Multiplicative excellent families of elliptic surfaces of type E_7 or E_8

TL;DR

This paper constructs explicit families of rational elliptic surfaces with Galois groups matching Weyl groups of types E_7 and E_8, linking their coefficients to invariant rings and providing examples with specific reduction and Galois properties.

Contribution

It introduces explicit multiplicative excellent families of elliptic surfaces associated with E_7 and E_8 Weyl groups, connecting their coefficients to invariant rings and Galois group realizations.

Findings

Explicit families with Galois group W(E_7) or W(E_8)

Examples of elliptic surfaces with rational sections and specific fiber configurations

Construction of polynomials with prescribed Galois groups

Abstract

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).