  We address a question posed by Ono, prove a general result for powers of an
arbitrary prime, and provide an explanation for the appearance of higher
congruence moduli for certain small primes. One of our results coincides with a
recent result of Edixhoven, and we hope that the comparison of the methods,
which are entirely different, may reveal a connection between the p-adic
geometry and the arithmetic of half-integral weight Hecke operators.
