Byungchul Cha

Chebyshev's bias in function fields

We study a function field analog of Chebyshev’s bias. Our results, as well as their proofs, are similar to those of Rubinstein and Sarnak in the case of the rational number field. Following Rubinstein and Sarnak, we introduce the grand simplicity hypothesis (GSH), a certain hypothesis on the inverse zeros of Dirichlet L-series of a polynomial ring over a finite field. Under this hypothesis, we investigate how primes, that is, irreducible monic polynomials in a polynomial ring over a finite field, are distributed in a given set of residue classes modulo a fixed monic polynomial. In particular, we prove under the GSH that, like the number field case, primes are biased toward quadratic nonresidues. Unlike the number field case, the GSH can be proved to hold in some cases and can be violated in some other cases. Also, under the GSH, we give the necessary and sufficient conditions for which primes are unbiased and describe certain central limit behaviors as the degree of modulus under consideration tends to infinity, all of which have been established in the number field case by Rubinstein and Sarnak.

Special units and ideal class groups of extensions of imaginary quadratic fields

Let K be an imaginary quadratic field, and let F be an abelian extension of K, containing the Hilbert class field of K. We fix a rational prime p > 2 which does not divide the number of roots of unity in the Hilbert class field of K. Also, we assume that the prime p does not divide the order of the Galois group G:=Gal(F/K). Let AF be the ideal class group of F, and EF be the group of global units of F. The purpose of this paper is to study the Galois module structures of AF and EF.

Transcendental Functions and Initial Value Problems: A Different Approach to Calculus II.

Byungchul Cha (cha@muhlenberg.edu) teaches at Muhlenberg College in Allentown, Pennsylvania, after working for three years at Hendrix College. He received his B.S. from the Korea Advanced Institute of Science and Technology, and his Ph.D. from Johns Hopkins University. His area of specialization is number theory. When he is not teaching calculus, he enjoys engaging his young son in doing other mathematics, such as counting from 1 to 10 and drawing octagons.