Small solutions to linear congruences and Hecke equidistribution

Abstract

The purpose of the present note is to give an explicit application of equidistribution of Hecke points to the problem of small solutions of linear congruences modulo primes. Let p be an odd prime. For a random system of d linear congruences in n variables modulo p, we shall ask how many “small” solutions it has, i.e. solutions in Z of size about pd/n, the natural limiting point. We will show that the answer has a limit distribution as p→∞. Questions of a similar flavour for varieties of higher degree have been studied, i.e. given an affine variety V ⊂ An over Z/pZ of codimension d, it may be regarded as defining a system of polynomial congruences for n integers (x1, . . . , xn) ∈ Zn. One can ask (if very optimistic) whether there always exists an integral solution so that max15i5n |xi| 5 Cpd/n, for some constant C depending only on the invariants of V , e.g. degree. This is easily seen to be false, but it turns out that one may prove weaker assertions of this nature: see, for example, [L]. The present note shows that in the seemingly easy case of linear varieties, the small solutions exhibit interesting statistical properties which are, somewhat surprisingly, related to automorphic forms. The question of small solutions of linear congruences is also closely related to the study of fractional parts of linear forms (cf. Section 7 below), which have been investigated by a number of authors using ergodic theory: see, e.g., Marklof [M1]. In our context, we shall use automorphic forms and spectral theory in place of ergodic theory; we are therefore able to give explicit error estimates. The application of the spectral theory of automorphic forms to certain Diophantine questions in number theory is not new; see, for instance, Sarnak’s ICM address [Sa2]. Hecke equidistribution in particular has been studied in great generality by Clozel, Oh and Ullmo in [COU]. Our problem involves passing from smooth to sharp cutoff test functions in the equidistribution results; for this we give a simple way of optimizing the harmonic