Dr. Katz DEq Homework Solutions 37

Dr. Katz DEq Homework Solutions 37 - 1 in f(f2~rTs2...

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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 ( is to satisfy every differential equation satisfied by the differential o~=co(1 ..... 1) (in the notation ( considered as a section of R a- 2f. (f2]./s)" By Dwork's congruence (, this is equivalent to the formal series G(2) ( 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: ( Corollary. Consider the hypersurface smooth over the spectrum S of Z[2][1/d(1-2d)] given by the equation ( For every dif- ferential equation d i ( 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 ( ~ai(2) ~ (G(2))=0 in Z [[2-1]], whereG(2)EZ[1][[2-1]]istheseries ( a(R)=(d2)-I ~ (1/d)a'"(d- 1/d)a a! . ..a! ( ( 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 ( ~-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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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.

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