Multiplicative excellent families of elliptic surfaces of type E_7 or E_8

We describe explicit multiplicative excellent families of rational elliptic surfaces with Galois group isomorphic to the Weyl group of the root lattices E_7 or E_8. The Weierstrass coefficients of each family are related by an invertible polynomial transformation to the generators of the multiplicative invariant ring of the associated Weyl group, given by the fundamental characters of the corresponding Lie group. As an application, we give examples of elliptic surfaces with multiplicative reduction and all sections defined over Q for most of the entries of fiber configurations and Mordell-Weil lattices in [Oguiso-Shioda '91], as well as examples of explicit polynomials with Galois group W(E_7) or W(E_8).Comment: 23 pages. Final versio

Lines on Fermat surfaces

We prove that the Neron-Severi groups of several complex Fermat surfaces are generated by lines. Specifically, we obtain these new results for all degrees up to 100 that are relatively prime to 6. The proof uses reduction modulo a supersingular prime. The techniques are developed in detail. They can be applied to other surfaces and varieties as well.Comment: 29 pages; v3: major extension thanks to RvL who joined as third author; results and techniques strengthened, paper reorganise