Weierstrass points on certain modular groups

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• 作者：Bo-Hae Im^a ; ¹ ; ^{imbh@cau.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Daeyeol Jeon^b ; ² ; ^{dyjeon@kongju.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Chang Heon Kim^c ; ³ ; ^{chhkim@skku.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 14H55 ; secondary ; 11F06 ; 11G18
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：160
• 期：Complete
• 页码：586-602
• 全文大小：353 K

文摘

We investigate Weierstrass points of the modular curve X_Δ(N)$X_{\mathrm{\Delta }}(N)$ of genus ≥2 when Δ is a proper subgroup of (Z/NZ)^⁎${(\mathbb{Z}/N\mathbb{Z})}^{⁎}$. Let N=p²M$N=p^{2}M$ where p is a prime number and M   is a positive integer. Modifying Atkin's method in the case ±(1+pM)∈Δ$\pm (1+pM)\in \mathrm{\Delta }$, we find conditions for the cusp 0 to be a Weierstrass point on the modular curve X_Δ(p²M)$X_{\mathrm{\Delta }}(p^{2}M)$. Moreover, applying Schöneberg's theorem we show that except for finitely many N  , the fixed points of the Fricke involutions W_N$W_N$ are Weierstrass points on X_Δ(N)$X_{\mathrm{\Delta }}(N)$.