At (about) the time Mumford was giving his lectures at Harvard, Ax was lecturing on his work with Kochen in which they proved a conjecture of Artin for almost all p by using ultrafilters. This is clearly what Mumford was thinking of. The reference for the Ax-Kochen work is:
MR0184930 Ax, James; Kochen, Simon Diophantine problems over local fields. I. Amer. J. Math. 87 1965 605--630. Ib. 87 1965 631--648.
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