  We introduce and study invariants of singularities in positive characteristic
called F-thresholds. They give an analogue of the jumping coefficients of
multiplier ideals in characteristic zero. We discuss the connection between the
invariants of an ideal in characteristic zero and the invariants of the
different reduction mod p of this ideal. Our main point is that this relation
depends on arithmetic properties of p. We also describe a new connection
between invariants mod p and the roots of the Bernstein-Sato polynomial.
