Explicit natural correspondence between cusps of $X(N)$ and isomorphism classes of level $N$ structures on Tate($q^N$)

In Katz' paper Antwerp III, section 1.4 (Ka-14) one reads (we assume $$n \geq 3$$ integer):

''The scheme $$\overline{M}_n - M_n$$" over $$\mathbb{Z}[1/n]$$ is finite and étale, and over $$\mathbb{Z}[1/n,\zeta_n]$$, it is a disjoint union of sections, called the cusps of $$\overline{M}_n$$,''

I would be interested to see a detailed proof of the next part of that sentence, namely:

"which in a natural way are the set of isomorphism classes of level $$n$$ structures on the Tate curve $$\text{Tate}(q^n)$$ viewed over $$\mathbb{Z}((q)) \otimes_{\mathbb{Z}} \mathbb{Z}[1/n,\zeta_n]$$."

What I did: I tried to extract the relevant information in Deligne-Rapoport and Katz-Mazur but in each case, certainly for a lack of understanding on my part, I'm not able to establish this correspondence explicitly. I found the discussion of formal completion at (the divisor of) cusps well explained in both references (something which is also addressed in Katz' paper Antwerp III on page Ka-14), but I couldn't connect the dots for the natural correspondence above and thus my question. Feel free to ask if you need more details.

In Deligne-Rapoport?

I first looked in Deligne-Rapoport (DR), which was in Antwerp II (and so the natural place to look for the arguments): http://smtp.math.uni-bonn.de/ag/alggeom/preprints/Lesschemas.pdf

The motivating example on pages DeRa-7 and beginning of page 8 hint to that fact. But it seems that's not the point of view (DR) take.

"Dans le texte, nous précisons cette interprétation modulaire de l'ensemble des points à l'infini de $$\mathcal{H}/\Gamma(n)$$ en une interprétation modulaire de la courbe projective compactifiée $$\overline{\mathcal{H}/\Gamma(n)}$$ de $$\mathcal{H}/\Gamma(n)$$."

Here: $$\mathcal{H} = \{ z \in \mathbb{C} \mid \Im(z) > 0 \}$$ is the upper half-plane.

On page DeRa-10 they do say that $$M_n$$ can be defined as the normalization, in the field of functions of $$M_n^0[1/n]$$, of the projective $$j$$-line over $$\mathbb{Z}[\zeta_n]$$. (That's what Katz and Mazur do in their book on Chapter 8.) (DR) say among other things that they prove that there exists a finite family of points $$\mathbb{Z}[\zeta_n]$$-points $$f_i : M_n \to Spec(\mathbb{Z}[\zeta_n])$$ such that the sections $$f_i$$ are disjoint (incongruent modulo any prime ideal of $$\mathbb{Z}[\zeta_n]$$) and that $$M_n^0$$ is the complement in $$M_n$$ of the union of the ''sections at infinity'' $$f_i$$.

The Tate curve is only constructed in chapter VII of (DR). But I don't find it immediate to deduce initial assertion by Katz in Antwerp III.

In Chapter VII (sections 1 and mostly 2 seem relevant to my question), DeRa-156, (1.16.4) gives me the description of the level $$r$$-structure of the Tate curve with $$r$$ edges over $$\mathbb{Z}[[q^{1/r}]]$$.

Moreover, $$\text{Tate}(q)$$ over $$\mathbb{Z}[[q]]$$ induces a morphism $$\tau: Spec(\mathbb{Z}[[q]]) \to \mathcal{M}_1$$ which identifies $$\mathbb{Z}[[q]]$$ with the formal completion of $$\mathcal{M}_1$$ along the section at infinity $$f_1$$ (Theorem 2.1).

The Néron $$n$$-gon $$C$$ over $$\mathbb{Z}[\zeta_n]$$ equipped with its structure of generalized elliptic curve and the natural isomorphism $$C[n] = \mu_n \times \mathbb{Z}/n\mathbb{Z}$$ defines a section at infinity $$f_n : Spec(\mathbb{Z}[\zeta_n]) \to \mathcal{M}_n$$. We also obtain an isomorphism between the $$n$$-torsion of the Tate curve with $$n$$ edges and $$\mu_n \times \mathbb{Z}/n \mathbb{Z}$$ and then we geta morphism $$Spec(\mathbb{Z}[\zeta_n][[q^{1/n}]]) \to \mathcal{M}_n$$. This latter morphism identifies $$\mathbb{Z}[\zeta_n][[q^{1/n}]]$$ with the formal completion of $$\mathcal{M}_n$$ along the section at infinity $$f_n$$.

Finally, Corollary 2.5 says that the completion of $$\mathcal{M}_n$$ along infinity is sum of copies of $$Spec(\mathbb{Z}[\zeta_n][[q^{1/n}]]$$ indexed by $$SL_2(\mathbb{Z}/n\mathbb{Z})/\pm U$$, where $$U$$ is the group of upper unipotent matrices.

It feels like the desired correspondence is there but I couldn't extract it explicitly.

In Katz-Mazur? I turned to the book of Katz and Mazur (see https://web.math.princeton.edu/~nmk/katz-mazur.djvu). Again, I feel I'm getting closed, but I'm not sure how to tie up the loose ends.

The point of view in (KM) doesn't deal (explicitly?) with stacks (as in (DR)). They consider the moduli problem (contravariant functor)

$$[\Gamma(N)] : \textbf{Ell} \to \textbf{Set}$$

which classifies elliptic curves (proper smooth curves $$\pi : E \to S$$ with geometrically connected fibers all of genus one, given with a section $$0$$, and here $$S$$ is any scheme.) equipped with a $$\Gamma(N)$$-structure (KM 3.1, page 98).

