# zbMATH — the first resource for mathematics

A family of K3 surfaces and $$\zeta(3)$$. (English) Zbl 0541.14007
In Apéry’s proof of the irrationality of $$\zeta(3)$$ a crucial role is played by the differential operator $$L=(t^ 4-34t^ 3+t^ 2)(d/dt)^ 3+(6t^ 3-153t^ 2+3t)(d/dt)^ 2+(7t^ 2-112t+1)d/dt+(t-5).$$ The authors prove here that $$Ly=0$$ is Picard-Fuchs equation, namely that its solutions are the periods of the (unique) holomorphic two form $$S_ t$$ of the surface $$S_ t:1-(1-XY)Z-tXYZ(1-X)(I-Y)(1-Z)=0$$ viewed as functions of the parameter t. - The authors also prove that, for generic t, $$S_ t$$ is birationally equivalent to a K3 surface with Picard number 19.
Reviewer: F.Baldassarri

##### MSC:
 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) 14J25 Special surfaces 11J81 Transcendence (general theory) 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 14H25 Arithmetic ground fields for curves
##### Keywords:
Picard-Fuchs equation
Full Text: