Description
Batyrev constructed a large but finite list of Calabi-Yau threefolds (CY3s) obtained from suitable triangulations of four-dimensional reflexive polytopes; all such polytopes were classified by Kruezer and Skarke. Working with these CY3s has been extremely fruitful, and has enabled significant recent progress in the explicit construction of flux vacua with interesting physical properties. Nevertheless, it also comes with practical and computational challenges: the Kreuzer-Skarke list hosts large redundancies, and the same CY3 can be realized from many different triangulations. In recent work (arXiv:2310.06820, with N. Gendler, N. MacFadden, L. McAllister, J. Moritz, A. Schachner, and M. Stillman), we enumerated CY3s obtained in this way, obtaining exact irredundant counts of CY3s with 1<= h^{1,1} <=5. In this talk, I will emphasize the key role techniques from arithmetic geometry played in this classification. I will also comment on ongoing work (with F. Abbasi and W. Taylor) on classifying elliptic fibrations in this class of CY3s and the search for elliptic curves of high rank therein.
