Dwork, B. On the Tate constant. (English) Zbl 0622.14016 Compos. Math. 61, 43-59 (1987). The author’s interest in a p-adic analytic approach to the congruence zeta function originated from an unpublished theorem of Tate of 1958. It asserted integrality of a series exp(Cu(t)) where u(t) is an integral of the first kind on an elliptic curve E with good reduction mod p and C is a certain (”Tate”) constant closely related to the zeta function of E mod p. This theorem was explained by the author via the p-adic theory of the hypergeometric equation with parameters (1/2,1/2,1). More recent results of the author on general hypergeometric operators and their (p-adic) ”normalized solution matrices” permit a wide generalization of Tate’s theorem, as explained in the present paper. Reviewer: F.Baldassarri Cited in 1 ReviewCited in 2 Documents MSC: 14G20 Local ground fields in algebraic geometry 12H25 \(p\)-adic differential equations 33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) Keywords:Tate constant; congruence zeta function; p-adic theory of the hypergeometric equation PDFBibTeX XMLCite \textit{B. Dwork}, Compos. Math. 61, 43--59 (1987; Zbl 0622.14016) Full Text: Numdam EuDML References: [1] B. Dwork : Norm residue symbol in local number fields . Hamb. Abh. 22 (1958) 180-190. · Zbl 0083.26001 [2] B. Dwork : A deformation theory for the zeta function of a hypersurface . Proc. Inst. Cong. Math. (1962) 247-259. · Zbl 0196.53302 [3] B. Dwork : p-adic cycles . Pub. Math. IHES 39 (1969) 27-111. · Zbl 0284.14008 [4] B. Dwork : Normalized period matrices I . Ann. of Math 94 (1971) 337-388. · Zbl 0241.14011 [5] B. Dwork : On p-adic differential equations IV . Ann. Sc. Ecole Norm Sup. 6 (1973) 295-316. · Zbl 0309.14020 [6] B. Dwork : Lectures on p-adic differential equations . Springer-Verlag (1982). · Zbl 0502.12021 [7] B. Dwork : Boyarsky principle . Amer. J. Math. 105 (1983) 115-156. · Zbl 0517.12012 [8] L. Lutz : Sur l’équation y2 = x3 - Ax - B dans les corps p-adiques . J. Reine Angew. Math. 177 (1937) 238-247. · Zbl 0017.05307 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.