Self-intersection of the relative dualizing sheaf on modular curves X 1 (N)
Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 1, pp. 111-161.

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)/. Nous en déduisons une formule asymptotique pour la hauteur stable de Faltings de la Jacobienne J 1 (N)/ de X 1 (N)/ ainsi qu’une version effective de la conjecture de Bogomolov pour X 1 (N)/ pour N suffisamment grand.

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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DOI : https://doi.org/10.5802/jtnb.862
Classification : 14G35,  14G40
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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, Tome 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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