Modularity of Calabi-Yau Varieties

Abstract

In this paper we discuss recent progress on the modularity of Calabi-Yau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the L-function in connection with mirror symmetry. Finally, we address some questions and open problems.

On the Chow Ring of Cynk–Hulek Calabi–Yau Varieties and Schreieder Varieties

“…Proof. This is [7,Theorem 3.3] (the construction of X is also explained in [16,Section 5.3]). In fact, the crepant resolution X can be constructed explicitly, inductively on the number of elliptic curves, cf.…”

A database of number fields

“…Other complete lists of fields were posted sporadically in the next ten years, while most fields and the new interface are recent additions. Results from the predecessors of the present database have occasionally been used as ingredients of formal arguments, as in for example [12,15,31]. The more common use of our computational results has been to guide investigations into number fields in a more general way.…”

Mesure de Mahler et série L d’une surface K3 singulière

“…D'après les conjectures de Weil (1949) prouvées par Dwork (1960) et Deligne (consulter par exemple [15]), Z V (T ) est une fonction rationnelle à coefficients dans Q satisfaisant l'équation fonctionnelle…”