This functor is relatively representable and flat over $$\textbf{Ell}$$ of constant rank $$\geq 1$$, and regular of dimension $$2$$. As a functor with source $$\textbf{Ell}/\mathbb{Z}[1/N]$$ it is étale on the source. (First Main Theorem 5.1.1, page 129).

When $$N \geq 3$$, $$[\Gamma(N)]$$ is in fact representable by some universal elliptic curve $$E_\text{univ}/Y(N)$$, where $$Y(N)$$ is a smooth affine curve (We have a rigidity.) [See (KM) Cor 2.7.2, 4.7.0 and 4.7.1)

Following (KM 8.6.3 and 8.6.8) we normalize $$Y(N)$$ near infinity to obtain $$X(N)$$ (we obtain a smooth proper curve over $$\mathbb{Z}[1/N]$$ which is the normalization of the projective $$j$$-line in $$Y(N)$$).

The Tate curve $$\text{Tate}(q)$$ itself represents an appropriate moduli problem $$\mathcal{S}$$. Applying corollary 8.4.4 (p.235) to this and to the moduli problem $$[\Gamma(N)]$$ over an excellent noetherian regular ring $$R$$, we obtain an isomorphim of $$R((q))$$-schemes

$$\left( [\Gamma(N)]_{\text{Tate}(q)/R((q))} \right)/ \pm 1 \xrightarrow{\simeq} Y(N)_{R((q))}$$

where $$Aut(\text{Tate}(q)/R((q))) = \pm 1$$ (see Proposition 8.11.7).

Moreover, the formal completion of $$X(N)$$ along the (divisor of) cusps $$X(N) - Y(N)$$, which is a finite $$R[[q]]$$-scheme, is the normalization of $$R[[q]]$$ in the finite normal $$R((q))$$-scheme $$\left( [\Gamma(N)]_{\text{Tate}(q)/R((q))} \right)/ \pm 1$$.

Finally, we have

Theorem 10.8.2

There is a canonical isomorphism of $$\mathbb{Z}[\zeta_N]((q))$$-schemes

$$[\Gamma(N)]_{\text{Tate}(q)/\mathbb{Z}[\zeta_N]((q))} \simeq \coprod_{\text{Hom Surj }((\mathbb{Z}/N\mathbb{Z})^2,\mathbb{Z}/N\mathbb{Z})} Spec(\mathbb{Z}[\zeta_N]((q^{1/N})))$$

and

Theorem 10.9.1

(1) $$\text{Cusps}(X(N))$$ is the disjoint union of $$\mid \text{Hom Surj }((\mathbb{Z}/N\mathbb{Z})^2,\mathbb{Z}/N\mathbb{Z}) \mid$$ sections of $$X(N)$$ over $$\mathbb{Z}[\zeta_N]$$.

(2) There exists an open neighborhood $$V$$ of the cusps $$\text{Cusps}([\Gamma(N)]) \subset V \subset X(N)$$ which is smooth over $$\mathbb{Z}[\zeta_N]$$.

(3) The formal completion of $$X(N)$$ along its cusps is the $$\mathbb{Z}[\zeta_N]$$-formal scheme

$$\coprod_{\text{Hom Surj }((\mathbb{Z}/N\mathbb{Z})^2,\mathbb{Z}/N\mathbb{Z})/\pm 1} Spf\left( \mathbb{Z}[\zeta_N][[q^{1/N}]] \right)$$.

• To define this correspondence you don't want to write down either side explicitly, but rather do it abstractly. Then you can calculate what it is in explicit terms if desired. The first steps are (1) Send each cusp in $\overline{M}_n$ to its formal neighborhood in $\overline{M}_n$, and then to its punctured formal neighborhood in $M_n$. (2) Observe that these punctured formal neighborhoods are connected components of the inverse image in $M_n$ of the punctured formal neighborhood of infinity in $\mathcal M_{1,1}$. – Will Sawin May 6 at 2:52
• (3) Observe that the punctured formal neighborhood of infinity in $\mathcal M_{1,1}$ is the Tate curve, and that the fiber of the map $M_n \to \mathcal M_{1,1}$ over any point parameterizes level $n$ structures on the corresponding elliptic curve. – Will Sawin May 6 at 2:56
• That said, if you want to present the bijection explicitly, can't you use Theorem 10.8.2 and Theorem 10.9.1? – Will Sawin May 6 at 2:58
• Maybe this is helpful : sections 2.7, 2.12 and 8.9 of Saito's "Fermat's Last Theorem", the "Basic Tools" for the first two and "The proof" for the last. – Chris Wuthrich May 6 at 8:30
• I'm a bit puzzled with Katz's statement when read literally. Namely, the cusps are in bijection with the isomorphism classes of level structures on the Néron $n$-gon. These are easy to describe but one should quotient by the automorphism group of the Néron $n$-gon, which is $\mu_n \rtimes \{\pm 1\}$. On the other hand, the Tate curve seen over $\mathbb{Z}((q))$ is an elliptic curve, and we are in characteristic 0 so the only automorphisms are $\pm 1$. My guess is that in Katz's paper one should replace "iso classes" by "equivalence classes" under the relation described above. – François Brunault May 6 at 18:54

(1) Send each cusp in $$\overline{M}_n$$ to its formal neighborhood in $$\overline{M}_n$$, and then to its punctured formal neighborhood in $$M_n$$.
(2) Observe that these punctured formal neighborhoods are connected components of the inverse image in $$M_n$$ of the punctured formal neighborhood of infinity in $$\mathcal M_{1,1}$$.
(3) Observe that the punctured formal neighborhood of infinity in $$\mathcal M_{1,1}$$ is the Tate curve, and that the fiber of the map $$M_n\to \mathcal M_{1,1}$$ over any point parameterizes level $$n$$ structures on the corresponding elliptic curve.