A NOTE ON REPRESENTATION NUMBERS OF QUADRATIC FORMS MODULO PRIME POWERS

Let f be an integral quadratic form in k variables, F the Gram matrix corresponding to a Z-basis of Z (k) . For r is an element of F (- 1) Z (k) , a rational number n with f (r) r ) n mod Z and a positive integer c , set N (f) ( n, r ; c ) := #{x { x is an element of Z(k)/cZ(k) : f (x + r ) equivalent to n mod c } . Siegel showed that for each prime p , there is a number w depending on r and n such that N (f) ( n, r ; p (nu +1) ) = p (k - 1) N (f) ( n, r ; p (nu) ) holds for every integer nu > w and gave a rough estimation on the upper bound for such w . In this short note, we give a more explicit estimation on this bound than Siegel's.