Dwork family

In algebraic geometry, a Dwork family is a one-parameter family of hypersurfaces depending on an integer n, studied by Bernard Dwork. Originally considered by Dwork in the context of local zeta-functions, such families have been shown to have relationships with mirror symmetry and extensions of the modularity theorem.^[1]

Definition

The Dwork family is given by the equations

${\displaystyle x_{1}^{n}+x_{2}^{n}+\cdots +x_{n}^{n}=-n\lambda x_{1}x_{2}\cdots x_{n}\,,}$

for all ${\displaystyle n\geq 1}$.

History

The Dwork family was originally used by B. Dwork to develop the deformation theory of zeta functions of nonsingular hypersurfaces in projective space.^[2]