Folding height

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In the mathematical field of arithmetic geometry , the folding height is a measure of the (arithmetic) complexity of Abelian varieties . It is named after the mathematician Gerd Faltings . She played an essential role in Faltings' proof of the Mordell conjecture , which states that a curve of gender has only a finite number of rational points . ${\displaystyle \geq 2}$

For an elliptic curve with a fixed isomorphism , the Faltings height is precisely the reciprocal of the area of a fundamental region of the lattice . ${\displaystyle E}$ ${\displaystyle E\cong \mathbb {C} /\Lambda }$ ${\displaystyle \Lambda }$

construction

The folding height measures the "size" of an Abelian variety over a number field . One looks at the Néron models from above all the completions of . The vector space of global sections of the top outer potency of the canonical bundle in the sense of Arakelov theory is a metrisierter - module and thus contributes a canonical standard . The product of the Haarschen dimensions of the basic meshes of the canonical lattices in this vector space (almost all are 1) is the folding height of . ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle A_{v}}$ ${\displaystyle A}$ ${\displaystyle K_{v}}$ ${\displaystyle K}$ ${\displaystyle O_{A_{v}}}$ ${\displaystyle A}$

meaning

The Faltings height is a height function on the set of Abelian varieties over number fields , which was introduced by Gerd Faltings in his famous article Finiteness theorems for Abelian varieties over number fields .

There are only finitely many polarized Abelian varieties with a limited fold height. This is an essential step in the proof of the Shafarevich conjecture and thus the Mordell conjecture .

literature

• Gerd Faltings : Finiteness theorems for Abelian varieties over number fields . Inventiones Mathematicae 73, 1983, pp. 349-366. doi : 10.1007 / BF01388432 . Free online access via DigiZeitschriften .

Web links

• Chapter 4.6 in Milne: Abelian Varieties
• Heights and Finiteness theorems of abelian varieties