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show/hide this revision's text 3 added 12 characters in body

Speaking of "connected components" is a delicate thing since you really mean in a relative sense, and more specifically the etale quotient $H$ can have its open and closed non-identity part with very nontrivial $\pi_1$-action (so more subtle than on geometric fibers over the base). But even if the $\pi_1$-action is trivial over whatever base, there are generally no no nontrivial sections through the non-identity part, so you can't do translation arguments, so there's no reason to expect intuition about "homogeneity" to have any relevance. Likewise for any property which isn't local for whatever topology the thing admits local sections. (In this case the fppf topology, for which regularity is not a local property, and ditto for reducedness.)

In this case Katz-Mazur (or better: Kummer!) did all of the deformation theory work. If we pass to the complete local ring at a geometric point of the base curve then (by the Serre-Tate deformation theorem) you're really asking a question about something over the universal deformation ring $R = k[[x]]$ of the $p$-divisible group

$$\mu_{p^{\infty}} \times (\mathbf{Q}_ p)/\mathbf{Z}_ p$$

over which the universal $p$-divisible group $\Gamma$ has finite flat $p$-torsion $G = \Gamma[p]$ with a connected etale sequence that is described explicitly in Katz-Mazur. So there you can stare at the non-identity factors, and if those are regular then you're done. And if not regular somewhere then likewise for the global case over the modular curve.

If you look at (8.7.1.1) in KM (with $i \ne 0$) and then the proof of Prop. 8.10.5 there, or instead think about Kummer theory for group schemes, you'll see that there is a unit $q$ unique up to $p$-power unit multiple which "classifies" (not quite uniquelyup to isomorphism) the extension structure on $G$ (as $p$-torsion extension of $\mathbf{Z}/p \mathbf{Z}$ by $\mu_p$). We can scale so $q$ is a 1-unit.

Now I claim that $q-1$ has ord equal to 1 in the deformation ring $k[[x]]$. Indeed, otherwise it would say that every first-order deformation of the $p$-divisible group has split $p$-torsion, which we know is nonsense (since we can use the unit $1+x$ to build a deformation violating that).

So then we can change $x$ so $q = 1 + x$ and the equations of the non-identity components are $T^p - (1+x)^i$ for $1 \le i \le p-1$ (from K-M, or thinking on our own). I claim the quotient in each case is a discrete valuation ring. Since $(1+x)^i = 1 + ix + x^2(\dots)$, by change of $x$ it is always the same as $T^p - (1+y)$ over $k[[y]]$, and writing it as $(T-1)^p - y$ since in characteristic $p$ we see it is Eisenstein, so we're done.

Since function fields of the modular curves you had in mind are not perfect, perhaps a more amusing example for you of funny behavior is to give a reduced group scheme over a field which is not smooth, and a non-reduced group scheme whose underlying reduced scheme is not a subgroup scheme (affine groups of finite type, but ground field must be imperfect of course).

show/hide this revision's text 2 I made the Eisenstein argument a little bit clearer

Speaking of "connected components" is a delicate thing since you really mean in a relative sense, and more specifically the etale quotient $H$ can have its open and closed non-identity part with very nontrivial $\pi_1$-action (so more subtle than on geometric fibers over the base). But even if the $\pi_1$-action is trivial over whatever base, there are generally no no nontrivial sections through the non-identity part, so you can't do translation arguments, so there's no reason to expect intuition about "homogeneity" to have any relevance. Likewise for any property which isn't local for whatever topology the thing admits local sections. (In this case the fppf topology, for which regularity is not a local property, and ditto for reducedness.)

In this case Katz-Mazur (or better: Kummer!) did all of the deformation theory work. If we pass to the complete local ring at a geometric point of the base curve then (by the Serre-Tate deformation theorem) you're really asking a question about something over the universal deformation ring $R = k[[x]]$ of the $p$-divisible group

$$\mu_{p^{\infty}} \times (\mathbf{Q}_ p)/\mathbf{Z}_ p$$

over which the universal $p$-divisible group $\Gamma$ has finite flat $p$-torsion $G = \Gamma[p]$ with a connected etale sequence that is described explicitly in Katz-Mazur. So there you can stare at the non-identity factors, and if those are regular then you're done. And sure enough, if not regular somewhere then likewise for the global case over the modular curve.

