arXiv subjects
Publications and source records attributed to Kenneth Chung Tak Chiu.
We prove the Ax-Schanuel property of the derivatives of mixed period mappings. We also prove the jet space reformulation of this result. The proofs use the Ax-Schanuel result for principal bundles with flat connections obtained by Blázquez-Sanz, Casale, Freitag, and Nagloo.
Let $X$ be a smooth irreducible quasi-projective algebraic variety over a number field $K$. Suppose $X$ is equipped with a $p$-adic étale local system compatible with an admissible graded-polarized variation of mixed Hodge structures on the complex analytification of $X_{\mathbb{C}}$. We prove that the $S$-integral points in $X$ are covered by subpolynomially many geometrically irreducible $K$-subvarieties, each lying in a fiber of the mixed period mapping arising from the variation of mixed Hodge structures. This is based on recent works by Brunebarbe-Maculan and Ellenberg-Lawrence-Venkatesh. As an application, we prove that there are subpolynomially many $S$-integral Laurent polynomials with fixed reflexive Newton polyhedron $Δ$ and fixed non-zero principal $Δ$-determinant. Our results answer a question asked by Ellenberg-Lawrence-Venkatesh.
We use the determinant method of Bombieri-Pila and Heath-Brown and its Arakelov reformulation by Chen utilizing Bost's slope method to estimate the number of hypersurfaces required to cover the regular rational points with bounded Arakelov height on a projective variety. Using complex cellular structures introduced by Binyamini-Novikov, we replace the usual subpolynomial factor by a polylogarithmic factor in the estimation.
We give properties of the real-split retraction of the mixed weak Mumford-Tate domain and prove the Ax-Schanuel property of period mappings arising from variations of mixed Hodge structures. An ingredient in the proof is the definability of the mixed period mapping obtained by Bakker-Brunebarbe-Klingler-Tsimerman. In comparison with preceding results, in the point counting step, we count rational points on definable quotients instead.
We first introduce hyperbolic analogues of Belyi maps, Shabat polynomials and Grothendieck's dessins d'enfant. In particular we introduce and study the Shabat-Blaschke products and the size of their hyperbolic dessin d'enfants in the unit disk. We then study a special class of Shabat-Blaschke products, namely the Chebyshev-Blaschke products. Inspired by the work of Ismail and Zhang (2007) on the coefficients of the Ramanujan's entire function, we will give similar arithmetic properties of the coefficients of the Chebyshev-Blaschke products and then use them to prove some Landen-type identities for theta functions.