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]

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 2641188

1. Totaro, Burt (2007). "Euler and algebraic geometry" (PDF). Bulletin of the American Mathematical Society. 44 (4): 541–559. doi:10.1090/S0273-0979-07-01178-0. MR 2338364. p. 545
2. Movasati, Hossein; Nikdelan, Younes (2021-09-01). "Gauss-Manin Connection in Disguise: Dwork Family". Journal of Differential Geometry. 119 (1). arXiv:1603.09411. doi:10.4310/jdg/1631124264. ISSN 0022-040X.