  This is the first of two papers about collisionless, electrostatic
micro-instabilities in stellarators, with an emphasis on trapped-particle
modes. It is found that, in so-called maximum-$J$ configurations,
trapped-particle instabilities are absent in large regions of parameter space.
Quasi-isodynamic stellarators have this property (approximately), and the
theory predicts that trapped electrons are stabilizing to all eigenmodes with
frequencies below the electron bounce frequency. The physical reason is that
the bounce-averaged curvature is favorable for all orbits, and that trapped
electrons precess in the direction opposite to that in which drift waves
propagate, thus precluding wave-particle resonance. These considerations only
depend on the electrostatic energy balance, and are independent of all
geometric properties of the magnetic field other than the maximum-$J$
condition. However, if the aspect ratio is large and the instability phase
velocity differs greatly from the electron and ion thermal speeds, it is
possible to derive a variational form for the frequency showing that stability
prevails in a yet larger part of parameter space than what follows from the
energy argument. Collisionless trapped-electron modes should therefore be more
stable in quasi-isodynamic stellarators than in tokamaks.
