The weights of irreducible $SL_3(q)$-modules in the defining characteristic

This is a final step in solving the problem of recognition of the simple groups $L_3(p^k)$ by element orders. It is proven that when $L_3(p^k)$ acts on an elementary abelian $p$-group, there always appears an element of new order. A model is proposed for constructing the absolutely irreducible $p$-modular representations of $L_3(p^k)$ in polynomial spaces.