Plectic Jacobians and Hodge theory

Abstract: Gehrmann, Guitart, Masdeu and myself recently proposed, and gave evidence for, plectic generalizations of Stark-Heegner points. The construction is p-adic, cohomological, and unfortunately lacking a satisfying geometric interpretation. Nevertheless, we formulated precise conjectures on the algebraicity of plectic points and their significance for the arithmetic of higher rank elliptic curves. In this talk I will report on work in progress on the Archimedean side of the story where geometry has a prominent role: I will describe a collection of complex tori — called plectic Jacobians — associated with the plectic Hodge structure appearing in the middle degree cohomology of Hilbert modular varieties. Interestingly, the Oda-Yoshida conjecture can be used to prove that plectic Jacobians are modular abelian varieties defined over $\overline{\mathbb{Q}}$. Moreover, the existence of exotic Abel-Jacobi morphisms (mapping zero-cycles to plectic Jacobians) further highlights the arithmetic appeal of the construction.

algebraic geometrynumber theory

Audience: researchers in the topic

Number Theory Seminars at Università degli Studi di Padova