Least lengths of Aoki's exceptional Hodge classes on Fermat varieties, and degrees with four prime factors

Aoki reduced the Hodge conjecture for Fermat varieties of degree m to the algebraicity of the classes of exceptional Hodge elements ξ_q, one for each divisor q of m that is a product of a positive even number of distinct elements of {4, 3, 5, 7, 11, …}. Miranda, Movasati, Rufino and Villaflor deduced from a result of Kang that such a class is algebraic once it is congruent, modulo linear and standard elements in some degree divisible by q, to an element with at most six entries. We study these lengths. For q = lp and 4p, with l and p odd primes, we describe the Hodge elements of degree q by pairs of odd functions and find the least length of the class of ξ_q in closed form; so the 105 values in Table 1 of Miranda et al. are exact. Lower bounds for lifts, partly from an exhaustive computer search, show that ξ_q or one of its lifts to the degrees 2q and rq (r a prime not dividing q) has length at most six exactly for the q of this form below 65 other than 44, 51 and 52. For q with four prime factors the parity of the number of entries in a coset of {±1}^4 is an invariant that vanishes on linear and standard elements and on lifts; the least length of a Hodge element on which it is 1 is 15 at 420, by results of Peterson, attained in a single class (by computation), and 16 at 924. Aoki's recursion for ξ_q has no solution, with either sign convention, when q has four or more prime factors. Kubert's work on the universal ordinary distribution gives, in other terms, the parity criterion for two prime factors, the nonvanishing of the invariant, and elements with the properties that Aoki's proof uses. Code and data are at github.com/sallerk/HC-fermat-parity.