GCD results for certain divisibility sequences of polynomials and a conjecture of Silverman

Abstract: A divisibility sequence is a sequence of integers $d_n$ such that, if $m$ divides $n$, then $d_m$ divides $d_n$. Bugeaud, Corvaja, Zannier showed that pairs of divisibility sequences of the form $a^n-1$ have only limited common factors. From a geometric point of view, this divisibility sequence corresponds to a subgroup of the multiplicative group, and Silverman conjectured that a similar behaviour should appear in (a large class of) other algebraic groups.

Extending previous works of Silverman and of Ghioca-Hsia-Tucker on elliptic curves over function fields, we will show how to prove the analogue of Silverman’s conjecture over function fields in the case of split semiabelian varieties and some generalizations. The proof relies on some results of unlikely intersections. This is a joint work with F. Barroero and A. Turchet.

number theory

Audience: researchers in the topic

Heilbronn number theory seminar

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