DATA : The Dimensions for the Atkin-Lehner eigenspaces

Let $\mbox{Ex}(N) = \{ e_{0} =1, e_{1}, e_{2}, \cdots \}$ be a group of exact divisors of $N$ under the opreration $e * e’ = ee’ / \gcd(e,e’)^2$.

By a sign pattern $\varepsilon$ for $N$ is meant a group homomorphism
$$\varepsilon : \mbox{Ex}(N) \to \{\pm 1\}, \qquad \varepsilon(1) =1.$$

For a sign pattern $\varepsilon$ of $N$, let $M_{k}^{\varepsilon}(N)$ (resp. $S_{k}^{\varepsilon}(N)$) be a subset of $M_{k}(N)$ (resp. $S_{k}(N)$) consisitng of all forms $f$ such that the Atkin-Lehner eigenvalue of $f$ is $\varepsilon(p)$ for every $p |N$, that is

\begin{eqnarray*}
M_{k}^{\varepsilon}(N) &:=& \{ f \in M_{k}(N) : f \mid_{k} W_{p} = \varepsilon (p) f \quad \mbox{for every prime}~ p|N\}, \cr
S_{k}^{\varepsilon}(N) &:=& \{ f \in S_{k}(N) : f \mid_{k} W_{p} = \varepsilon (p) f \quad \mbox{for every prime}~ p|N\}.
\end{eqnarray*}

Similarly we can define $M_{k}^{!,\varepsilon}(N)$ by the subspace of the weakly holomorphic modular form space $M_{k}^{!}(N)$ with a sign pattern $\varepsilon$.

For $f \in M_{k}^{!}(N)$, it follows easily that $f$ can be expressed as $f = \sum_{\varepsilon} \frac{\left( \sum_{d \mid N} \varepsilon(d) \cdot f \mid_{k} W_{d} \right) }{2^{r}}$, where $\varepsilon$ runs over the all sign patterns of $N$.

Then we easily see that

\begin{eqnarray*}
\left( \sum_{d \mid N} \varepsilon(d) \cdot f \mid_{k} W_{d} \right) \mid_{k} W_{e} &=& \sum_{d \mid N} \varepsilon(d) \cdot f \mid_{k} W_{d} W_{e}, \cr
&=& \varepsilon(e) \sum_{d \mid N} \varepsilon(d) \varepsilon(e) \cdot f \mid_{k} W_{d} W_{e} ,\cr
&=& \varepsilon(e) \sum_{d \mid N} \varepsilon \left( \frac{de}{(d,e)^{2}}\right) \cdot f \mid_{k} W_{\frac{de}{(d,e)^{2}}} ,\cr
&=& \varepsilon(e) \sum_{d \mid N} \varepsilon \left( d*e\right) \cdot f \mid_{k} W_{d*e}.
\end{eqnarray*}

Hence each numerator $\left( \sum_{d \mid N} \varepsilon(d) \cdot f \mid_{k} W_{d} \right) $ of summand lies in $M_{k}^{!,\varepsilon}(N)$.

Now we have following decomposition

$$M_{k}^{!}(N) =\bigoplus_{\varepsilon} M_{k}^{!,\varepsilon}(N)$$

for a squarefree integer $N$ with $r$ distinct prime factors, where $\varepsilon$ runs over the all sign patterns $\varepsilon$ of $N$.

To denote a specific sign pattern $\varepsilon$, we also use a tuple of the signs of the Atkin-Lehner eigenvalues for each of the prime factors.

Let $N=p_{1}p_{2}\cdots p_{r}$ be a squarefree integer with $r$ distinct prime factors $p_{1}, p_{2}, \ldots, p_{r}$, where we assume that $p_{i}< p_{j}$ if $i<j$.

Then, we denote the sign pattern $\varepsilon$ by $(\mathfrak{s}_{p_{1}}, \ldots, \mathfrak{s}_{p_{r}})$ where $\mathfrak{s}_{p_{i}}$ represents the sign of $\varepsilon(p_{i})$.

For concrete example, let $N=21=3 \times 7$. Then we have

$$M_{k}^{!}(21)=M_{k}^{!,(+,+)}(21) \oplus M_{k}^{!,(+,-)}(21) \oplus M_{k}^{!,(-,+)}(21) \oplus M_{k}^{!,(-,-)}(21).$$

For $f \in M_{k}^{!}(21)$,

\begin{align*}
f= &\frac{f+f \mid W_{3} + f \mid W_{7} +f \mid W_{21} }{4} +\frac{f-f \mid W_{3} + f \mid W_{7} -f \mid W_{21} }{4} \cr &+\frac{f+f \mid W_{3} – f \mid W_{7} -f \mid W_{21} }{4}+\frac{f-f \mid W_{3} – f \mid W_{7} +f \mid W_{21} }{4},
\end{align*}

and

\begin{align*}
&\frac{f+f \mid W_{3} + f \mid W_{7} +f \mid W_{21} }{4} \in M_{k}^{!,(+,+)}(21) ,
& \frac{f-f \mid W_{3} + f \mid W_{7} -f \mid W_{21} }{4} \in M_{k}^{!,(-,+)}(21), \cr
&\frac{f+f \mid W_{3} – f \mid W_{7} -f \mid W_{21} }{4} \in M_{k}^{!,(+,-)}(21),
&\frac{f-f \mid W_{3} – f \mid W_{7} +f \mid W_{21} }{4} \in M_{k}^{!,(-,-)}(21).
\end{align*}

We exhibit the dimensions for the space $S_{k}^{\varepsilon}(N)$ for squarefree integers $N$ for which the genus of $X_{0}^{*}(N)$ is 0 or 1 : See;

Last modified : 2024. 8. 9.