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Additional resources for p-adic Analysis: Proceedings of the International Conference held in Trento, Italy, May 29–June 2, 1989

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T. group associated to a 1-motive [M ¢ * T] , ,~ p where ¢ is given by the Serre--Tate parameters [K] . __Atfunctorial, as is easily seen from the case of complex multiplication (¢ = 1). e) The construction of Frobenius generalizes easily to the case of 1-motives. This allows to construct p-adic Betti lattices for 1-motives whose Abelian part has multiplicative reduction. ] 52 3. Computation of periods. a) We shall compute the matrix of the restriction of {d#i/l+#i} g - " in F 1 , {#~ = ~ . t, the bases mV}g=l in H~, assuming that T splits over j=l K.

The map natural Let Hom(M(K),Qp) ~ can be identified with H~t [M the injection of fi-modules : *G] . 7. M. is tautological for the quotient 1-motive [M(K) ......... ,~1] (whose associated p--divisible group is ~ (~p/~p)n) . ~Wp(Gm)(= ~p(1)) . By using an orientation of ~:p and duality, we get from c) an injection: Jet: M'v (1) := M 'v ® X,(Gm) Using the Raynaud 1-motive [M ~ G] pairing between H1 R and M~V(1) ~. ) . K ~" over 1~, it is then clear that the Fontaine-Messing takes its values in K't for some finite unramified P extension K / of K (even in Ktp if the torus part of the semi---stable reduction ~ splits), 40 2.

2. Construction of H ~ A ) • From now onwards, we shall assume that A has multiplicative reduction. , by prop. , M / (K) ®~ K ) . M. is described in 1 4 c). The splitting of BT (up to isogeny) reflects on H~, and translates into an isomorphism: ~fl : u r (~o acts trivially on W~I ~ ~1 n0~ no Gr 0 = MY(K)®K 0 , and by multiplication by g p on the image of Gr I = M r(K) ~ K 0 ) . Let us now choose an orientation of ~p (see III 10: g ( - 1) := X • (Gm) ~-~ ~tp 1 C BDR ,and let us consider the etale lattice A_:= M v ~ M ~ ( - 1) , and let A := A_(I~) = A__(R---)= _A(Knr) , where K nr denotes the maJdmal subfield of K non ramified over K .

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