In this paper we determine all modular curves X(N ) (with N ≥ 7) that are hyperelliptic or bielliptic. We also give a proof that the automorphism group of X(N ) is the group PSL 2 (Z/N Z), therefore it coincides with the normalizer of Γ(N ) in PSL 2 (R) modulo ±Γ(N ).
Abstract. We say that an abelian variety over a p-adic field K has anisotropic reduction (AR) if the special fiber of its Néron minimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the K-rational torsion subgroup of a g-dimensional AR variety depending only on g and the numerical invariants of K (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of g, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an AR abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.
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