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2005•Unpublished venueOpen access

Abelian varieties isogenous to a Jacobian

Frans Jeroen Oort

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Abstract

(0.1) Question Given an abelian variety A; does there exist an algebraic curve C such that \nthere is an isogeny between A and the Jacobian of C ? \n• If the dimension of A is at most three, such a curve exists; see (1.3). \n• For any g ≥ 4 there exists an abelian variety A of dim(A) = g over C such that there \nis no algebraic curve C which admits an isogeny A ∼ Jac(A), see (3.1). One of the \narguments which proves this fact (uncountability of the ground field) does not hold over \na countable field.

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What this paper is about

(0.1) Question Given an abelian variety A; does there exist an algebraic curve C such that \nthere is an isogeny between A and the Jacobian of C ? \n• If the dimension of A is at most three, such a curve exists; see (1.3). \n• For any g ≥ 4 there exists an abelian variety A of dim(A) = g over C such that there \nis no algebraic curve C which admits an isogeny A ∼ Jac(A), see (3.1). One of the \narguments which proves this fact (uncountability of the ground field) does not hold over \na countable field.

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Available abstract

(0.1) Question Given an abelian variety A; does there exist an algebraic curve C such that \nthere is an isogeny between A and the Jacobian of C ? \n• If the dimension of A is at most three, such a curve exists; see (1.3). \n• For any g ≥ 4 there exists an abelian variety A of dim(A) = g over C such that there \nis no algebraic curve C which admits an isogeny A ∼ Jac(A), see (3.1). One of the \narguments which proves this fact (uncountability of the ground field) does not hold over \na countable field.

Key concepts: Isogeny, Abelian variety, Mathematics, Abelian variety of CM-type, Abelian group, Variety (cybernetics), Arithmetic of abelian varieties, Jacobian matrix and determinant

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