# Why is this function a modular function of level $5$?

Suppose we have a function $$\phi\colon \mathfrak H \longrightarrow \mathbb C$$ such that

1. $$\phi^{24}$$ is a modular function of level $$5$$.
2. $$\phi(\tau)=\sum_{n=-1}^{\infty}a_{n}q^{n/5}$$, $$a_{-1}\neq 0,q=e^{2\pi i\tau}$$.

Does it follow that $$\phi$$ is a modular function of level $$5$$?

In particular, I am interested in the function $$\phi(\tau)=-\frac{1}{5^{1/2}}\frac{\eta(\tfrac{\tau}{5})\eta(\tfrac{\tau+1}{5})\eta(\tfrac{\tau-1}{5})}{\eta(5\tau)\eta(\tfrac{\tau+2}{5})\eta(\tfrac{\tau-2}{5})}.$$ Is there a a simple way to show that it is modular of level $$5$$? Siegel says that it is easy to show that it is so. We could compute the generators for $$\Gamma(5)$$ and then use the transformation formula for the Dedekind eta function, but this seems like a mess.

• $j(z/5)(1-1728/j(z/5))^{1/24}$ is not a modular function, it has a branch point at $5i$ Commented May 16, 2021 at 21:44
• Just a remark, I would say you dont' need generators of $\Gamma(5)$. Usually to show modularity, one just takes any $\gamma$ in $\Gamma(5)$, work out the transformation formula, and use the congruence conditions on the coefficients. You could also look at Yifan Yang, Transformation formulas for generalized Dedekind eta functions. This is probably still complicated. I don't know what "easy" means to Siegel. Commented May 17, 2021 at 9:01
• There's a nice answer here for a similar question: mathoverflow.net/questions/297049/… In particular there's reference to a book by Ken Ono containing some theorems that might interest you. Commented May 17, 2021 at 15:56

Here's a fairly straightforward way to show that $$\phi$$ is modular of level $$5$$ using Siegel functions.

Claim: The function $$f(\tau)$$ is a modular function for $$\Gamma(5)$$ if and only if $$f(5\tau)$$ is a modular function for $$\Gamma_{0}(25) \cap \Gamma_{1}(5)$$. (This is straightforward to prove using properties of the slash operator.)

Using this claim, it's not hard to see that $$\eta(\tau/5)/\eta(5\tau)$$ is a modular function of level $$5$$, using the standard result of Gordon, Hughes and Newman about when an eta quotient is modular of level $$N$$. (This result can be found in the Wikipedia article about the Dedekind $$\eta$$-function.)

It remains to show that $$\frac{\eta(\frac{\tau+1}{5})\eta(\frac{\tau-1}{5})}{\eta(\frac{\tau+2}{5})\eta(\frac{\tau-2}{5})}$$ is a modular function of level $$5$$.

The Siegel functions $$g_{(a_{1}/N,a_{2}/N)}$$ can be used to build modular functions of level $$N$$. (See Section 5 of the article here by Amanda Folsom that gives a product expansion of the Siegel functions and criteria of Kubert and Lang that indicate when a product of Siegel functions is modular.) The product formula implies that $$g_{(0,a_{2}/N)}(\tau) = c \eta(\tau + a_{2}/N) \eta(\tau - a_{2}/N)$$ for some constant $$c$$. The modularity criteria imply that $$h(\tau) = \frac{g_{(0,1/5)}(\tau)}{g_{(0,2/5)}(\tau)} = \frac{\eta(\tau + 1/5) \eta(\tau - 1/5)}{\eta(\tau + 2/5) \eta(\tau - 2/5)}$$ is a modular function for $$\Gamma(5)$$. Now $$h(\tau) = h(\tau + 1)$$ and so $$h(\tau)$$ is a modular function for $$\left\langle \Gamma(5), \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\right\rangle = \Gamma_{1}(5)$$. Since $$\Gamma_{0}(25) \cap \Gamma_{1}(5) \subset \Gamma_{1}(5)$$, the claim above implies that $$h(\tau/5)$$ is a modular function for $$\Gamma(5)$$ and this proves that $$\phi$$ is a modular function of level $$5$$.

As a note related to your first question about $$\phi^{24}$$ being modular of level $$5$$ implying $$\phi$$ is modular of level $$5$$, Theorem 1 of Kubert and Lang's paper here is the following. Let $$N$$ be a prime power and $$U_{N}$$ be the group of modular units of level $$N$$ that are generated by the Siegel functions. If $$g$$ is a modular function and there is a positive integer $$k$$ so that $$g^{k} \in U_{N}$$, then $$g \in U_{N}$$.

• Thank you for your answer! I appreciate the reference to Kubert-Lang. Commented May 23, 2021 at 21:52
• Do I understand correctly, that if $\ell>3$ is a prime, all modular units of level $\ell$ are generated by the Siegel functions, so we may apply this to the function $\phi$ and similar eta-quotients of higher (prime) level? Commented May 23, 2021 at 22:41
• Yes, that is correct. The modular units in $U_{\ell}$ have full rank within the group of level $\ell$ modular units. This forces $U_{\ell}$ to have finite index in the group of modular units, and Theorem 1 of Kubert and Lang's paper implies the index is $1$. Commented May 23, 2021 at 23:54