Decomposable forms and automorphisms

Abstract

In the late 19th century Jordan initiated the study of forms of higher degree and derived (see Memoire sur l'equivalence des formes, Oeuvres III, Gauthier Villars, Paris, 1962) the finiteness of the automorphism group Aut( f) of complex forms of degree ⩾3 and non-zero discriminant. This result has been extended to forms over arbitrary fields by Schneider (J. Algebra 27 (1973) 112), see also Curtis and Reiner (Representation Theory of Finite Groups and Associative Algebras, Wiley, New York, 1962) for related topics. Orlik and Solomon gave some bounds for the cardinality of Aut( f) using cohomological arguments in Orlik and Solomon (Math. Ann. 231 (1978) 229); besides this, little seems to be known about this group in general. In connection with his study (Monatsh. Math., submitted for publication) of representations of forms by linear forms, the author was led to an investigation of the group of automorphisms of decomposable forms f through the permutations of the linear factors these automorphisms induce. The main result (Theorem 4.2 in Chapter 4) states that almost all forms in k⩾2 variables of degree d⩾max{5, k+2} have only the trivial automorphisms that consist in multiplying each variable by the same dth root of unity. The case k=2, d=4 has already been studied (see Survey in Algebraic Geometry, Part 2, Invariant theory, Encyclopaedia of Mathematical Sciences, Vol. 55, Springer, Berlin, 1994); however, it is treated in full detail to illustrate the elaborated techniques. The first chapters are devoted to the proof of some general results concerning the structure of the permutation group associated to a form f which also help to understand the case of forms with non-trivial automorphisms. In a few special cases, this allows to determine this group explicitly; in general we give a bound for the cardinality of Aut( f) depending only on the degree of f which is relevant for some diophantine problems (see e.g. Ann. Math. 155 (2002) 553). The author is indebted to G. Wuestholz for his substantial help and encouragement during the redaction of the paper, he also wishes to thank V. Popov for several helpful remarks.