arXiv subjects
Publications and source records attributed to Akio Nakagawa.
We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms between certain algebraic varieties. The numbers of rational points on these varieties are hypergeometric functions over finite fields.
Ramanujan's theory of elliptic functions to alternative bases connects modular forms with hypergeometric series and has led to applications such as the modularity of certain hypergeometric Galois representations. In this paper, we relate special values of $L$-functions of certain CM Hecke eigenforms to Ramanujan's alternative bases via the modularity of hypergeometric Galois representations associated with hypergeometric series ${}_{3}F_{2}\!\left[ \genfrac{}{}{0pt}{}{\frac{1}{2} \ \frac{1}{d} \ \frac{d-1}{d}}{\ 1 \ \ \ \ 1} ;\ t \right]$, $d=2$, $3$, $4$, and $6$, arising from tensor products of CM elliptic curves over real quadratic fields. We also give a complete classification of these type of hypergeometric Galois representations.
We prove sum representations of Appell-Lauricella functions over a finite field using confluent hypergeometric functions over the finite field. As an application, we also prove transformation formulas, summation formulas and reduction formulas for Appell-Lauricella functions over the finite field.
Kampé de Fériet hypergeometric functions are two-variable hypergeometric functions, which are a generalization of Appell's functions. It is known that they satisfy many reduction and summation formulas. In this paper, we define Kampé de Fériet hypergeometric functions over finite fields and show analogous formulas.
We prove finite field analogues of integral representations of Appell- Lauricella hypergeometric functions in many variables. We consider certain hypersurfaces having a group action and compute the numbers of rational points associated with characters of the group, which will be expressed in terms of Appell-Lauricella functions over finite fields.
We compute the Artin $L$-function of a diagonal hypersurface D_λ over a finite field associated to a character of a finite group acting on D_λ , and under some condition, express it in terms of hypergeometric functions and Jacobi sums over the finite field. As an application, we consider the Dwork hypersurfaces and obtain relations among certain hypergeometric functions over different finite fields.