On the degree of subvarieties on abelian varieties

Let $(X,\mathrm{\Theta })$ be a very general principally polarized abelian variety of dimension $g$, and consider the minimal cohomology class ${\theta }_k=[\mathrm{\Theta }{]}^k\mathrm{/}k!$ for $k<g$. We show that the minimal positive multiple of ${\theta }_k$ which is algebraic is divisible by all primes $p\le (k+1)\mathrm{/}2$. 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.