Transcendence of ${}_3F_2(1)$ Hypergeometric Series and $L$-values of Modular Forms

Abstract: We examine a certain class of hypergeometric motives, recently associated to Hecke eigenforms of weight 3 by Allen, Grove, Long, and Tu. One aspect of their results allows us to express a special $L$-value of a specific Hecke eigenform as a combination of ${}_3F_2(1)$ hypergeometric series. First, we show that the $L$-function can be twisted in a predictable way so that the $L$-value of the twist can be also be written as a sum of hypergeometric series. Moreover, the hypergeometric series involved in the twist are related to a classical formula of ${}_3F_2(1)$ series due to Kummer, which we use the extend to relations between $L$-values of the eigenforms. In the process, we find a bound on the transcendence degree for certain ${}_3F_2(1)$ series. This suggests a generalization of a theorem due to W\"ustholz which provides a formula for the transcendence degree of periods on abelian varieties.