Showing papers by "Shinichi Mochizuki published in 2010"

On the combinatorial cuspidalization of hyperbolic curves

Abstract: In this paper, we continue our study of the pro-$\Sigma$ fundamental groups of configuration spaces associated to a hyperbolic curve, where $\Sigma$ is either the set of all prime numbers or a set consisting of a single prime number, begun in an earlier paper. Our main result may be regarded either as a combinatorial, partially bijective generalization of an injectivity theorem due to Matsumoto or as a generalization to arbitrary hyperbolic curves of injectivity and bijectivity results for genus zero curves due to Nakamura and Harbater--Schneps. More precisely, we show that if one restricts one's attention to outer automorphisms of such a pro-$\Sigma$ fundamental group of the configuration space associated to a(n) affine (respectively, proper) hyperbolic curve which are compatible with certain ``fiber subgroups'' (i.e., groups that arise as kernels of the various natural projections of a configuration space to lower-dimensional configuration spaces) as well as with certain cuspidal inertia subgroups, then, as one lowers the dimension of the configuration space under consideration from $n+1$ to $n \ge 1$ (respectively, $n \ge 2$), there is a natural injection between the resulting groups of such outer automorphisms, which is a bijection if $n \ge 4$. The key tool in the proof is a combinatorial version of the Grothendieck conjecture proven in an earlier paper by the author, which we apply to construct certain canonical sections.

Arithmetic elliptic curves in general position

Abstract: We combine various well-known techniques from the theory of heights, the theory of “noncritical Belyi maps”, and classical analytic number theory to conclude that the “ABC Conjecture”, or, equivalently, the so-called “Effective Mordell Conjecture”, holds for arbitrary rational points of the projective line minus three points if and only if it holds for rational points which are in “sufficiently general position” in the sense that the following properties are satisfied: (a) the rational point under consideration is bounded away from the three points at infinity at a given finite set of primes; (b) the Galois action on the l-power torsion points of the corresponding elliptic curve determines a surjection onto GL 2 (Zl), for some prime number l which is roughly of the order of the sum of the height of the elliptic curve and the logarithm of the discriminant of the minimal field of definition of the elliptic curve, but does not divide the conductor of the elliptic curve, the rational primes that are absolutely ramified in the minimal field of definition of the elliptic curve, or the local heights [i.e., the orders of the q-parameter at primes of [bad] multiplicative reduction] of the elliptic curve.