On the degree of subvarieties on abelian varieties

Let $(X,\Theta)$ be a very general principally polarized abelian variety of dimension $g$, and consider the minimal cohomology class $\theta_k=[\Theta]^k/k!$ for $k<g$. We show that the minimal positive multiple of $\theta_k$ which is algebraic is divisible by all primes $p\leq (k+1)/2$. In particular, these minimal multiples grow exponentially with $k$. Our main result follows from [EGFS25] together with a new combinatorial result about $\mathbb F_p$-solutions of certain graphic matroids in their own Albanese graphs.