Shou-Te Chang

A course on abstract algebra

Algebraic Structure of Numbers Sets and Functions Basic Group Theory Elementary Group Theory Group Homomorphisms Direct Products of Finite Abelian Groups Sylows Theorems and Applications Symmetric Groups Basic Ring Theory Special Properties of Rings Field Theory Basic Galois Theory.

Hilbert-Kunz functions and Frobenius functors

We study the asymptotic behavior as a function of e of the lengths of the cohomology of certain complexes. These complexes are obtained by applying the e-th iterated Frobenius functor to a fixed finite free complex with only finite length cohomology and then tensoring with a fixed finitely generated module. The rings involved here all have positive prime characteristic. For the zeroth homology, these functions also contain the class of HilbertKunz functions that a number of other authors have studied. This asymptotic behavior is connected with certain intrinsic dimensions introduced in this paper: these are defined in terms of the Krull dimensions of the Matlis duals of the local cohomology of the module. There is a more detailed study of this behavior when the given complex is a Koszul complex.

An arithmetic property of Fourier coefficients of singular modular forms on the exceptional domain

We shall develop the theory of Jacobi forms of degree two over Cayley numbers and use it to construct a singular modular form of weight 4 on the 27-dimensional exceptional domain. Such a singular modular form was obtained by Kim through the analytic continuation of a nonholomorphic Eisenstein series. By applying the results in a joint work with Eie, A. Krieg provided an alternative proof that a function with a Fourier expansion obtained by Kim is indeed a modular form of weight 4. This work provides a systematic and general approach to deal with the whole issue.