Bielliptic and hyperelliptic modular curves \(X(N)\) and the group \(\Aut(X(N))\) (Q2868212)

If \(N \geq 1\) is an integer, then the modular group \(\Gamma(N)\) consists of the Möbius transformations \(A(z)=(az+b)/(cz+d)\), where \(a,b,c,d \in {\mathbb Z}\), \(ad-bc=1\), \(a,d \equiv 1\) mod \(N\) and \(b,c \equiv 0\) mod \(N\). The modular group \(\Gamma(N)\) acts as a discrete group of holomorphic automorphisms of the upper-half plane \({\mathbb H}=\{z \in {\mathbb C}: {\mathrm{Im}}(z)>0\}\). Note that \(\Gamma(1)=\mathrm{PSL}(2,{\mathbb Z})\) and that \(\Gamma(N)\) is a normal subgroup of \(\Gamma(1)\) of index \(\delta_{N}= (N^{3}/2) \prod_{p/N}(1-p^{-2})\), \(N>2\), and \(\delta_{2}=6\).NEWLINENEWLINEIf \(N \geq 2\), then \(Y(N)={\mathbb H}/\Gamma(N)\) is an analytically finite Riemann surface with punctures and \(Y(1)\) is an orbifold of genus zero with two cone points, one of order two and the other of order three, and exactly one puncture. If \(N \geq 2\), by filling the punctures of \(Y(N)\), one obtains a closed Riemann surface \(X(N)\); of genus \(g_{N}=1+\delta_{N}(N-6)/12N\). In particular, \(X(N)\) has genus zero for \(N \in \{2,3,4,5\}\), \(X(6)\) has genus one and, for \(N \geq 7\), \(X(N)\) has genus \(g_{N}>1\).NEWLINENEWLINESet \(G_{N}=\Gamma(1)/\Gamma(N) \cong \mathrm{PSL}_{2}({\mathbb Z}/N{\mathbb Z})\); a group of holomorphic automorphisms of \(X(N)\). Theorem 3.1 of the paper under review, states that \(G_{N}\) is the full group of holomorphic automorphisms of \(X(N)\), for \(N \geq 7\). The proof of this fact is done over three and half pages. Another proof can be done as follows. The quotient \(X(N)/ G_{N}\) is the Riemann sphere with exactly three cone points, whose orders are \(2\), \(3\) and \(N \geq 7\). Since the triangular signature \((2,3,N)\), \(N \geq 7\), is maximal [\textit{D. Singerman}, J. Lond. Math. Soc., II. Ser. 6, 29--38 (1972; Zbl 0251.20052)], there are no more holomorphic automorphisms of \(X(N)\) outside \(G_{N}\).NEWLINENEWLINETheorem 4.1 states that, for \(N \geq 7\), the Riemann surface \(X(N)\) is non-hyperelliptic. The main idea is to observe that if \(X(N)\) is assumed to be hyperelliptic, say with hyperelliptic involution \(\iota\), then the group \(H_{N}=G_{N}/\langle \iota \rangle\) is a finite group of the Möbius group \(\mathrm{PSL}_{2}({\mathbb C})\) that must preserve a collection of \(2g_{N}+2\) points of order two. Since the total quotient must produce only three cone points, of orders \(2\), \(3\) and \(N \geq 7\), and the quotient of the Riemann sphere by a finite subgroup of \(\mathrm{PSL}_{2}({\mathbb C})\) has either two cone points with the same order, or three cone points with tuples of orders either \((2,2,m)\) or \((2,3,3)\) or \((2,3,4)\) or \((2,3,5)\), one gets a contradiction.NEWLINENEWLINETheorem 4.2 states that, for \(N \geq 7\), the Riemann surface \(X(N)\) is bielliptic only for \(N\in \{7,8\}\). A closed Riemann surface \(S\) is bielliptic if it has an automorphism \(h\) of order two so that \(S/\langle h \rangle\) has genus one. The proof is also obtained by looking at the fixed points the involution \(h\). The Riemann surface \(X(7)\) is the first Hurwitz curve; this being the Klein curve of genus three defined by the quartic equation NEWLINE\[NEWLINEx^{3}y+y^{3}z+z^{3}x=0.NEWLINE\]NEWLINE The Riemann surface \(X(8)\) is the Wiman curve defined by the equations NEWLINE\[NEWLINEx_{0}^{2}=x_{3}x_{4}, \; x_{3}^{2}=4x_{1}^{2}+x_{2}^{2}, \; x_{4}^{2}=x_{1}x_{2}.NEWLINE\]NEWLINE The last part of paper is concerned with rational points on \(X(N)\). For example, Theorem 5.2 states that, for \(N \geq 7\), the set of points of \(X(N)\) that are rational over quadratic extensions of \({\mathbb Q}(e^{2 \pi i/N})\) is always finite.