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Let $N \in \mathbb{N}$, and let $Q \Vert N$ be a prime power dividing $N$ and such that $(Q,N/Q)=1$.

Let $x,y,z,t \in \Z$ be such that the determinant of \[ w_Q=\left( \begin{matrix} Qx & y \\ Nz & Qt\end{matrix} \right) \] is $Q$. Then $w_Q$ defines an involution $W_Q$ of the modular curve $X_0(N)$ which does not depend on the choice of $x,y,z$ and $t$. Equivalently, $w_Q$ acts by conjugation on $\Gamma_0(N)$.

This involution commutes with the Hecke operators, and so any eigenform for these operators is also an eigenform for $W_Q$, with eigenvalue $\pm 1$.

If a cusp of $\Gamma_0(N)$ is of the form $Q/N$ and $\mathrm{gcd}(Q,N/Q)=1$ we can choose an Aktkin-Lehner map $W_Q$ as a cusp-normalizing map for this cusp, that is $W_Q(\infty)=Q/N$. If $h$ denotes the width of the cusp $\frac{Q}{N}$ then $W_Q T^h W_{Q}^{-1}$ is a generator of the stabilizer of $\frac{Q}{N}$ in $\Gamma_{0}(N)$.

Knowl status:
  • Review status: reviewed
  • Last edited by Andrew Sutherland on 2020-10-22 14:45:58
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