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Let $X$ be a general proper and smooth curve of genus $2$ (resp. of genus $3$) defined over an algebraically closed field of characteristic $p$. When $3\le p\le 7$, the action of Frobenius on rank $2$ semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order $2$ line bundle over $X$. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian...