Motivation: It seems that the splitting of a hypergeometric motive is closely related to some highly non-trivial hypergeometric identities discovered by Ramanujan, Guillera et al. The splitting of hypergeometric motives can be treated as a conceptual idea for discovering other Ramanujan-type formulas, such as (using notations from Magma Handbook)
(of course, one can obtain their supercongruence counterparts as well.)
Question: We notice that the rational arguments $t$ attached to the hypergeometric motives $H(\alpha,\beta,t)$ of Ramanujan-type formulas share some interesting properties. In other words, the prime factors of the denominator and the numerator of $t$ are quite small.
are hypergeometric motives that (highly likely) split. Even the formula (31) and (38) in Guillera's paper from quadratic fields seem to fit well with the observation.
- It seems that one can usually succeed in finding out a hypergeometric motive that splits with such $t$. Why is it so?
- Is there any known algorithm for locating an algebraic $t$ that $H(\alpha,\beta,t)$ splits?