Binomial Coefficients and Quadratic Fields

Abstract

Let E be a real quadratic field with discriminant d and let p be an odd prime not dividing d. For ρ=1 or -1, we determine $\prod_{0<c<d, (d/c)=ρ} binomial coeff.{p-1}{\lfloor pc/d\rfloor}$ modulo p^2 in terms of Lucas numbers, the fundamental unit and the class number of E, where (d/c) is the Kronecker symbol.

10 pages; final version, accepted by Proc. AMS