The theory of Mordell-Weil lattices is applied to a specific example of a rational elliptic surface. This provides a complete description of the sections of this surface, and of the sections which are defined over Q. The surface is related to the diophantine problem of expressing squares as a sum of consecutive squares. Some consequences which our description has to this problem are discussed.

We explore higher-form symmetries of M- and F-theory compactified on elliptic fibrations, determined by the topology their asymptotic boundaries. The underlying geometric structures are shown to be equivalent known characterizations gauge group in via Mordell--Weil torsion string junctions. further study dimensional reductions 11d Chern--Simons term presence torsional boundary $G_4$-fluxes, whi...

We analyze K3 surfaces admitting an elliptic fibration E and a finite group G of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration E/G comparing its properties to the ones of E. We show that if E admits an n-torsion section, its quotient by the group of automorphisms induced by this section admits again an n-torsion section. Considering au...

A bstract We prove that, for every 6D supergravity theory that has an F-theory description, the property of charge completeness connected component gauge group (meaning all charges in corresponding lattice are realized by massive or massless states theory) is equivalent to a standard assumption made how geometry encodes global means Mordell-Weil elliptic fibration. This result also holds 4D con...

We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and Ellenberg, when the base field has characteristic zero and the supports of the conductor of the elliptic curve and of the ramification divisor of the extension a...