Showing papers by "Shinichi Mochizuki published in 2003"

Topics Surrounding the Anabelian Geometry of Hyperbolic Curves

Abstract: Introduction 119 1. The Tate Conjecture as a Sort of Grothendieck Conjecture 122 1.1. The Tate conjecture for non-CM elliptic curves 122 1.2. Some pro-p group theory 126 2. Hyperbolic Curves As Their Own Anabelian Albanese Varieties 128 2.1. A corollary of the Main Theorem of [Mzk2] 128 2.2. A partial generalization to nite characteristic 129 3. Discrete Real Anabelian Geometry 132 3.1. Real complex manifolds 132 3.2. Fixed points of antiholomorphic involutions 137 3.3. Hyperbolic curves and their moduli 139 3.4. Abelian varieties and their moduli 140 3.5. Pro nite real anabelian geometry 141 4. Complements to the p-adic Theory 147 4.1. Good Chern classes 147 4.2. The group-theoreticity of a certain Chern class 152 4.3. A generalization of the main result of [Mzk2] 157 References 163

The Absolute Anabelian Geometry of Canonical Curves

Abstract: In this paper, we continue our study of the issue of the ex- tent to which a hyperbolic curve over a finite extension of the field of p-adic numbers is determined by the profinite group structure of its ´ fundamental group. Our main results are that: (i) the theory of corre- spondences of the curve — in particular, its arithmeticity — is completely determined by its fundamental group; (ii) when the curve is a canonical lifting in the sense of "p-adic Teichmuller theory", its isomorphism class is functorially determined by its fundamental group. Here, (i) is a conse- quence of a "p-adic version of the Grothendieck Conjecture for algebraic curves" proven by the author, while (ii) builds on a previous result to the effect that the logarithmic special fiber of the curve is functorially determined by its fundamental group.