# Explicit expressions for "weakly holomorphic" modular forms of weight 1

I am mainly thinking about the group $$\Gamma(N)$$. A weakly modular form of weight one is a holomorphic function $$f: \mathfrak{H} \to \mathbb{C}$$ satisfying $$f(\gamma \tau) = (c\tau+d)f(\tau), \qquad \gamma \in \Gamma(N),$$ and which is also "meromorphic at the cusps". It corresponds to an algebraic section of the Hodge line bundle $$\underline{\omega}$$ over $$Y(N) = \mathfrak{H}/\Gamma(N)$$.

I believe that, at least for small values of $$N$$, the Hodge line bundle is actually trivial (in other words, the universal elliptic curve over $$Y(N)$$ admits a Weierstrass equation). In particular, the trivialization of the $$\underline{\omega}$$ should correspond to some explicit weakly modular form of weight 1 with no zeros.

Where can I find explicit formulas for such modular forms? For instance, can they be defined by a series on $$\mathfrak{H}$$ (much like Eisenstein or Poincaré series)?

• Y(N) is affine, so the Hodge bundle is trivial automatically (modulo stackiness issues for N = 1 or 2 which mean the bundle doesn't exist) Commented May 30, 2022 at 19:12
• @DavidLoeffler Is this a particular property of the Hodge bundle? I don't think it's true in general that every line bundle over an affine curve is trivial (algebraically). Commented May 30, 2022 at 23:47
• Oops, that was wrong, sorry. Here is a fix. There is a construction of Tate giving a moduli space for pairs $(E, P)$, over an arbitrary base scheme, such that $P$ does not have order 1, 2 or 3. Over this space the universal elliptic curve has a Weierstrass equation. So we can pull back this universal Weierstrass equation to any modular curve. Commented May 31, 2022 at 6:03
• For the construction of Tate I referred to, see $\S9$ of these seminar notes: math.stanford.edu/~conrad/vigregroup/vigre03/moduli.pdf Commented May 31, 2022 at 6:15

Completing David's answer, it is clear that any eta quotient of level $$N$$ and weight 1 satisfies the required condition. If you ask in addition that the form be holomorphic, it is not difficult to write a program giving all holomorphic eta quotients of a given level. For instance there are none in prime level $$N\equiv1\pmod4$$ (and more generally 50 levels out of the first 100 have none), but otherwise some levels have hundreds. The smallest level is of course level $$4$$ with the eta quotient $$\eta(\tau)^{-4}\eta(2\tau)^{10}\eta(4\tau)^{-4}=\theta(\tau)^2$$.
This begs for instance the additional question: do there exist weakly modular forms of weight $$1$$ and level $$N\equiv1\pmod4$$ prime? I assume this is known.
I'm wary of speaking about "the trivialisation of $$\omega$$", since the unit group $$\mathcal{O}(Y(N))^\times$$ (the group of modular units) is a pretty big group, and your trivialization will only be well-defined up to a modular unit, so there is no reason to expect any particularly canonical choice.
That said, for certain explicit $$N$$ you can find one. For $$N = 23$$, there is a holomorphic cusp form of weight 1 which vanishes nowhere on $$\mathcal{H}$$, given by the rather elegant formula $$q \prod_{n \ge 1} (1 - q^n)(1 - q^{23n})$$ where $$q = e^{2\pi i z}$$ as usual. (That is, this form is just $$\eta(z) \eta(23z)$$ where $$\eta$$ is Dedekind's eta function.) Note the non-vanishing is obvious from the convergence of the infinite product.
I don't know of straightforward examples for general values of $$N$$; probably one wants to look for products, rather than sums. Maybe the theory of Borcherds products might be of use here, but I'm not an expert on that theory so I'll leave it to others to comment on that.