  A linear gyrokinetic particle-in-cell scheme, which is valid for arbitrary
perpendicular wavelength $k_\perp\rho_i$ and includes the parallel dynamic
along the field line, is developed to study the local electrostatic drift modes
in point and ring dipole plasmas. We find the most unstable mode in this system
can be either electron mode or ion mode. The properties and relations of these
modes are studied in detail as a function of $k_\perp\rho_i$, the density
gradient $\kappa_n$, the temperature gradient $\kappa_T$, electron to ion
temperature ratio $\tau=T_e/T_i$, and mass ratio $m_i/m_e$. For conventional
weak gradient parameters, the mode is on ground state (with eigenstate number
$l=0$) and especially $k_\parallel\sim0$ for small $k_\perp\rho_i$. Thus,
bounce averaged dispersion relation is also derived for comparison. For strong
gradient and large $k_\perp\rho_i$, most interestingly, higher order eigenstate
modes with even (e.g., $l=2,4$) or odd (e.g., $l=1$) parity can be most
unstable, which is not expected by previous studies. High order eigenstate can
also easily be most unstable at weak gradient when $\tau>10$. This work can be
particularly important to understand the turbulent transport in laboratory and
space magnetosphere.
