Here are a few ideas for final project material, inspired by things we've come across already:

• Use the Weil conjectures to study exponential sums, Kloosterman sums, etc.
• Prove the Riemann hypothesis for curves using intersection theory on surfaces. (You could also try following arguments due to Nick Katz or Tony Scholl, but these are even more involved.)
• Prove the Riemann hypothesis for elliptic curves using properties of the Frobenius endomorphism, and maybe even the analogous statement for abelian varieties.
• Give Dwork's proof of the rationality of zeta functions.
• Explain the Lang-Trotter conjecture.
• Develop higher reciprocity laws for polynomial rings, say as in Michael Rosen's book.
• Show that for a curve X over a finite field q, the genus of X is the supremum of |#X(F_q^r)-(q^r+1)|/sqrt(q^r). (You should be able to give a proof using Weyl's equidistribution theorem and the fact that the eigenvalues of Frobenius on X/F_q^r are the rth powers of those of X.)