Timeline for How is Taylor-Wiles patching "horizontal Iwasawa theory"?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 12 at 10:45 | comment | added | Wojowu | @MarsaultChabat That makes sense, thank you for clarifying! | |
| Feb 12 at 9:27 | comment | added | Marsault Chabat | [...] In Taylor-Wiles systems, one takes the limit of $H^{1}(Y(Qp),V)/I_{Q}$ denoted $H_{\infty}$ and the limit of the corresponding Hecke algebra denoted $T_{\infty}$. Again, $T_{\infty}$ is really nice (a formal series algebra), and quotienting it by appropriate primes restores the Hecke algebra acting faithfully on $H^{1}(Y(Qp),V)$. This process is what I refer to as the "control theorem" in this scenario. | |
| Feb 12 at 9:24 | comment | added | Marsault Chabat | [...] In the context of Hida theory, as David mentioned, one examines the Hecke algebra of the limit of ordinary cusp forms of level $Np^{r}$ (taking the limit over $r$). Consequently, this Hecke algebra becomes $\mathcal{O}[[X]]$-free of finite rank, and quotienting by particular primes $P_{k}$ yields the finite (ordinary) Hecke algebra. This process is what we refer to as the "control theorem." It yields numerous consequences, mainly due to the favorable properties of the "infinite" ordinary Hecke algebra. [...] | |
| Feb 12 at 9:24 | comment | added | Marsault Chabat | Whoops, I forgot to respond. I can't elaborate much more than David's answer, but I can refine my comment. When I use the terminology "control theorem," it's more of an intuitive sense rather than a formal theory. Specifically concerning TW systems, my intuition is this: "If you aim to prove a property about an algebra dependent on a set of primes, then consider taking the limit with respect to this set of primes. This results in an algebra of formal series, facilitating the proof of the property. Subsequently, you can deduce the property at a finite level". [...] | |
| Feb 9 at 13:54 | comment | added | Wojowu | @MarsaultChabat Thanks for these comments. Which control theorem do you have in mind? I'm not sure I'm familiar with the results of Hida theory you mention; is "Hida algebra" here an inverse limit of (ordinary) Hecke algebras I mentioned in my post? | |
| Feb 9 at 8:35 | answer | added | David Loeffler | timeline score: 7 | |
| Feb 9 at 3:49 | comment | added | Marsault Chabat | [...] "horizontal" means (I think) that you consider a projective system of algebras and modules ordered by sets of primes q different from p (where in classical Iwasawa theory, you let the power of p grow and fix the integer N, which is vertical). | |
| Feb 9 at 2:28 | comment | added | Marsault Chabat | Dear @Wojowu, You may have already understood what I'm going to say, but in case you haven't, I'll take the opportunity to explain it to you. Among the several reasons why this comparison holds, I believe one might be the control theorem. As you've already noticed, you deduce the freeness at the finite level Q from the freeness at the infinite level. This is also what happens in Hida theory, where it is shown that the limit space of p-adic automorphic forms is free over the Hida algebra. Furthermore, [...] | |
| Feb 9 at 0:43 | history | asked | Wojowu | CC BY-SA 4.0 |