Reduced norms and the Riemann-Roch theorem for Abelian varieties

We explain how the Riemann-Roch theorem for divisors on an abelian variety $A$ is related to the reduced norms of the Wedderburn components of $\operatorname{End}^0(A)$ the $\mathbb{Q}$-endomorphism algebra of $A$. We then describe consequences and examples.