Dwork family

Short description: Family of hypersurfaces in algebraic geometry

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.

Definition

The Dwork family is given by the equations

$x_1^n+x_2^n+\cdots +x_n^n=-n\lambda x_1{x}_2\cdots x_n,$

for all $n\ge 1$.

History

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

References

• Katz, Nicholas M. (2009), "Another look at the Dwork family", Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Progress in Mathematics, 270, Boston, MA: Birkhäuser Boston, pp. 89–126, http://www.math.princeton.edu/~nmk/dworkfam64.pdf