Finite coverings of algebraic varieties and group schemes over a ring of mixed characteristics

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(1) Define a formal power series E_p(U, LAMBDA ; T) epsilonQ[U,LAMBDA][[T]] byE_p(U, LAMBDA ; T)=(1+$KT)^<U/(bda)>II^^*__(1+LAMBDA^<pk>T^<pk>)^<pk/{((LAMBDA)/)^<pk>-((LAMBDAk)/)pk-1>}The Artin-Hasse exponential seiries E_p(T) was defined byE<@D2p@>D2(T)=exp(SIGMA<@D6*(/)k=0@>D6<@D7T<@D1pk@>D1(/)p<@D1k@>D1@>D7) (Artin-Hasse exponential series).We have proved the equality<<numerical formula>>E_p(T) corresponds to exp t and E_p (U, LAMBDA ; T) to (1+lambdat)^<1/lambda> in the well known formula<lambda * 0>___(1+lambdat)^<1/lambda> = expt(2) We have improved some results by Gouvea and Yui on special values of the congruence zeta function and the discriminant on the intersection forms of algebraic cycles for a diagonal hypersurfaces, correcting defects of their method.