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TheoremStatement: Literature-sourcedProof: Literature-sourced
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Divisibility of rational points of an abelian variety

PDF p. 13: Picard group of a smooth affine curve

Statement

If A is an abelian variety over an algebraically closed field k, then for every positive integer n the multiplication map n: A(k) → A(k) is surjective, including when the characteristic divides n.

An abelian variety is a proper geometrically integral group scheme of finite type over k; its k-rational points form an abelian group. Multiplication by n is repeated use of the group law.

External input

This is the positive-integer consequence of Proposition 39.9.11(4), with part (3) recording the corresponding scheme-level surjectivity. It is imported from the literature; no local reconstruction of the multiplication theorem is claimed.

Depends on

Used by · 1

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

The Stacks Project, Proposition 39.9.11: Abelian varieties · Proposition 39.9.11, parts (3)–(4), and opening proof of part (4)

For an abelian variety A over an algebraically closed field k, A(k) is a divisible abelian group; multiplication by every positive integer is surjective as a morphism.

Hypotheses: A is an abelian variety over the algebraically closed field k; no restriction on the characteristic or on the positive integer is imposed.

Reading limits: Read the proposition statement and the opening proof explaining part (4) from part (3); the proof of the preceding multiplication theorem was not reconstructed here.

Sources