All nuts (with the exception of peanuts) can be eaten if they have special Passover supervision.
Corn is kitniot. What is kitniot? Kitniot are pulses that are usually ground up and soaked in water, at which time they swell up. A decree was enacted by certain Rabbis to forbid kitniot on Passover because it shares this characteristic in common with leaven. This decree was accepted by Ashkenazic Jewery, hence the prohibition. However, there is a fundamental difference between kitniot and leaven, and that is the amount of the protein, Beta Amylase, that each contains. In the five species of grains there is a very large concentration of this protein, whereas in kitniot products there is not, and this is the vital point that distinguishes kitniot from chametz. What is clear is that the prohibition against eating chametz is from the Torah, whereas kitniot is from the Rabbis.
Peanuts are the only nuts that cannot be eaten as they are also kitniot. All legumes (peas, corn, rice etc.) are forbidden for Ashkenazim to eat on Pesach.