We study a class of curves which includes the hyperelliptic curves and the Fermat curves of prime degree. We compute their Hasse-Witt matrix when the curves are defined over an algebraically closed field of positive characteristic. In particular, we get a formula for the Hasse-Witt invariant of the Fermat curves at each prime not equal to the degree. These invariants depend only on residue degrees.
We show that there are infinitely many Fermat curves for which there exists a set of primes p with positive density such that the geometric fibre at p of the Fermat Jacobian is not isogenous to a power of a supersingular elliptic curve, but the Hasse-Witt invariant at p is equal to zero.

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We regard a closed 2-form on a Riemannian manifold as a magnetic field and define a magnetic flow which is a perturbation of a geodesic flow. A sufficient condition is given for a magnetic flow to become an Anosov flow.

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Smooth actions of non-compact semi-simple Lie groups have been considered by Asoh, Mukoyama and others, in the case where the actions restricted to the maximal compact subgroup have codimension-one principal orbits. In this paper, we consider such actions on the (ρ+1)-sphere for Lorentz group of type (ρ, 2).

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For any totally real number field k and any prime number p, the Iwasawa lambda-invariant and the mu-invariant are conjectured to be both zero. We give a new efficient method to verify this conjecture for certain real abelian fields. The new features of our method compared with other existing ones are that we use effectively cyclotomic units and that we introduce a new way to apply p-adic L-functions to the conjecture.

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We consider here a general Lotka-Volterra type n-dimensional periodic functional differential system. Sufficient conditions for the existence, uniqueness and global asymptotic stability of periodic solutions are established by combining the theory of monotone flow generated by FDEs, Horn's asymptotic fixed point theorem and linearized stability analysis. These conditions improve and generalize the recent ones obtained by Freedman and Wu (1992) for scalar equations. We also present a nontrivial application of our results to a delayed nonautonomous predator-prey system.

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We will give an explicit description of extensions of the group scheme of Witt vectors of length n (resp. the formal group of Witt vectors of length n) by the multiplicative group scheme (resp. the multiplicative formal group) over an algebra for which all prime numbers except a given prime p is invertible.

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We study self-duality of pseudo-Riemannian metrics of type (2, 2). We show correspondences between self-duality of Riemannian metrics and that of pseudo-Riemannian metrics of type (2, 2) under appropriate conditions. Moreover we give global constructions for four-dimensional manifolds with self-dual metrics of type (2, 2).

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Lagrangian H-umbilical submanifolds are the "simplest" Lagrangian submanifolds next to totally geodesic ones in complex-space-forms. The class of Lagrangian H-umbilical submanifolds in complex Euclidean spaces includes Whitney's spheres and Lagrangian pseudo-spheres. For each submanifold M of Euclidean n-space and each unit speed curve Fin the complex plane, we introduce the notion of the complex extensor of M in the complex Euclidean n-space via F. The main purpose of this paper is to classify Lagrangian H-umbilical submanifolds of the complex Euclidean n-space by utilizing complex extensors. We prove that, except the flat ones, Lagrangian H-umbilical submanifolds of complex Euclidean n-space with n greater than 2 are Lagrangian pseudo-spheres and complex extensors of the unit hypersphere of the Euclidean n-space. For completeness we also include in the last section the classification of flat Lagrangian H-umbilical submanifolds of complex Euclidean spaces.

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