Degree
Doctor of Philosophy (PhD)
Department
Mathematics
Document Type
Dissertation
Abstract
We study two families of ${}_3F_2(1)$ and ${}_2F_1(1)$ hypergeometric motives introduced by Allen, Grove, Long, and Tu, as well as the author for the ${}_2F_1(1)$ cases, in the process of developing the explicit hypergeometric modularity method. The method enables us to study the explicit modularity of these motives, and produces explicit modular forms attached to them using Ramanujan's theory of elliptic functions to alternative bases. In particular, formulas for the Fourier coefficients of the modular forms are obtained using finite field hypergeometric functions and the critical $L$-values of the same modular forms at 1 and 2 are written using classical hypergeometric series. In this thesis, the $L$-values obtained from these two families are completely classified, and many relationships among them are determined. In addition, formulas for the Fourier coefficients of the modular forms in the ${}_2F_1(1)$ family are found. Finally, Allen, Grove, Long, and Tu focused on ${}_3F_2(1)$ hypergeometric motives whose Hodge numbers are $(1,0,1)$, which we call a regular case. When we say Hodge numbers, we are always referring to the vector $(h^{2,0},h^{1,1},h^{0,2})$, where $h^{p,q}$ denotes the dimensional of the Dolbeault cohomology group $H^{p,q}$. In this thesis, ${}_3F_2(1)$ we describe how to construct modular forms attached to the irregular cases, which have Hodge numbers $(2,0,0)$. The modularity and $L$-values for the irregular motives are also proved.
Date
7-3-2026
Recommended Citation
Rosen, Esme, "Explicit Modularity of Hypergeometric Motives and L-values of Modular Forms" (2026). LSU Doctoral Dissertations. 7126.
https://repository.lsu.edu/gradschool_dissertations/7126
Committee Chair
Ling Long
LSU Acknowledgement
1
LSU Accessibility Acknowledgment
1