Number-theoretic properties of the binomial distribution with applications in arithmetic geometry - Schmidt, Eric, author; Achter, Jeffrey, advisor; Pries, Rachel, committee member; Cavalieri, Renzo, committee member; Bohm, Wim, committee member (Colorado State University. Libraries, 2014)
Alina Bucur et al. showed that the distribution of the number of points on a smooth projective plane curve of degree d over a finite field of order q is approximated by a particular binomial distribution. We generalize their arguments to obtain a similar theorem concerning hypersurfaces in projective m-space. We briefly describe Bucur and Kedlaya's generalization to complete intersections. We then prove theorems concerning the probability that a binomial distribution yields an integer of various certain properties, such as being prime or being squarefree. Finally, we show how to apply such a theorem, concerning a property P, to yield results concerning the probability that the numbers of points on random complete intersections possess property P.
Keywords: binomial distribution; squarefree; complete intersection