The Hodge numbers of generic elliptically fibered Calabi-Yau threefolds over toric base surfaces fill out the “shield” structure previously identified by Kreuzer and Skarke. The connectivity structure of these spaces and bounds on the Hodge numbers are illuminated by considerations from F-theory and the minimal model program. In particular, there is a rigorous bound on the Hodge number h21 ≤ 49...

The purpose of the present manuscript is to give a survey of the Hodge-Arakelov theory of elliptic curves (cf. [Mzk1,2]) — i.e., a sort of “Hodge theory of elliptic curves” analogous to the classical complex and p-adic Hodge theories, but which exists in the global arithmetic framework of Arakelov theory — as this theory existed at the time of the workshop on “Galois Actions and Geometry” held ...

Hodge integrals are a class of intersection numbers on moduli spaces of curves involving the tautological classes λi, which are the Chern classes of the Hodge bundle E. In recent years Hodge integrals have shown a great amount of interconnections with Gromov-Witten theory and enumerative geometry. The classical Hurwitz numbers, counting the numbers of ramified Covers of a curve with an assigned...

We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their L-functions extend meromorphically to complex plane satisfy a functional equation conjectured by Broadhurst Roberts. Although in question turn out be “classical,” we compute Hodge means irregular filtration on realizations as exponential mixed structures. show all are...

Stringy Hodge numbers are introduced by Batyrev for a mathematical formulation of mirror symmetry. However, since the stringy Hodge numbers of an algebraic variety are defined by choosing a resolution of singularities, the well-definedness is not clear from the definition. Batyrev proved the well-definedness by using the theory of motivic integration developed by Kontsevich, Denef-Loeser. The a...

This thesis aims to expose the amazing sequence of ideas, concerning p-adic representations coming from geometry, that form the heart of what was called Hodge-Tate theory. This subject, initiated by Tate in the late ’60s in analogy to classical Hodge theory, leads in to the now vast and highly fruitful program of p-adic Hodge Theory. The central result of the theory is the Hodge-Tate decomposit...

This expository paper is an expanded version of a talk given at the joint meeting of the Edinburgh and London Mathematical Societies in Edinburgh to celebrate the centenary of the birth of Sir William Hodge. In the talk the emphasis was on the relationship between Hodge theory and geometry, especially the study of algebraic cycles that was of such interest to Hodge. Special attention will be pl...