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Arithmetic of cyclic quotients of the Fermat quintic. (English) Zbl 0923.11096

Let \(F\) be the Fermat quintic projective curve given by the equation \(X^5+Y^5+Z^5=0\) and \(J\) as its Jacobian. Let \(\zeta\) be a primitive 5th root of 1 and \(K=\mathbb{Q}(\zeta)\). We denote by \(\sigma\) and \(\tau\) the automorphisms of \(F\) given by \(\sigma(X,Y,Z)=(\zeta X,Y,Z)\) and \(\tau(X,Y,Z)= (X,\zeta Y,Z)\) respectively, and by \(F_s\) the quotient of \(F\) by the group of automorphisms generated by \(\sigma\tau^{-s}(s=1,2,3)\). Let \(J_s\) be the Jacobian of \(F_s\) and \(f_s:J\to J_s\), \(f^*_s:J_s\to J\) be the natural projection map and its dual respectively \((s=1,2,3)\). The map \[ f=\prod^3_{s=1} f_s:J\to \prod^3_{s=1}J_s \] is an isogeny and the dual isogeny is given by \[ f^*=\sum^3_{s=1} f^*_s:\prod^3_{s=1}J_s\to J. \] In the paper under review the author gives explicit generators for the Mordell-Weil group \(J_s(K)\) and for the kernel of \(f^*\). Furthermore, using a result of C.-H. Lim [J. Number Theory 41, 102-115 (1991; Zbl 0791.14010)], he describes the action of \(\text{End}(J)\) on \(\text{Ker}(f^*)\).

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H25 Arithmetic ground fields for curves
14H40 Jacobians, Prym varieties

Citations:

Zbl 0791.14010
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