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I'm trying to better understand some results contained in Deligne and Rapoport's paper on the moduli spaces of elliptic curves.

For convenience, I'll briefly summarize the parts of the paper that I'm interested in (sorry for the long intro):

One has the compactified moduli stacks $\mathcal{M}_1$ and $\mathcal{M}_{\Gamma_0(p)}$ (p is a prime), the first classifying generalized elliptic curves and the other adds the $\Gamma_0(p)$ structure.

(Chapter VI, section 4.3, page 277-278) This defines the degree of an invertible sheaf $\omega$ on an algebraic stack $\mathcal{C}$ (of Cohen-Macaulay, dimension 1, over an algebraically closed field $k$) like this:

  • Take a global section $f$ of $\omega$
  • Let $x$ be a geometrically closed point of $\mathcal{C}$, $\tilde{\mathcal{O}_x}$ the local henselian ring of $\mathcal{C}$ at $x$. Define $\deg_x(f)$ as either $\dim_k \tilde{\mathcal{O}_x}/(f)$ or $-\dim_k \tilde{\mathcal{O}_x}/(f^{-1})$, depending on whether or not $f$ is regular at $x$ respectively.
  • Define $\deg \omega = \sum_{x} \frac{1}{|Aut(x)|}\deg_x (f)$.

Then they proceed to calculate the degree of $\omega$, the dual of the Lie algebra of the universal elliptic curve, on $\mathcal{M}_1$, using the existence of the discriminant $\Delta \in \omega^{12}$, and since the point at infinity has 2 automorphisms, $\deg \omega = \frac{1}{24}$.

(Definition 3.1-3.2, page 302) A modular form of weight $k$ and level $H$ is then an element of $H^0(\mathcal{M}_H,\omega^{\otimes k})$ (i'm interested in $\mathcal{M}_H = \mathcal{M}_1$ and $\mathcal{M}_{\Gamma_0(p)}$).

(3.16 - 3.20 pages 307-309) One takes $f \in H^0(\mathcal{M}_{\Gamma_0(p)},\omega^{k})$. It has 2 q-expansions: $f_1$ at infinity and $f_2$ at the cusp zero. Let $v_1$ and $v_2$ be the $p$-adic valuations of $f_1$ and $f_2$ respectively (the inf of the $p$-adic valuations of the coefficients of $f_1$ and $f_2$). The case I'm interested in is $v_1 = 0$ and $v_2 = v \geq 0$.

(3.19) The reduction $\mathcal{M}_{\Gamma_0(p)}\otimes \mathbb{F}_p$ has two irreducible components $N_1$ and $N_2$, where $N_1$ contains the cusp infinity and $N_2$ contains the cusp $0$. The divisor of $f$ is $div(f) = D + v N_2$, where $D$ is a divisor that meets $N_1$ and $N_2$ a finite number of times.

Then they calculate some intersection numbers:

  • $(N_2, N_1) = \frac{p-1}{24}$ (this is the degree of vanishing of the Hasse invariant, counting supersingular points weighted by the sizes of automorphism groups)
  • $N_1$ is isomorphic to $\mathcal{M}_1\otimes \mathbb{F}_p$ so the degree of $\omega$ on $N_1$ is $\frac{1}{24}$
  • Thus $(D,N_1) + v (N_2,N_1) = \frac{k}{24}$.

Here are my questions:

  1. On $\mathcal{M}_1$, the degrees of $\omega^2$ and $\omega^k$ for odd $k$ are strictly positive, even though there are no modular forms of level 1 in these weights; is there a simple explanation for this?
  2. How are the intersection numbers $(D,N_1)$ and $(N_1, N_2)$ really defined? What is there relation to the degree of an invertible sheaf on a stack defined in VI 4.3?
  3. What do these intersection numbers really mean? My intuition is that $(D,N_1)$ counts the zeroes of $f$ (is it in characteristic 0 or mod $p$ ?) on $N_1$, and $v(N_2, N_1)$ counts the forced vanishing of $f$ at the supersingular points (coming from the fact that $f$ vanishing to order $v$ on the component $N_2$).
  4. Is it possible to have $0 < (D,N_1) \leq \frac{1}{12}$? my intuition says this can't happen if $p > 3$, because this should be counting points x by weighting them with $\frac{1}{|Aut(x)|}$, but we know that elliptic curves in characteristic $p$ will have at most $6$ automorphisms.

Sorry for the long question, but I would really like to have a lucid understanding of this, and confirmation of my intuition, so thanks in advance!

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If the degree of $\omega$ weren't strictly positive, then not only would there be no weight 1 cusp forms for $\Gamma_0(1)$, but there would be no weight 1 cusp forms for any level whatsoever, because pullback via a finite flat morphism will just multiply the degree of $\omega$ by a positive constant. Does that answer your (1)? – David Loeffler Oct 8 '13 at 10:05
@DavidLoeffler: Thanks, yes that sounds quite reasonable. I just found it surprising that on stacks, an invertible sheaf can have positive degree without having global sections. – Nadim Rustom Oct 8 '13 at 11:21
@NadimRustom: You can also have D1-D2 on a usual curve with degree 1 but no sections (i.e., not equivalent to an effective divisor). – David Zureick-Brown Dec 9 '13 at 23:04
@NadimRustom: see section B.2.2 of the preprint – user27920 Aug 9 '14 at 10:49

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