Timeline for Galois Representations and Rational Points
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Jan 19, 2017 at 11:51 | comment | added | Kevin Buzzard | @JasonStarr: there is also Grothendieck's approach -- stick to the "anabelian" setting; then $\pi_1$ might determine the rational points purely group-theoretically. user00000: now there's a question! Let me show my ignorance by saying that of course one could ask this for e.g. compact complex manifolds first -- is it true then? Probably not -- you'll maybe just get something like a homotopy equivalence or something, rather than an isomorphism. It would not surprise me if one could then patch up an example in this more arithmetic setting. | |
| Jan 19, 2017 at 9:38 | comment | added | user00000 | What if we also have a morphism which induces the isomorphisms on cohomology? | |
| Jan 18, 2017 at 12:09 | comment | added | Jason Starr | Given your examples, maybe it makes sense to ask the question only for those projective varieties $X$ over a number field that have "large fundamental group" in the sense that for every normal subvariety $W$, the induced homomorphism $\pi_1^{alg}(W)\to \pi_1^{alg}(X)$ has infinite image. One could also ask for a comparison of rational points after a finite extension of the ground field. That would address the examples coming from torsors. | |
| Jan 18, 2017 at 8:20 | history | answered | Kevin Buzzard | CC BY-SA 3.0 |