Computing the determinant of the operator \(U_{p}\)
Graduate project, Sun Yat-sen University, 2025
X. Huang, T. Huber and D. Ye, On the determinant of \(U_{p}\) on \(M_{k}(p,\chi)\) Ramanujan J. 67 (2025), no. 2, 39.
Project description
Denote by \(M_{k}(p,\chi)\) the vector space over \(\mathbb{C}\) spanned by the holomorphic modular forms of weight~\(k\) and level~\(\Gamma_{0}(p)\) with Dirichlet character~\(\chi\). It is known by the Riemann–Roch Theorem that such a vector space is of finite dimension. Now for a given modular form
\[f(\tau)=\sum_{n=0}^{\infty}a_{n}q^{n},\]define an operator \(U_{p}\) on \(f(\tau)\) by
\(U_{p}(f)=\sum_{n=0}^{\infty}a_{pn}q^{n}.\) Such an operator actually defines a linear automorphism of \(M_{k}(p,\chi)\). In this project, we explicitly formulate the determinant of \(U_{p}\) in terms of the data \((k,p,\chi)\). As an implication, we fully confirmed all the conjectures of the second author Tim Huber posed in 2014; for example, in one of the conjectures it was asserted that
\[|\det(U_{3})|=3^{w(d)},\] where
$$
w(d)=\begin{cases}
\frac{(3d-1)(d+1)}{2} &\text{ if } 2\nmid d,\\
\frac{6d^2+3d}{4} &\text{ if } d\equiv0\pmod{4},\\
\frac{6d^2+3d-2}{4} &\text{ if } d\equiv2\pmod{4}.
\end{cases}
$$