Characteristic numbers of elliptic fibrations with non-trivial Mordell-Weil groups

We compute characteristic numbers of elliptically fibered fourfolds with multisections or non-trivial Mordell-Weil groups. We first consider the models of type E${}_{9-d}$ with $d=1,2,3,4$ whose generic fibers are normal elliptic curves of degree $d$. We then analyze the characteristic numbers of the $Q_7$-model, which provides a smooth model for elliptic fibrations of rank one and generalizes the E${}_5$, E${}_6$, and E${}_7$-models. Finally, we examine the characteristic numbers of $G$-models with $G=\text{SO}(n)$ with $n=3,4,5,6$ and $G=\text{PSU}(3)$ whose Mordell-Weil groups are respectively $\mathbb{Z}\mathrm{/}2\mathbb{Z}$ and $\mathbb{Z}\mathrm{/}3\mathbb{Z}$. In each case, we compute the Chern and Pontryagin numbers, the Euler characteristic, the holomorphic genera, the Todd-genus, the L-genus, the A-genus, and the eight-form curvature invariant from M-theory.