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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
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