Let \(G\) be a finite group of Lie type in prime characteristic \(p\) and let \(\chi\) be an irreducible complex character of \(G\). A basic question in the representation theory of \(G\) is to decide whether or not \(\chi\) is irreducible modulo \(p\). Most of the experimental evidence suggests that \(\chi\) is rarely irreducible modulo \(p\) and the results of this paper provide theoretical confirmation of this observation. The authors' main results are not a complete solution of the question raised, as certain technical difficulties arise, but nonetheless the theorems proved are quite striking. The authors require a certain integrality property of \(\chi\) to make their approach succeed. We say that \(\chi\) is unramified above \(p\) if the values taken by \(\chi\) lie in an unramified extension of the field \(\mathbb{Q}_p\) of \(p\)-adic numbers. (This is equivalent to saying that \(\chi\) is \(p\)-rational in the usual terminology.) Lemma 3.3 of the paper shows that if \(\chi\) is unramified above \(p\), it may be realized as the character of an \(RG\)-lattice, \(L\), say, where \(R\) is the ring of integers of a finite unramified extension of \(\mathbb{Q}_p\). (This is a general result for all finite groups.) A clever idea used by the authors is then to apply Diederichsen's classification of indecomposable \(R\mathbb{Z}_p\)-lattices, work that dates back to 1938. Here, \(\mathbb{Z}_p\) denotes a cyclic group of order \(p\). Up to isomorphism, there are three indecomposable \(R\mathbb{Z}_p\)-lattices: the trivial one, the regular one, and one of rank \(p-1\). If now \(\overline L\) denotes the reduction of \(L\) modulo \(p\), the action on \(\overline L\) of any element of order \(p\) in \(G\) involves only Jordan unipotent blocks of size \(1\), \(p-1\) or \(p\) (in the language of the paper under review such an action is said to be coreless). The authors use the representation theory of algebraic groups to show that, under fairly general hypotheses, when \(G\) is a finite group of Lie type in characteristic \(p\), an irreducible module for \(G\) over a field of the same characteristic, on which all elements of order \(p\) in \(G\) are coreless, is essentially derived from the Steinberg module. Thus, in virtually all cases, the reduction modulo \(p\) of an \(RG\)-lattice must be reducible. The rough sketch of the main working methods of the paper that we have just given can be supplemented by the following precise theorem. Let \(G\) be a finite connected reductive group in characteristic \(p>3\) with no simple component of type \(A_1\). Let \(\Theta\) be an irreducible complex representation of \(G\) that is unramified above \(p\) (this means that its character is unramified). Suppose that \(\ker\Theta\) is solvable. Then \(\Theta\) is irreducible modulo \(p\) if and only if it has defect 0 modulo \(p\). This latter condition means that \(\Theta\) is essentially the Steinberg representation. It should be remarked that, in the absence of the unramified property of \(\Theta\) required for the theorem, the main result just described need not be true. The following is an immediate application of this theory to the study of globally irreducible lattices. Suppose that \(G\) is as above and let \(\Lambda\) be an absolutely irreducible \(\mathbb{Z} G\)-lattice of rank greater than 1 and with solvable kernel. Then \(\Lambda\) is not globally irreducible. This certainly severely restricts the supply of globally irreducible lattices and suggests that there are very few new cases to be discovered.