Timeline for p-adic period map in Lawrence and Venkatesh
Current License: CC BY-SA 4.0
9 events
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| Feb 13, 2023 at 16:07 | comment | added | Wojowu | @kindasorta I am not! I wanted to add some remarks about that in the answer, but I was not confident in their validity and they didn't seem necessary. My idea was that perhaps after showing that this set is locally contained in a vanishing locus, perhaps one can argue cohomologically (using essentially resolutions of Cousin's problems over $K_v$) that one can glue the functions defining respective loci into a single one. However it is not at all obvious that the functions I produce can be chosen "compatibly" (it is true a posteriori by finiteness, but that doesn't seem clear from the argument) | |
| Feb 13, 2023 at 16:02 | comment | added | kindasorta | Yeah, you are right. I guess there is no harm done thinking of the problem one residue disk at a time. Thanks for writing a detailed reply. BTW, while I understand YOUR argument, are you sure that this is precisely what they do in THEIR paper? | |
| Feb 13, 2023 at 15:58 | comment | added | Wojowu | @kindasorta In this case, unless $X(K_v)$ is empty, then properness should be equivalent to compactness. And while you are right that general hyperbolic curves are not always proper, it ought to be an assumption in your problem - non-proper hyperbolic curves can have infinitely many rational points over $K$, it is integral points over $O_K$ which they have only finitely many of. And anyway, considering $X$ proper is enough to see the statement is not true in general. | |
| Feb 13, 2023 at 15:07 | comment | added | kindasorta | I think that the property you were referring to was properness and not compactness as in the complex analytic category. However, being hyperbolic does not imply being proper, so for example, in the case of the punctured line, there is no obstruction as the one you mentioned for the existence of a global analytic function. | |
| Feb 12, 2023 at 21:46 | history | edited | Wojowu | CC BY-SA 4.0 |
added 1093 characters in body
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| Feb 12, 2023 at 19:48 | comment | added | kindasorta | The disk $\Omega$ is a residue disk on the $p$-adic curve $X(K_v)$. My worry is that $V_K$ and $V_{K_v}$ could be projective, in which case it is unclear to me how one comes up with such a function on $V_{K_v}$. If such a function exists then it is clear that it may be pulled back to a nonzero function on $X(K_v)$. | |
| Feb 12, 2023 at 19:29 | comment | added | Wojowu | @kindasorta Oops, you are right, my argument would have worked if we were working with affine varieties, but that is not the case here. By working locally on $V$'s I believe we can deduce that $X(K)$ is locally contained in a vanishing locus of such an analytic function. I have looked at the paper and it seems that Lemma 3.3 isn't claiming anything more - they only make a claim about subsets of the disk $\Omega_v$, not the entire projective variety | |
| Feb 12, 2023 at 19:15 | comment | added | kindasorta | $\mathbb{P}^1$ is a proper subvariety of $\mathbb{P}^n$, but there is no non-zero regular algebraic or analytic function $f$ on $\mathbb{P}^n$ that vanishes on your favorite embedding of $\mathbb{P}^1$. Could you elaborate on why such $f$ exists? In which category does it live? | |
| Feb 12, 2023 at 18:19 | history | answered | Wojowu | CC BY-SA 4.0 |