  Motivated by an application to LDPC (low density parity check) algebraic
geometry codes described by Voloch and Zarzar, we describe a computational
procedure for establishing an upper bound on the arithmetic or geometric Picard
number of a smooth projective surface over a finite field, by computing the
Frobenius action on p-adic cohomology to a small degree of p-adic accuracy. We
have implemented this procedure in Magma; using this implementation, we exhibit
several examples, such as smooth quartics over F_2 and F_3 with arithmetic
Picard number 1, and a smooth quintic over F_2 with geometric Picard number 1.
We also produce some examples of smooth quartics with geometric Picard number
2, which by a construction of van Luijk also have trivial geometric
automorphism group.
