Let $M$ be the moduli stack of ordinary but possibly nodal elliptic curves over the field $\overline{\mathbf{F}_p}$. Then $M$ has a $\mathbb{Z}_p^{\times}$-torsor over it, given by the moduli scheme of "trivialized" elliptic curves: that is, elliptic curves equipped with an isomorphism between the formal group and $\widehat{\mathbf{G}_m}$. A theorem of Igusa states that this cover (or rather, inverse system of covers) is connected.

Away from $p$-primary information, is this the universal cover of $M$, or is there an additional portion of the fundamental group? (In other words: what is the fundamental group of Katz's ring of $p$-adic modular forms?)


1 Answer 1


For $p>13$, there are at least two supersingular $j$ invariants, say $a$ and $b$, and adjoining $\sqrt[N]{\frac{j-a}{j-b}}$ is always an etale cover for $N$ prime to $p$. This gives an additional portion of the fundamental group.

To compute the full fundamental group, we can first take an etale cover that kills the elliptic points. To do this, if $a \neq 1728$ is a supersingular $j$ invariant, adjoin $\sqrt{ \frac{j-a}{j-1728}}$ and if $a \neq 0$ is a supersingular $j$ invariant, adjoin $\sqrt[3]{\frac{j-a}{j}}$. If $1728$ or $0$ is supersingular, just ignore that steps.

Now our stack is basically just a scheme - an algebraic curve minus finitely many points. We can compute its fundamental group (or the prime-to-$p$ part?) the usual way. Then there is an extra central extension by $\mathbb Z/2$, coming from the automorphism $y \to -y$ shared by all elliptic curves.

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    $\begingroup$ Is it easy to see that the cover obtained by taking roots of (a linear fractional transformation of) the $j$-invariant is not already split by the Igusa tower? $\endgroup$ Commented Nov 14, 2013 at 21:05
  • $\begingroup$ Just compare the Galois group. If $N$ is prime to $p$ and does not divide $(p-1)$, its Galois group cannot possibly be a quotient of the Igusa tower group. $\endgroup$
    – Will Sawin
    Commented Nov 15, 2013 at 2:16

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