Let H be a semisimple Hopf algebra over an

algebraically closed field k of characteristic 0, then H is commutative if and only if [z.sub.2] [member of] k.

Let n be a positive integer, and let V be a 2n-dimensional vector space over an

algebraically closed field K of characteristic 0, with a fixed basis [e.sub.1], [e.sub.2], ..., [e.sub.n], [e.sub.1]*, [e.sub.2]*, ..., [e.sub.n]*.

A Lie algebra G over an

algebraically closed field of characteristic 0 is said to be Frobenius if there exists a linear form f [member of] [G.sup.*] such that the bilinear form [[PHI].sub.f] f on G is nondegenerate.

In this paper we shall consider a moduli space [M.sub.G] of stable principal G-bundles over a smooth projective variety X of dimension n, defined over an

algebraically closed field k of characteristic 0, and we shall describe a natural procedure which leads to the construction of closed differential forms on [M.sub.G] starting with some cohomology classes on X.

Our approach was suggested by a reexamination of the trace map Tr for projective curves X over

algebraically closed fields: Note that for n invertible on X, Tr is the composition:

4.2 hold true over any

algebraically closed field k.

Let k be an

algebraically closed field of characteristic 0, let A be a reduced Noetherian k-algebra, and let (M, [nabla]) be a finitely generated torsion free A-module of rank one with a (not necessarily integrable) connection.

This graduate textbook introduces the theory of error-correcting codes, working over an arbitrary finite field; algebraic curves, working over an

algebraically closed field; geometry over a finite field; and constructions of algebraic geometry codes.

Keywords abc-theorem; abc-conjecture;

algebraically closed field; Wronskian; Diophantine equations.

Let [G.sub.0] be a one-parameter formal group law of height 2 over the

algebraically closed field k of characteristic p.

Let k be an

algebraically closed field of characteristic zero, which is complete for a non-trivial non-Archimedean absolute value [absolute value of *].

A numerical Godeaux surface is a minimal surface X of general type over an

algebraically closed field with [K.sup.2.sub.X] = 1 and [chi][O.sub.X] = 1.