The arithmetic of Calabi-Yau motives and mobile higher regulators

Abstract

We construct elements in the motivic cohomology of certain rank 4 weight 3 Calabi--Yau motives, and write down explicit expressions for the regulators of these elements in the context of conjectures on $L$-values such as those of Beilinson or Bloch-Kato. We apply a combination of three ideas: (i) that a motive can be made to vary in a family in such a way that a desired motivic cohomology class is realized by relative cohomology; (ii)~that there are ways to construct higher-rank (such as $2\times 2$) regulators from a single family; and (iii) that one can arrange elements in $H^4_{\text{Mot}}(X,\mathbb{Z}(p))$ with different $p$'s by choosing hypergeometric families with different local exponents. Following background material on Hodge theory, algebraic cycles, differential equations, and hypergeometric variations, we work out two cases in detail where $p=3,4$. Regarding our Calabi-Yau motives $X_t$ as fibers in a suitable total space $\mathcal{X}_U\to U\subset \mathbb{P}^1$, each Hodge class in $\mathrm{Hom}_{\mathrm{MHS}}(\mathbb{Q}(0),H^4(\mathcal{X}_U,\mathbb{Q}(p)))$ produces a family of extension classes in $\mathrm{Ext}^1_{\mathrm{MHS}}(\mathbb{Q}(0),H^3(X_t,\mathbb{Q}(p)))$ called a normal function. Our main results for these cases, which are essentially independent, are the explicit computation of the normal functions, and the construction of motivic cohomology cycles realizing the Hodge classes, thereby proving Beilinson's Hodge-type conjecture and providing the first numerical checks of Beilinson's conjectures on special values of $L$-functions for such motives.