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.

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 $n\geq 1$.

History

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

References

• Katz, Nicholas M. (2009), "Another look at the Dwork family", Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II (PDF), Progress in Mathematics, vol. 270, Boston, MA: Birkhäuser Boston, pp. 89–126, MR 264118889-126&rft.pub=Birkhäuser Boston&rft.date=2009&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=2641188#id-name=MR&rft.aulast=Katz&rft.aufirst=Nicholas M.&rft_id=http://www.math.princeton.edu/~nmk/dworkfam64.pdf&rfr_id=info:sid/en.wikipedia.org:Dwork family" class="Z3988">