Cusp forms of weight 1 associated to Fermat curves
Let \(\Delta\) be the free subgroup of the principal congruence subgroup \(\Gamma (2)\) generated by \(A= \left(\begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix} \right)\) and \(B = \left (\begin{smallmatrix} 1 & 0 \\ 2 & 1 \end{smallmatrix} \right)\) (one has \(\Gamma(2) = \{\pm I\} \Delta)\), and for \(N\in\mathbb{N}\) let \(\Phi(N)\) be the subgroup of \(\Delta\) generated by \(A^N\) and \(B^N\) and the commutator \([\Delta,\Delta]\). Then the associated modular curve \(X(\Phi (N))\) is isomorphic to the Fermat curve \(X^N + Y^N=Z^N\). As is well-known, \(\Phi(N)\) is not a congruence subgroup unless \(N=1,2,4,8\). Using the above isomorphism the author constructs a canonical basis for the space of modular forms and cusp forms of weight 1 for \(\Phi(N)\), and, as a consequence, obtains explicit dimension formulas for these spaces. One should note that the classical Riemann-Roch theorem in general only gives explicit dimension formulas for spaces of cusp forms of weight \(k\geq 2\).
canonical basis for the space of cusp forms of weight 1
canonical basis for the space of modular forms
scientific article; zbMATH DE number 912123