If you look at (8.7.1.1) in KM (with $i \ne 0$) and then Prop. 8.10.5 there, or instead think about Kummer theory for group schemes, you'll see that there is a unit $q$ unique up to $p$-power unit multiple which "classifies" (not quite uniquely) the extension structure on $G$ (as $p$-torsion extension of $\mathbf{Z}/p \mathbf{Z}$ by $\mu_p$). We can scale so $q$ is a 1-unit.

Now I claim that $q-1$ has ord equal to 1 in the deformation ring $k[[x]]$. Indeed, otherwise it would say that every first-order deformation of the $p$-divisible group has split $p$-torsion, which we know is nonsense (since we can use the unit $1+x$ to build a deformation violating that).

So then we can change $x$ so $q = 1 + x$ and the equations of the non-identity components are $T^p - (1+x)^i$ for $1 \le i \le p-1$ (from K-M, or thinking on our own). Over $k[[x]]$ this is an Eisenstein polynomial, so I claim the quotient by it in each case is a discrete valuation ring. QED

KevinSince $(1+x)^i = 1 + ix + x^2(\dots)$, by change of $x$ it is always the same as $T^p - (1+y)$ over $k[[y]]$, and writing it as $(T-1)^p - y$ since in characteristic $p$ we see it is Eisenstein, so we're done.

Since function fields of the modular curves you had in mind are not perfect, perhaps a more amusing example for you of funny behavior is to give a reduced group scheme over a field which is not smooth, and a non-reduced group scheme whose underlying reduced scheme is not a subgroup scheme (affine groups of finite type, but ground field must be imperfect of course).

show/hide this revision's text 1

Speaking of "connected components" is a delicate thing since you really mean in a relative sense, and more specifically the etale quotient $H$ can have its open and closed non-identity part with very nontrivial $\pi_1$-action (so more subtle than on geometric fibers over the base). But even if the $\pi_1$-action is trivial over whatever base, there are generally no no nontrivial sections through the non-identity part, so you can't do translation arguments, so there's no reason to expect intuition about "homogeneity" to have any relevance. Likewise for any property which isn't local for whatever topology the thing admits local sections. (In this case the fppf topology, for which regularity is not a local property, and ditto for reducedness.)

In this case Katz-Mazur (or better: Kummer!) did all of the deformation theory work. If we pass to the complete local ring at a geometric point of the base curve then (by the Serre-Tate deformation theorem) you're really asking a question about something over the universal deformation ring $R = k[[x]]$ of the $p$-divisible group

$$\mu_{p^{\infty}} \times (\mathbf{Q}_ p)/\mathbf{Z}_ p$$

over which the universal $p$-divisible group $\Gamma$ has finite flat $p$-torsion $G = \Gamma[p]$ with a connected etale sequence that is described explicitly in Katz-Mazur. So there you can stare at the non-identity factors, and if those are regular then you're done. And sure enough, if you look at (8.7.1.1) in KM (with $i \ne 0$) and then Prop. 8.10.5 there, or instead think about Kummer theory for group schemes, you'll see that there is a unit $q$ unique up to $p$-power unit multiple which "classifies" (not quite uniquely) the extension structure on $G$ (as $p$-torsion extension of $\mathbf{Z}/p \mathbf{Z}$ by $\mu_p$). We can scale so $q$ is a 1-unit.

Now I claim that $q-1$ has ord equal to 1 in the deformation ring $k[[x]]$. Indeed, otherwise it would say that every first-order deformation of the $p$-divisible group has split $p$-torsion, which we know is nonsense (since we can use the unit $1+x$ to build a deformation violating that).

So then we can change $x$ so $q = 1 + x$ and the equations of the non-identity components are $T^p - (1+x)^i$ for $1 \le i \le p-1$ (from K-M, or thinking on our own). Over $k[[x]]$ this is an Eisenstein polynomial, so quotient by it is a discrete valuation ring. QED

Kevin, since function fields of the modular curves you had in mind are not perfect, perhaps a more amusing example for you of funny behavior is to give a reduced group scheme over a field which is not smooth, and a non-reduced group scheme whose underlying reduced scheme is not a subgroup scheme (affine groups of finite type, but ground field must be imperfect of course).