Prompted by this MO question, I have the following question about modular forms which do not vanish on the upper-half plane.

Q1. Let $N \geq 1$ be an integer and let $\Gamma(N)$ be the principal congruence subgroup of $\mathrm{SL}_2(\mathbf{Z})$ of level $N$. What is the minimal weight $k \geq 1$ such that there exists a meromorphic modular form of weight $k$ on $\Gamma(N)$ which does not vanish on the upper-half plane?

Note that I only require the modular form to be meromorphic at the cusps, so the set of such weights is of the form $k_0 \mathbb{N}$ for some $k_0 \geq 1$. Moreover, the example of the modular discriminant $\Delta$ shows that $k_0$ divides 12. Jeremy Rouse's answer shows additionally that if $12$ divides $N$ then we have $k_0=1$.

Another motivation for asking this question is given by universal elliptic curves. If $E/S$ is an elliptic curve over an arbitrary scheme $S$, then the $12$-th power of the line bundle $\omega_{E/S}$ on $S$ is trivial, see e.g. this MO post by Brian Conrad. In the case $S=Y(N)$ and $E=E(N)$ is the universal elliptic curve over $Y(N)$ with $N \geq 3$, then I think the order of $\omega_{E(N)/Y(N)}$ in the Picard group of $Y(N)$ coincides with the integer $k_0$ introduced above, because weight $k$ modular forms correspond to sections of $\omega_{E(N)/Y(N)}^{\otimes k}$, and non-vanishing modular forms correspond to trivializations.

Q2. Let $N \geq 3$ be an integer. Let $\mathcal{Y}(N)$ be the modular curve over $\mathbf{Z}[1/N]$ representing the moduli problem of elliptic curves with full level $N$ structure as in Katz-Mazur. Let $\mathcal{E}(N)$ be the universal elliptic curve over $\mathcal{Y}(N)$. What is the order of $\omega_{\mathcal{E}(N)/\mathcal{Y}(N)}$ in the Picard group of $\mathcal{Y}(N)$?

Note that Q1 makes sense for arbitrary congruence subgroups, and Q2 makes sense as soon as the corresponding moduli problem is representable. I would be interested as well by answers to Q1 and Q2 for congruence subgroups other than $\Gamma(N)$.


1 Answer 1


I will answer Q2:

N=2: Denote by $Y^1(2)$ the moduli of elliptic curves with point of order 2 and fixed invariant differential. It is not hard to show that $Y^1(2) = \mathrm{Spec}\, \mathbb{Z}[\frac12][x_2, y_2, \Delta^{-1}]$ with $\Delta = 16x_2^2y_2^2(x_2-y_2)^2$. We obtain $Y(2)$ as the $\mathbb{G}_m$-stack quotient of $Y^1(2)$ -- on the other hand $Y^1(2)$ arises as the relative spec of $\bigoplus_{k\in\mathbb{Z}} \omega^{\otimes k}$. Thus, $x_2$ defines a trivialization of $\omega^{\otimes 2}$. But there is no non-zero section of $\omega$ itself.

N=3: An equally explicit description might be given for $Y(3)$ that shows that $\omega$ is trivial in this case. (See e.g. Katz--Mazur, p.71-73, or Stojanoska, p.7).

N>=4: In these cases, even on $Y_1(N)$ the line bundle $\omega$ is trivial. Indeed, the $Y_1(N)$ are schemes and as soon as we can write down a Weierstraß equation for the universal elliptic curves over them, the usual formula for an invariant differential defines a trivialization of $\omega$. Such a Weierstraß equation is often called the Tate or Kubert--Tate normal form. This is explained in a number of sources, e.g. over a field in Husemoller's book on elliptic curves. A detailed exposition in the case $N=5$ is also in Theorem 1.1.1 of the article On the homotopy of $Q(3)$ and $Q(5)$ at the prime 2 of Behrens and Ormsby.


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