Timeline for The congruence subgroup property for mapping class groups and a conjecture of Grothendieck
Current License: CC BY-SA 4.0
13 events
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| Oct 1 at 14:57 | comment | added | HJRW | Thanks again for this. On re-reading the discussion, it sounds like contacting Hoshi might be my best bet! | |
| Sep 22 at 20:07 | comment | added | HJRW | Thanks -- I appreciate the references, although I don't see how any of them answer my question (apart from Voeodsky's original paper, in a very special case). To be clear, I am not interested in alternative proofs of the Grothendieck conjecture; I am interested in applications of the congruence subgroup property. | |
| Sep 22 at 19:38 | comment | added | coLaideronnette | The role of CSP was played by $p$-adic Hodge theory in Mochizuki’s proof, apart from that, CSP is still open anyway. So this has not been discussed much after Voevodsky, although his philosophy should hold in general. Yuichiro Hoshi (kurims.kyoto-u.ac.jp/~yuichiro/papers_e.html) has a series of papers on the topic (especially he treated the conjecture for the moduli space of hyperbolic curves of genus 1) but I’ve hardly read any of them. Maybe you can try to find something in his article or contact him directly. | |
| Sep 22 at 14:00 | comment | added | HJRW | Boggi's paper is about proving the CSP for hyperelliptic surfaces. He doesn't explicitly mention proving that any curves are anabelian. (I just checked to make sure.) If this is implicit in his work, could you give an explicit reference? | |
| Sep 21 at 15:35 | comment | added | coLaideronnette | @HJRW Boggi has proved certain results for higher $g$ and $n$ in his paper The congruence subgroup property for the hyperelliptic modular group: the open surface case and On the procongruence completion of the Teichmüller modular group, but he used the theory of complexes of profinite curves rather than the anabelian geometry. The anabelian proof of the same results is due to Hoshi–Tamagawa, but as far as I know there is no published paper on that. | |
| Sep 21 at 15:00 | comment | added | HJRW | So, in §4 of the linked paper, Voevodsky states a certain "combinatorial conjecture", that a certain morphism is injective. This is plausibly equivalent to the CSP, though I didn't check yet. (CSP is the statement that the natural map $\widehat{\operatorname{Mod}(S)}\to\operatorname{Out}(\widehat{\pi_1(S)})$ is injective.) He then reduces Grothendieck's conjecture to this statement in genus 0. But the CSP was already known in genus 0 when Voevodsky wrote his paper (due to Diaz--Donagi--Harbater)! What about higher genera? | |
| Sep 21 at 14:21 | comment | added | HJRW | Ah, many thanks for the reference. I’ll take a look at Voevodsky’s paper and asked if it answers my question. | |
| Sep 21 at 13:23 | history | edited | coLaideronnette | CC BY-SA 4.0 |
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| Sep 21 at 12:57 | comment | added | coLaideronnette | @HJRW This has been discussed in Voevodsky’s paper, I added the link. | |
| Sep 21 at 12:56 | history | edited | coLaideronnette | CC BY-SA 4.0 |
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| Sep 21 at 12:24 | comment | added | HJRW | Thanks for this. Please forgive me -- anabelian geometry is far from my area of expertise, so it will take me some work to understand your answer. At the moment, I don't see how this answers my question. As I understand it, you are combining the CSP with a result of Hoshi--Mochizuki to show that (A) the natural representation of the étale fundamental group of moduli space is injective. However, my question was how to use CSP to deduce that (B) hyperbolic curves are anabelian (proved by Mochizuki as you say). Does (A) imply (B)? If so, could you explain how? | |
| Sep 21 at 10:57 | history | edited | coLaideronnette | CC BY-SA 4.0 |
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| Sep 20 at 19:51 | history | answered | coLaideronnette | CC BY-SA 4.0 |