Introduction. If f and g are forms with integer coefficients and n and m variables respectively (n > m) the results of the last chapter give us methods of finding whether or not some form in the genus of f represents g integrally. Corollary 44a or 44b may be used to show that the existence of such a representation depends on the solvability of the congruence f ≡ g (mod 8 | g | P) where P is the product of the odd primes in |g|·|f|, or on the existence of representations in R(p) for all p dividing 2|f||g|. When there is only one class in the genus of f, the same criteria serve to determine the existence of representations of g by the form f. However, when there is more than one class in the genus except for certain very special cases and asymptotic results there are no known criteria for existence of representations.

When it comes to determining the number of representations of g by f, known results, except for those in section 38 below, depend on analytic theory which is beyond the scope of this book. However we shall describe such conclusions.

For the case m = n two fundamental problems arise. First, the question of equivalence cannot in general be elegantly resolved in the ring of rational integers. Faced with such a problem one would first test for semi-equivalence by methods of the previous chapter; then employ a reduced form such as is described in theorem 23.