Timeline for Katz's $\ell$-adic Airy sheaf
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 12, 2021 at 21:46 | vote | accept | Pulcinella | ||
| Nov 10, 2021 at 23:32 | answer | added | Will Sawin | timeline score: 7 | |
| Nov 4, 2021 at 18:12 | comment | added | Pulcinella | The coefficients all have simple poles for $\varepsilon\ne 0$, so $$(1+\varepsilon x^n)y''(x)\ =\ xy(x)$$ is a family of regular ODEs degenerating to the Airy equation, whenever $n>4$. | |
| Nov 4, 2021 at 18:10 | comment | added | Pulcinella | @WillSawin Using the coordinate $u=1/x$ about the irregular point $x=\infty$, we have $$y''(x)\ +\ xy(x)\ =\ (u^4y''(u)+2u^3y'(u))\ +\ (y(u)/u)$$ which is irregular at $u=0$ because when you divide through by $u^4$, the $y$ term has a pole of order $>2$. However, we have $$(1+\varepsilon x^n) y''(x)\ +\ xy(x)\ =\ (u^n+\varepsilon)(u^4y''(u)+2u^3y'(u))/u^n\ +\ (y(u)/u)$$ so if we normalise we now get $$y''(u)\ +\ \frac{2u^{n-1}}{u^n+\varepsilon}y'(u)\ +\ \frac{u^{n-5}}{u^n+\varepsilon}y(u)\ =\ 0.$$ | |
| Nov 4, 2021 at 15:32 | comment | added | Will Sawin | For 2, what limiting family are you thinking of? | |
| Nov 4, 2021 at 10:29 | history | asked | Pulcinella | CC BY-SA 4.0 |