Arakelov geometry associates to an arithmetic surfaces an intrinsic invariant: the arithmetic self-intersection number of the dualizing sheaf. In this talk a result that implies upper bounds for this real number in particular for Fermat curves and modular curves will be presented.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/246