A "height-free" effective isogeny estimate for GL(2)-type abelian varieties

Abstract: We prove a ”height-free” effective isogeny estimate for abelian varieties of GL(2)-type. More precisely, let g ∈ Z +, K a number field, S a finite set of places of K, and A, B/K g-dimensional abelian varieties with good reduction outside S which are K-isogenous and of GL(2)-type over $\bar \text{Q}$. We show that there is a K-isogeny A → B of degree effectively bounded in terms of g, K, and S only. We deduce an effective open image theorem for these abelian varieties, as well as an effective upper bound on the number of S-integral K-points on a Hilbert modular variety

number theory

Audience: researchers in the topic

Columbia CUNY NYU number theory seminar