Dr. Katz DEq Homework Solutions 37

Dr. Katz DEq Homework Solutions 37 - .., 1) in f,...

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Algebraic Solutions of Differential Equations 37 (2.3.8) 7he example, continued. According to (2.3.6), the Hasse invariant of the hypersurface (2.3.7.19) is to satisfy every differential equation satisfied by the differential o~=co(1 ..... 1) (in the notation (2.3.7.10)) considered as a section of R a- 2f. (f2]./s)" By Dwork's congruence (2.3.7.21), this is equivalent to the formal series G(2) (2.3.7.21) satisfying every such differential equation. Because the formal series G(2) is universal, i.e., independent of p, and the formation of the Hodge and De Rham cohomologies of a smooth hypersurface commutes with arbitrary change of base, it follows that we have: (2.3.8.1) Corollary. Consider the hypersurface smooth over the spectrum S of Z[2][1/d(1-2d)] given by the equation (2.3.7.19). For every dif- ferential equation d i (2.3.8.2) 2 ai(2) 17 (d-f)(r in Rd-2f,(~/s) satisfied by
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Unformatted text preview: .., 1) in f, (f2~rTs2), ai(2)~Z[~.] [1/d(l-Aa)], we have (2.3.8.3) ~ai(2) ~ (G(2))=0 in Z [[2-1]], whereG(2)EZ[1][[2-1]]istheseries (2.3.8.4) a(R)=(d2)-I ~ (1/d)a'"(d- 1/d)a a! . ..a! (2.3.8.5) (2.3.8.6) 1~ -ad . Remark. In fact, "the" differential equation satisfied by co is (Z(-' d,"-' as may be deduced from 1-23 a] and an immediate calculation shows that indeed [ d ~ d-1 (2 d ~ d-1 (2.3.8.7) ~-d2-! (G(2)) = -d2-! (2G(2)). (2.3.9) We refer to forthcoming works of B.Mazur for the congruence relations between the higher Hasse-Witt matrices and the zeta function, which generalize the "ordinary" congruence formula [251. 2.4. The Question of Quasieoherence of the Conjugate Spectral Sequence (2.4.0) We return now to the geometric situation of 1.0, and, to fix ideas, we suppose that S is affine. The conjugate spectral sequence (2,4.0.1) E~2'b=Raf,(Jfb(f2[/s(logD))) =~ Ra+bf, (~/s(1Og D))...
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