2
$\begingroup$

Let $\gamma_2(\tau)=j(\tau)^{1/3}$. The modular equation shows that the functions $$j\left(\frac{a\tau+b}{c\tau+d}\right),\qquad ad-bc=n$$ are integral over $\mathbf Z[j]$. Under what conditions is the function $$\gamma_2\left(\frac{a\tau+b}{c\tau+d}\right)$$ integral over $\mathbf Z[\gamma_2]$?

Weber shows in Lehrbuch der Algebra that this is true for the functions $$\gamma_2\left(\frac{a\tau+b}{d}\right)$$ where $$\begin{pmatrix} a & b\\ 0 &d\end{pmatrix}\in\mathcal S(n,d)=\bigg\lbrace \begin{pmatrix} r & s\\ 0 &t\end{pmatrix}\colon rt=n,r>0,(r,s,t)=1,3\mid s,0\leq s<3t\bigg\rbrace,$$ for $d=3$ and $(3,n)=1$.

The set $\mathcal S(n,d)$ is a set of representatives for the orbits of the action $SL_2(\mathbf Z)\backslash\lbrace \text{matrices with determinant n}\rbrace$ provided $(n,d)=1$. Unfortunately the function $\gamma_2$ is not invariant under $SL_2(\mathbf Z)$ but only under the subgroup

$$\bigg\lbrace \begin{pmatrix} a & b\\ c &d\end{pmatrix}\colon a\equiv d\equiv 0 \text{ or }b\equiv c \text{ mod 3}\bigg\rbrace$$ so we cannot proceed as with the $j$-invariant.

How did Weber come up with the set $S(n,d)$? For $d=16$ this set also occurs in Stark's On the “gap” in a theorem of Heegner. Is there some other reference for $\mathcal S(n,d)$?

$\endgroup$
1
$\begingroup$

Since nobody has answered yet, I'll try it. I think that if $j(M \cdot \tau)$ satisfies the $n$-th modular equation $\Phi_n(X) = 0$ over $\mathbb{Z}[j]$ (where $\mathrm{det}(M) = n$) then $\gamma_2(M \cdot \tau)$ satisfies $\Phi_n(X^3) = 0$, also over $\mathbb{Z}[j]$ and therefore over the larger ring $\mathbb{Z}[\gamma_2]$. So it is always algebraic over $\mathbb{Z}[\gamma_2]$.

It may be better to view $\gamma_2$ as a modular form of level 1 with character. You can read the character off of the character of $\eta(\tau)$ because $\gamma_2(\tau) = E_4(\tau) \eta(\tau)^{-8}$. In particular the subgroup that you wrote under which $\gamma_2$ is invariant is too small; it should be the group generated by $S$ and $T^3$.

I do not know whether there is a name for elements of $\mathcal{S}(n,d)$ in general. If $(n,d) = 1$ then these are the matrices in Hermite normal form.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.