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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