some of the many open problems that are reasonably "next" in the area. I'll use [GLC] to refer to the recent papers proving the unramif …
- 24k