Torsion finite problems

Abstract: Consider an abelian variety A over a number field K. The torsion subgroup of A(K) is finite; a result of Ribet shows that this finiteness persists over the cyclotomic extension of K.

Now consider a second abelian variety B/K, and the infinite extension K_B generated by the coordinates of its torsion points. Conditional on the Mumford-Tate conjecture (and up to a finite extension of K), I will give a criterion for the finitude of the torsion subgroup of A(K_B). I'll also describe a motivic generalization of this story, which in retrospect explains certain algebraic cycles we discovered on torsion-infinite pairs of CM abelian varieties. (Joint work with Lian Duan and Xiyuan Wang.)

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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