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2014•Pacific Journal of MathematicsOpen access

On the torsion anomalous conjecture in CM abelian varieties

Sara Checcoli, Evelina Viada

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Abstract

The torsion anomalous conjecture (TAC) states that a subvariety V of an abelian variety A has only finitely many maximal torsion anomalous subvarieties.In this work we prove, with an effective method, some cases of the TAC when the ambient variety A has CM, generalising our previous results in products of CM elliptic curves.When V is a curve, we give new results and we deduce some implications on the effective Mordell-Lang conjecture.

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The torsion anomalous conjecture (TAC) states that a subvariety V of an abelian variety A has only finitely many maximal torsion anomalous subvarieties.In this work we prove, with an effective method, some cases of the TAC when the ambient variety A has CM, generalising our previous results in products of CM elliptic curves.When V is a curve, we give new results and we deduce some implications on the effective Mordell-Lang conjecture.

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Available abstract

The torsion anomalous conjecture (TAC) states that a subvariety V of an abelian variety A has only finitely many maximal torsion anomalous subvarieties.In this work we prove, with an effective method, some cases of the TAC when the ambient variety A has CM, generalising our previous results in products of CM elliptic curves.When V is a curve, we give new results and we deduce some implications on the effective Mordell-Lang conjecture.

Key concepts: Subvariety, Conjecture, Mathematics, Abelian variety, Torsion (gastropod), Abelian group, Elliptic curve, Pure mathematics

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