Cubic elliptic functions

Elliptic fibrations on cubic surfaces

We classify elliptic fibrations birational to a nonsingular, minimal cubic surface over a field of characteristic zero. Our proof is adapted to provide computational techniques for the analysis of such fibrations, and we describe an implementation of this analysis in computer algebra.

Trigonometric functions, elliptic functions, elliptic modular forms

Ranks of Elliptic Curves in Cubic Extensions

For an elliptic curve over the rationals, Goldfeld’s conjecture [4] asserts that the analytic rank ords=1 L(Ed/Q, s) of quadratic twists Ed of E is positive for squarefree d’s with density 1/2. In other words, the analytic rank of E goes up in quadratic extensions Q( √ d)/Q half of the time. In particular, for every E/Q there are (a) infinitely many quadratic extensions where the rank goes up, ...

Torsion of elliptic curves over cubic fields

Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as S. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves ...

Semilinear Elliptic Equations with Generalized Cubic Nonlinearities

A semilinear elliptic equation with generalized cubic nonlinearity is studied. Global bifurcation diagrams and the existence of multiple solutions are obtained and in certain cases, exact multiplicity is proved.