  We prove the remaining part of the conjecture by Denef and Sperber [Denef, J.
and Sperber, S., \textit{Exponential sums mod $p^n$ and {N}ewton polyhedra},
Bull. Belg. Math. Soc., {\bf{suppl.}} (2001) 55-63] on nondegenerate local
exponential sums modulo $p^m$. We generalize Igusa's conjecture of the
introduction of [Igusa, J., \textit{Lectures on forms of higher degree}, Lect.
math. phys., Springer-Verlag, {\bf{59}} (1978)] from the homogeneous to the
quasi-homogeneous case and prove the nondegenerate case as well as the modulo
$p$ case. We generalize some results by Katz of [Katz, N. M., \textit{Estimates
for "singular" exponential sums}, Internat. Math. Res. Notices (1999) no. 16,
875-899] on finite field exponential sums to the quasi-homogeneous case.
