Timeline for How can I see the relation between shtukas and the Langlands conjecture?
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| Apr 3, 2020 at 19:56 | comment | added | Will Sawin | @GTA To relate these two notions, justifying why locally symmetric spaces are the special fiber, you have to specialize at $\infty$, then turn the Galois representations into Hodge structures, then split the Hodge structures into 1-forms and conjugate 1-forms, then view the $1$-forms as generalized functions. This Hodge theory story doesn't work the same way as function fields, so there's no reason for the two kinds of spaces to have such a direct relationship. | |
| Apr 3, 2020 at 19:54 | comment | added | Will Sawin | @GTA From the point of view of Langlands, the key property of modular curves (for concreteness, but the same applies to other Shimura varieties) is that their cohomology splits as a sum over automorphic forms of two-dimensional Galois representations associated to them. On the other hand, the key property of locally symmetric spaces is that (generalized) functions on them are modular curves. | |
| Apr 3, 2020 at 19:52 | comment | added | Will Sawin | @GTA No, the fibers will all have the same dimension, which will almost never be the dimension of $\operatorname{Bun}_G$. The analogy might not be perfect in all respects - although maybe there is an explanation here. | |
| Apr 3, 2020 at 19:32 | comment | added | GTA | @WillSawin I am confused how this could happen in the context of shtukas. Can a fiber of the moduli of shtukas with some number of legs at a point of the base be just Bun_G? | |
| Apr 3, 2020 at 16:33 | comment | added | Will Sawin | @GTA Shimura varieties are not locally symmetric spaces. They are schemes over the integers whose fiber over the complex numbers has an underlying manifold which is a locally symmetric space. Shtukas with one leg at a point $p$ are the analogue of the reduction of the Shimura variety mod $p$, and Shtukas with one varying leg are the analogue of the Shimura variety as a scheme over $\mathbb Z$. For questions about automorphic forms which only depend on the manifold, not on the Shimura variety, shtukas with no legs can be used. | |
| Apr 3, 2020 at 4:08 | comment | added | GTA | Shimura varieties are locally symmetric spaces. How are the two analogies (Shimura variety = one leg, locally symmetric space = no leg) consistent with each other? | |
| Apr 3, 2020 at 1:21 | history | edited | David Ben-Zvi | CC BY-SA 4.0 |
Added section on physics perspective
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| Apr 2, 2020 at 21:08 | history | bounty ended | Mr. Palomar | ||
| Apr 2, 2020 at 19:36 | history | answered | David Ben-Zvi | CC BY-SA 4.0 |