The Arithmetic of Reducible Rank 2 Hypergeometric Motives

Abstract: We study the arithmetic of the realizations for hypergeometric motives attached to ${}_3F_2(1)$ hypergeometric series defined over number fields, focusing on the case where the étale realization is reducible and irregular. We then use motivic techniques to prove new evaluation formulas for certain ${}_3F_2(1)$ series, understood as a period of the hypergeometric motive. In addition, we provide examples which are closely related to the exact values of $L$-functions for modular forms in special cases.