Faltings height

The Faltings height is a height function on the set of abelian varieties over number fields, by Gerd Faltings in his famous article finite sets for abelian varieties over number fields ( Invent. Math 73 (3): 349-366 ) has been introduced.

The Faltings height measures the " size " of an abelian variety over a number field. We consider the Néron models of over all completions of. The vector space of global sections of the canonical bundle of the highest external power in the sense of Arakelov theory is a metrisierter module and thus carries a canonical norm. The product of the Haar measures of the basic loop of the canonical mesh in this vector space ( almost all 1 ) is the Faltings height of.

There are only a finite number of polarized abelian varieties with limited Faltings height. This is a major proof step in the proof of the Shafarevich conjecture and thus the Mordell conjecture.

• Number Theory
• Algebraic Geometry