  Motivated by applications in point counting algorithms using p-adic
cohomology, we give an explicit description of integral lattices in rigid
cohomology spaces that p-adically approximate logarithmic crystalline
cohomology modules. These lattices are expressed in terms of the global
sections of the twisted logarithmic de Rham complex. We prove the main theorem
for smooth proper hypersurfaces with a smooth hyperplane section, then deduce
the result for the quotient of such a pair in weighted projective space. We
show how these results may be used to reduce the necessary p-adic precision
with which one must work to compute zeta functions of varieties over finite
fields.
