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Extra resources for p-adic Analysis: Proceedings of the International Conference held in Trento, Italy, May 29–June 2, 1989
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Because v is discrete, so is is a complete discretely valued field of unequal By construction of the Raynaud extension, the Barsotti-Tate groups associated to A / $ resp. to the 1-motive [M/$ ~P~G/$] coincide. t. { t Gauss ' Since A / $ has good reduction modulo the valuation ideal of $ (indeed its reduction is the generic fiber of the reduction of A modulo v , which is proper when v ~ v ) , this pairing has to be trivial: {qij{ Gauss = 1. g) An example: let us consider the Legendre elliptic pencil with parameter x = A , given by the affine equation v2=u(u-1)(u-x).
The comparison theorem for Abelian varieties Let A = A K be an Abelian variety over K . S. BDR. [Faltings and Wintenberger have generalized this pairing to the relative case [W] ; the relative . e. extends to a semi-abelian scheme A R over R. (By a fundamental result of Grothendieck this always happens after replacing K by a finite extension). Let '~R be the formal group attached to A R ; then Tp(,~R)(K----) is the "fixed part" of Tp(AI~ ) [G2]. Now the restricted pairing H 1 R(A) ® Tp(~R)(lk~ ® K is easily described as follows: ~Bcris K0 a) It factorizes through the quotient H1R(AR)K ® Tp(,~R)(K---) .
E) More generally, let us consider a 1-motive [M the universal extension splits canonically: ¢ . , T] , where T is a torus. In this case G b = T x Hom(M,~a )v ; this induces a canonical splitting of the Hodge filtration: H~)R[M , T] = F 1 ¢ Hom(M,K). ~. Gm the bilinear form induced by . ~. Het ~T] ~ M' ®i/Qp(-1) ,0 Now assume that M and M ' are constant. ; let et R (#i). denote a basis of M ' , let d/~i/l+/~ j . be the corresponding basis in F 1 , and let ~j lift /~i/tp. inside Hlet" At last, let (mi) denote the basis of qij = q(mi'#j) .