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Self-intersection of the relative dualizing sheaf on modular curves X 1 ( N )

Hartwig Mayer — 2014

Journal de Théorie des Nombres de Bordeaux

Let N be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than 4 . Our main theorem is an asymptotic formula solely in terms of N for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves X 1 ( N ) / . From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian J 1 ( N ) / of X 1 ( N ) / , and, for sufficiently large N , an effective version of Bogomolov’s conjecture for X 1 ( N ) / .

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