Given a finite metacyclic group G, a central extension F having the projective lifting property over all fields is constructed. This extension and its group rings are used to investigate the faithful irreducible projective representations of G and the fields over which they can be realized. A full description of the finite metacyclic groups having central simple twisted group rings over fields ...

We show that each Jordan homomorphism R→ R′ of rings gives rise to a harmonic mapping of one connected component of the projective line over R into the projective line over R′. If there is more than one connected component then this mapping can be extended in various ways to a harmonic mapping which is defined on the entire projective line over R. Mathematics Subject Classification (2000): 51C0...

Let X C ]p'N be a projective variety defined over the Galois field GF(q). Denote by X(q) the set of GF(q)-rational points of X. Let k be an integer. We say that the pair (X, X(q)) satisfies the Finite Field Nullstellensatz of order k, (the FFN(k), for short), if every homogeneous form of degree :::; k on r N (J<) vanishing on X (q), vanishes on X (J<). Here, we prove the Finite Field Nullstelle...

1. Rational points. A classical conjecture of Mordell states that a curve of genus ^ 2 over the rational numbers has only a finite number of rational points. Let K be a finitely generated field over the rational numbers. Then the same statement should hold for a curve defined over K, and a specialization argument due to Néron shows in fact that this latter statement is implied by the correspond...

We determine for all simple simply connected reductive linear algebraic groups defined over a finite field all irreducible representations in their defining characteristic of degree below some bound. These also give the small degree projective representations in defining characteristic for the corresponding finite simple groups. For large rank l our bound is proportional to l3 and for rank 11 m...

In this note, we prove that for any finite dimensional vector space V over an algebraically closed field k, and for any finite subgroup G of GL(V ) which is either solvable or is generated by pseudo reflections such that the |G| is a unit in k, the projective variety P(V )/G is projectively normal with respect to the descent of O(1)⊗|G|.

Let X be a smooth projective variety, defined over an algebraically closed field of positive characteristic, such that the tangent bundle TX is trivial. Let FX : X −→ X be the absolute Frobenius morphism of X. We prove that for any n ≥ 1, the n–fold composition Fn X is a torsor over X for a finite group–scheme that depends on n. For any vector bundle E −→ X, we show that the direct image (Fn X)...

The third part of J.-P. Serre’s book Linear Representations of Finite Groups discusses of modular representations, i.e., representations of finite groups over a field of nonzero characteristic. The aim of this work is to complement Serre’s work by introducing the reader to some noncommutative algebra used in the study of modular representations, and in particular to the theory of semisimple and...

Fix a number field k. We prove that if there is an algorithm for deciding whether a smooth projective geometrically integral k-variety has a k-point, then there is an algorithm for deciding whether an arbitrary k-variety has a k-point and also an algorithm for computing X(k) for any k-variety X for which X(k) is finite. The proof involves the construction of a one-parameter algebraic family of ...