Timeline for Iwasawa theory over function fields - How do eigenvalues vary in $\mathbb Z_\ell$ towers?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 22, 2021 at 3:18 | comment | added | Will Sawin | Yes, I'm including the Frobenius. The way that Frobenius acts on the geometric Galois group imposes some strong restrictions about the eigenvalues of Frobenius, just based on the fact that Frobenius and this group both act on cohomology. It's possible that, beyond this, not much is known. | |
| Jan 22, 2021 at 3:06 | comment | added | Asvin | I am not sure what to call this Galois group but maybe let's just restrict to the class of examples I mentioned. | |
| Jan 22, 2021 at 3:06 | history | edited | Asvin | CC BY-SA 4.0 |
added 46 characters in body
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| Jan 22, 2021 at 3:04 | comment | added | Asvin | Sorry, could you explain more? You get the automorphisms that send $x \to x\zeta_{\ell^n}$ on $C_n$ but what other automorphisms do you get? Are you including the Frobenius? If so, I guess I meant the geometric Galois group should be $\mathbb Z_\ell$ - sorry for being unclear. | |
| Jan 22, 2021 at 3:00 | comment | added | Will Sawin | That has Galois group $\mathbb Z_\ell \rtimes \mathbb Z_\ell$, at least if $q$ contains the $\ell$th roots of unity. | |
| Jan 22, 2021 at 2:58 | comment | added | Asvin | (I have been thinking about this problem and want to make sure that my results are new.) | |
| Jan 22, 2021 at 2:52 | comment | added | Asvin | Yes, you can consider $f(x^{\ell^n},y) = 0$ for instance | |
| Jan 22, 2021 at 2:48 | comment | added | Will Sawin | Do such towers exist? | |
| Jan 21, 2021 at 22:14 | history | asked | Asvin | CC BY-SA 4.0 |