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For any two elements f, g ∈ Can (D), we define their distance as ∞ dist(f, g) := m=1 2−m ||f − g||cm . 1 + ||f − g||cm One can show (a), that with respect to dist the C∞ -linear space Can (D) is a complete linear metric space, and (b), that the topology on this space does not depend on the choice of the sequence (cn ) (provided that cn → c and cn > c for all n). In the following, Zp is equipped with its usual locally compact topology. Proposition 7. If L(X, F, (z, w))) converges on Hc , then the function w → (z → L(X, F, (z, w))) defines a continuous function Zp → Can (Dc∗ ).

For any x ∈ |X|, the stalk F x is flat, and so is F x ⊗B Q(B). By Theorem 9 the latter is canonically represented by the locally free τ -sheaf (F x ⊗B Q(B))ss . The endomorphism τ dx is kx ⊗ Q(B)-linear. By Lemma 1, part 1, the following definition makes sense. Definition 11. The L-function of F at x is L(x, F, t) := det (id − t dx τ dx | (F x ⊗B Q(B))ss )−1 ∈ kx ⊗ Q(B)[[t dx ]]. kx ⊗Q(B) Lemma 2. 1. The power series L(x, F, t) lies in 1 + t dx B[[t dx ]]. 2. The assignment F → L(x, F, t) is multiplicative in short exact sequences.

2) i≥0 Then di = 0 for i ≥ d odd, Pi = 0 for i > r/2, while for d/2 ≤ i ≤ r/2 we have Pi = d2i ; for these i also α2i,j = pi . Proof. The lemma follows by induction on r. Indeed, assume that either r = d or n | ≤ d p in/2 , we have that the lemma holds for r − 1. As | 1≤j ≤di αi,j i r n = αi,j (−1)i i=0 1≤j ≤di n + o(p nr/2 ) αr,j (n → ∞) 1≤j ≤dr and also, using (2), that Pi = 0 for i > r/2. Note that if z is an element of a finite product (S 1 )s of complex unit circles, then the closure of {zn | n ≥ 1} contains the unit element.

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