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

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

for all $n\geq 1$ .

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

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

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