I'm reading the five chapter of the book of Katz-Mazur, Arithmetic moduli of elliptic curves, concerning regularity of the moduli problems of $\Gamma(N)$-structures, $\Gamma_1(N)$-structures and $\mbox{bal}.\Gamma_1(N)$-structures on the category $(\mbox{Ell})$. The theorem says that the three problems are finite flat over (Ell), and regular. Following the notation of Katz-Mazur, the previous means that for any representable étale moduli problem $\mathcal{S}$, with universal object $E/\mathfrak{M}(\mathcal{S})$, the morphism of schemes

$$ \mathcal{P}_{E/\mathfrak{M}(\mathcal{S})}\longrightarrow \mathfrak{M}(\mathcal{S}) $$ is finite flat, and the scheme $\mathcal{P}_{E/\mathfrak{M}(\mathcal{S})}$ is regular ($\mathcal{P}$ is one of the moduli problems $[\Gamma(N)]$, $[\Gamma_1(N)]$, $[\mbox{bal}.\Gamma_1(N)]$).

The heart of the proof lies in a more general proof which uses four axioms for a moduli problem $\mathcal{P}$ which are

Reg. 1. $\mathcal{P}$ is relatively representable and finite over $(\mbox{Ell})$.

Reg. 2. $\mathcal{P}\otimes \mathbb{Z}[1/p]$ is finite étale over $(\mbox{Ell}/ \mathbb{Z}[1/p])$.

Reg. 3. $\mathcal{P}$ depends only on the $p$-divisible group.

Reg. 4. An axiom which takes care the supersingular case over a field of characteristic $p$.

At some point of the proof they take an odd prime $\ell\neq p$ and they consider a representable moduli problem $\mathcal{S}_1$ which is étale and surjective over $(\mbox{Ell}/ \mathbb{Z}[1/\ell])$ (i.e. such that the morphism $\mathfrak{M}(\mathcal{S}_1)\rightarrow \mathbb{Z}[1/\ell]$ is étale and surjective). Then, they consider the finite morphism of schemes

$$ \mathcal{P}_{\mathfrak{M}(S_1)}\rightarrow \mathfrak{M}(\mathcal{S}_1). $$ It is étale when tensoring by $\mathbb{Z}[1/p]$. They consider the open subset $\mathcal{V}\subset \mathcal{P}_{\mathfrak{M}(\mathcal{S}_1)}$ of points where $\mathcal{P}_{\mathfrak{M}(\mathcal{S}_1)}$ is flat over $\mathfrak{M}(\mathcal{S}_1)$ and regular. My question is, why is the set $\mathcal{V}$ open? The flatness is not a problem since we have references in EGA or in an appendix of Görtz-Wedhorn's Algebraic Geometry I. What I cannot see is that the locus of regularity is in fact open (it would suffice to prove that it is constructible for a curve over $\mbox{Spec}\mathbb{Z}$ since it is stable by generization). Note that smoothness implies regularity, indeed it implies étale since the morphism is finite, but the converse is not true since we deal with curves over $\mathbb{Z}$.

  • 1
    $\begingroup$ The openess of the regularity part is studied in the Stacks project, section 28.18: stacks.math.columbia.edu/tag/07R2 $\endgroup$ – Xarles Apr 16 at 7:21
  • 1
    $\begingroup$ Thank you very much! That is the reference I needed. $\endgroup$ – Jrodri26 Apr 16 at 8:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.