The Hodge - Arakelov Theory of Elliptic Curves
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The Hodge-Arakelov Theory of Elliptic Curves in Positive Characteristic
The purpose of this paper is to study the Hodge-Arakelov theory of elliptic curves (cf. [Mzk1-4]) in positive characteristic. The first two §’s (§1,2) are devoted to studying the relationship of the Frobenius and Verschiebung morphisms of an elliptic curve in positive characteristic to the Hodge-Arakelov theory of elliptic curves. We begin by deriving a “Verschiebung-Theoretic Analogue of the H...
متن کاملA Survey of the Hodge-Arakelov Theory of Elliptic Curves I
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 ...
متن کاملAnabelian Geometry in the Hodge-Arakelov Theory of Elliptic Curves
The purpose of the present manuscript is to survey some of the main ideas that appear in recent research of the author on the topic of applying anabelian geometry to construct a “global multiplicative subspace”— i.e., an analogue of the well-known (local) multiplicative subspace of the Tate module of a degenerating elliptic curve. Such a global multiplicative subspace is necessary to apply the ...
متن کاملThe Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories
Introduction §1. Statement of the Main Results §2. Technical Roots: the Work of Mumford and Zhang §3. Conceptual Roots: the Search for a Global Hodge Theory §3.1. From Absolute Differentiation to Comparison Isomorphisms §3.2. A Function-Theoretic Comparison Isomorphism §3.3. The Meaning of Nonlinearity §3.4. Hodge Theory at Finite Resolution §3.5. Relationship to Ordinary Frobenius Liftings and...
متن کاملA Survey of the Hodge-Arakelov Theory of Elliptic Curves II
The purpose of the present manuscript is to continue the survey of the Hodge-Arakelov theory of elliptic curves (cf. [7, 8, 9, 10, 11]) that was begun in [12]. This theory is 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. In particular, in the present manuscript, we...
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