On the minus parts of classical Poincar\'e series

초록

Let $S_k(N)$ be the space of cusp forms of weight $k$ for ${\Gamma }_0(N)$. We show that $S_k(N)$ is the direct sum of subspaces $S_k^{+}(N)$ and $S_k^{-}(N)$. Where $S_k^{+}(N)$ is the vector space of cusp forms of weight $k$ for the group ${\Gamma }_0^{+}(N)$ generated by ${\Gamma }_0(N)$ and $W_N$ and $S_k^{-}(N)$ is the subspace consisting of elements $f$ in $S_k(N)$ satisfying $f{|}_k{W}_N=-f$. We find generators spanning the space $S_k^{-}(N)$ from Poincar\'e series and give all linear relations among such generators.

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