On a question of B. Mazur (Q5906627)

This paper is concerned with a question due to \textit{B. Mazur} with respect to the existence of a universal upper bound for \(\# (Y \cap \Gamma)\), where \(\Gamma\) is a finite rank subgroup of a complex torus \(T\) and \(Y\) is an embedded closed Riemann surface in \(T\). The meaning for ``universal'' is in the sense that such an upper bound depends only on the genus of \(Y\) and the rank of \(\Gamma\). -- The author shows the existence of such an universal upper bound in the case that \(Y\) carries no moduli, equivalently in the language of complex algebraic geometry, that the algebraic curve \(Y\) does not descend to the field \(\overline \mathbb{Q}\). The proof of this existence is essentially reduced to the case when \(T\) is the Jacobian of \(Y\).