Beukers, F.; Peters, C. A. M. A family of \(K3\) surfaces and \(\zeta(3)\). (English) Zbl 0541.14007 J. Reine Angew. Math. 351, 42-54 (1984). 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: Francesco Baldassarri (Padova) Cited in 5 ReviewsCited in 14 Documents 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 PDFBibTeX XMLCite \textit{F. Beukers} and \textit{C. A. M. Peters}, J. Reine Angew. Math. 351, 42--54 (1984; Zbl 0541.14007) Full Text: DOI EuDML