Timeline for Semistable model of product of modular curves

Current License: CC BY-SA 4.0

9 events

when toggle format	what		by	license	comment
Jan 24, 2022 at 11:06	history	edited	David Loeffler	CC BY-SA 4.0	added 449 characters in body
Jan 13, 2022 at 7:38	history	edited	David Loeffler	CC BY-SA 4.0	added 416 characters in body
Jan 12, 2022 at 16:52	history	edited	David Loeffler	CC BY-SA 4.0	added 174 characters in body
Jan 12, 2022 at 16:48	comment	added	David Loeffler		Yes, that certainly seems like the "natural" choice for the moduli space interpretation.
Jan 12, 2022 at 16:41	comment	added	Will Sawin		Blowing up along the component with ideal $(x,z)$ introduces a $\mathbb P^1$ with coordinate $x/z$. Similarly the other four introduce $\mathbb P^1$s with coordinates $x/w$, $y/z$, $y/w$. We have $x/z =w/y$ and $x/w=z/y$ so there are two pairs that give isomorphic blowups, but I don't think the blowups are isomorphic between the pairs. Specifically, I think you want to blowup either the component where both subgroups are $\mathbb Z/p$ or the component where both subgroups are $\mu_p$ so that the coordinate of your new $\mathbb P^1$ will get inverted as you swap the two elliptic curves.
Jan 12, 2022 at 16:17	comment	added	Will Sawin		The supersingular points will be nodes, with local equations of the form $xy = p^v$. So the product of two supersingular points will be given by an equation of the form $xy = zw = p^v$. So the resolution should be the same as the resolution for a space with equation $xy=zw$, where there's two different ways to blow it up to a $\mathbb P^1$. I think at most one of them can be this moduli space, but possibly one of them is.
Jan 12, 2022 at 16:03	history	edited	David Loeffler	CC BY-SA 4.0	added 109 characters in body
Jan 12, 2022 at 14:54	history	edited	David Loeffler	CC BY-SA 4.0	added 11 characters in body
Jan 12, 2022 at 14:24	history	asked	David Loeffler	CC BY-SA 4.0