Self-intersection of the relative dualizing sheaf on modular curves X 1 (N)
Hartwig Mayer1
1 Universität Regensburg Universitätsstrasse 31 93053 Regensburg, Germany
Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 1, pp. 111-161.

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)/ℚ. From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian J 1 (N)/ℚ of X 1 (N)/ℚ, and, for sufficiently large N, an effective version of Bogomolov’s conjecture for X 1 (N)/ℚ.

Soit N un entier naturel impair sans facteur carré ayant au moins deux diviseurs relativement premiers et supérieurs ou égaux à 4. Le théorème principal de cet article est une formule asymptotique exclusivement en termes de N pour l’auto-intersection arithmétique du dualisant relatif des courbes modulaires X 1 (N)/ℚ. Nous en déduisons une formule asymptotique pour la hauteur stable de Faltings de la Jacobienne J 1 (N)/ℚ de X 1 (N)/ℚ ainsi qu’une version effective de la conjecture de Bogomolov pour X 1 (N)/ℚ pour N suffisamment grand.

DOI: 10.5802/jtnb.862
Classification: 14G35, 14G40

Hartwig Mayer 1

1 Universität Regensburg Universitätsstrasse 31 93053 Regensburg, Germany 
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Hartwig Mayer. Self-intersection of the relative dualizing sheaf  on modular curves $X_1(N)$. Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 1, pp. 111-161. doi : 10.5802/jtnb.862. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.862/

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