On Hecke’s decomposition of the regular differentials on the modular curve of prime level

Abstract

In this paper, we review Hecke’s decomposition of the regular differentials on the modular curve of prime level p under the action of the group $${{\mathrm{SL}}}_2(p)/\langle {\pm 1}\rangle $$ SL 2 ( p ) / ⟨ ± 1 ⟩ . We show that his distinguished subspace corresponds to a factor of the Jacobian which decomposes as a product of conjugate, isogenous elliptic curves with complex multiplication.