  In this paper, we study the monodromy of Appell hypergeometric partial
differential equations, which lead us to find four derivatives which are
associated to the group GL(3). Our four derivatives have the remarkable
properties. We find that Appell hypergeometric partial differential equations
can be reduced to four nonlinear partial differential equations by the use of
our four derivatives. We investigate these four nonlinear partial differential
equations. We are interested in some particular cases. In these cases, the
system of linear partial differential equations has the remarkable properties.
We find the $\eta$-functions associated to the unitary group $U(2, 1)$, which
has the interesting properties. Our $\eta$-functions lead us to study the
arithmetic of Picard curves, especially the transform problems (moduli
problems) and the corresponding modular equations. We give a rational
transformation between two irrational integrals which are associated to some
algebraic surfaces of degree seven. We find a modular equation which give the
relation between the moduli of two Picard curves.
