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Calabi-Yau Manifolds Seminar

Speaker: Rikuto Ito (Nagoya University)

Title: Modularity for rank-21 cubic fourfolds over Q

Abstract: A cubic fourfold is a Fano variety defined by a homogeneous cubic equation in five-dimensional projective space. Cubic fourfolds share many structural similarities with K3 surfaces; in particular, when the second Picard number is at least two, Hassett (2000) showed that a cubic fourfold is cohomologically associated with a K3 surface. Under additional hypotheses, this correspondence is motivic in the sense of Chow motives (B\”ulles 2018). Consequently, one expects cubic fourfolds to exhibit arithmetic phenomena closely parallel to those of K3 surfaces.

In this talk, I will explain a proof of the modularity of cubic fourfolds of rank 21 defined over \mathbf{Q}. The corresponding statement for singular K3 surfaces was obtained by Livné, but under the very restrictive assumption that the Picard number is 20 over \mathbf{Q}. Our approach does not require any assumption on the Picard number over \mathbf{Q}; we only assume that the cubic fourfold has rank 21 over \mathbf{C}.

We also obtain the modularity of singular K3 surfaces without Livné’s Picard-number hypothesis, using a method that is essentially identical to the one applied to cubic fourfolds. In both settings, the resulting Galois representations are shown to arise from the same type of weight-3 CM newform. I will focus on the construction of this modular form and the shared mechanism underlying both proofs.