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Arithmetic Elliptic Curves in General Position英文教材.pdf

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ARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION Shinichi Mochizuki February 2009  We combine various well-known techniques from the theory of heights, the theory of “noncritical Belyi maps”, and classical analytic number theory to con- clude 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 po- sition” 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 GL2 ( ), for some prime number l which l 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. Introduction In the classical intersection theory of subvarieties, or cycles, on algebraic vari- eties, various versions of the “moving lemma” allow one to replace a given cycle by another cycle which is equivalent, from the poi
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