In algebraic geometry, a Dwork family is a one-parameter family of

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Eucl ...

s depending on an integer ''n'', studied by

Bernard Dwork Bernard Morris Dwork (May 27, 1923 – May 9, 1998) was an American mathematician, known for his application of ''p''-adic analysis to local zeta functions, and in particular for a proof of the first part of the Weil conjectures: the rationality ...

. Originally considered by Dwork in the context of

local zeta-function In number theory, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as :Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^m\right) where is a non-singular -dimensional projective alge ...

s, such families have been shown to have relationships with

mirror symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D t ...

and extensions of the

modularity theorem The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. ...

Definition

The Dwork family is given by the equations :

x_1^n + x_2^n +\cdots +x_n^n = -n\lambda x_1x_2\cdots x_n \, ,

for all

n\ge 1

References

* Algebraic geometry {{algebraic-geometry-stub