Gonality of the modular curve \(X_1(N)\)
For a curve \(C\) defined over a field \(K\), let \(K(C)\) be the associated function field and define the \textit{gonality} of \(C\) over \(K\) as NEWLINE\[NEWLINE \mathrm{Gon}_K(C):=\min\{\deg(f)\,:\,f\in K(C)-K\}\;.NEWLINE\]NEWLINE The paper deals with the modular curves \(X_1(N)\) over the fields \(\mathbb{Q}\) and \(\mathbb{F}_p\) (\(p\) any prime), and uses \texttt{MAGMA} computations and the relation NEWLINE\[NEWLINE \mathrm{Gon}_{\mathbb{F}_p}(C) \leqslant \mathrm{Gon}_{\mathbb{Q}}(C) \qquad\qquad\qquad\qquad\qquad (*)NEWLINE\]NEWLINE (valid for \(C\) defined over \(\mathbb{Q}\) and a prime of good reduction \(p\)) to compute their gonalities for \(N\leqslant 40\).NEWLINENEWLINE\noindent The authors look for low degree functions in the modular units of \(\mathbb{Q}(X_1(N))\) (i.e., functions whose zeroes and poles are cusps) and, building on earlier work of \textit{A. V. Sutherland} [Math. Comput. 81, No. 278, 1131--1147 (2012; Zbl 1267.11074)], find upper bounds for \(\mathrm{Gon}_{\mathbb{Q}}(X_1(N))\). Then they use some reduction steps (explicitly described for \(N=37\) and \(p=2\), the case with the highest gonality in the range \(N\leqslant 40\)) to make \texttt{MAGMA} computations for \(\mathrm{Gon}_{\mathbb{F}_p}(X_1(N))\) feasible for the lowest possible \(p\). By \((*)\), this provides lower bounds which coincide with the previous upper ones for all \(N\leqslant 40\).NEWLINENEWLINE\noindent The gonalities are used in the last section to provide a complete list of all the \(N\) for which \(X_1(N)\) has infinitely many places of degree \(d\) over \(\mathbb{Q}\) for \(5\leqslant d\leqslant 8\) (the values for \(1\leqslant d \leqslant 4\) where already known).
scientific article; zbMATH DE number 6339748