bielliptic

Bielliptic is an adjective used in algebraic geometry to describe objects that are related to elliptic curves in a specific two-to-one way. It has two principal uses in the study of curves and surfaces.

For algebraic curves, a smooth projective curve X is bielliptic if there exists a non-constant morphism of degree two from X to an elliptic curve E. Equivalently, X admits a nontrivial involution i such that the quotient X/i is an elliptic curve (genus one). This is a special property: many curves do not admit such a map, while others of genus g ≥ 2 do. A bielliptic curve has implications for its Jacobian, in that the Jacobian typically contains a quotient that is an elliptic curve up to isogeny.

For complex surfaces, the term bielliptic (also called a hyperelliptic surface in older literature) refers to a compact complex surface whose universal cover is a product of two elliptic curves and which can be written as a quotient (E1 × E2)/G by a finite group G acting freely. Bielliptic surfaces are a class of Kodaira dimension zero surfaces in the Enriques–Kodaira classification; they have irregularity q = 1, geometric genus p_g = 0, and a torsion canonical bundle. There are seven families in this class, classified by Bagnera and de Franchis, corresponding to the possible finite group actions and elliptic curves involved.

In summary, bielliptic indicates a special relationship with elliptic curves, either via a degree-two map from a curve to an elliptic curve or as a structured quotient of a product of elliptic curves in the setting of complex surfaces.