2021•Contemporary mathematics - American Mathematical SocietyOpen access

Ranks of abelian varieties and the full Mordell-Lang conjecture in dimension one

Arno Fehm, Sebastian Petersen

Abstract

Let A A be a non-zero abelian variety over a field F F that is not algebraic over a finite field. We prove that the rational rank of the abelian group A ( F ) A(F) is infinite when F F is large in the sense of Pop (also called ample). The main ingredient is a deduction of the 1-dimensional case of the relative Mordell-Lang conjecture from a result of Rössler.

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Key concepts: Abelian group, Conjecture, Arithmetic of abelian varieties, Abelian variety, Rank of an abelian group, Mathematics, Rank (graph theory), Dimension (graph theory)