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)
$$H([3,4,6],[2,1,1,1,1,1],729/4)$$
and
$$H([2,3,3],[1,1,1,6],-4)$$
give
$$\begin{equation}\sum_{k=0}^{\infty}\frac{(1/3)_k(2/3)_k(1/4)_k(3/4)_k(1/6)_k(5/6)_k}{(1)_k^5(1/2)_k}\frac{41760k^3+28512k^2+4264k+220}{2k+1}\left(\frac{4}{729}\right)^k=\frac{2187}{\pi^2}\end{equation}$$
and
$$\begin{equation}\sum_{k=0}^{\infty}\frac{(1/2)_k(1/3)_k^2(2/3)_k^2}{(1)_k^3(1/6)_k(5/6)_k}\frac{270k^3+333k^2+102k+10}{(k+1/6)(k+5/6)}\left(-\frac{1}{4}\right)^k=\frac{108\sqrt{3}}{\pi}\end{equation}.$$
(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.
Examples:
$$H([2,4],[1,1,1],99^4)$$
and
$$H([2,2,2,2,5],[1,1,1,1,1,1,1,1,1],-3125/1024)$$
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?
