GCD results for certain divisibility sequences of polynomials and a conjecture of Silverman
Heilbronn Number Theory Seminar
21st April 2021, 4:00 pm – 5:00 pm
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.