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By Francesco Baldassarri, Pierre Berthelot, Nick Katz, François Loeser

This two-volume publication collects the lectures given throughout the 3 months cycle of lectures held in Northern Italy among may possibly and July of 2001 to commemorate Professor Bernard Dwork (1923 - 1998). It offers a wide-ranging assessment of a few of the main lively parts of latest learn in mathematics algebraic geometry, with precise emphasis at the geometric functions of thep-adic analytic ideas originating in Dwork's paintings, their connection to numerous contemporary cohomology theories and to modular kinds. the 2 volumes include either vital new examine and illuminating survey articles written via best specialists within the box. The booklet willprovide an imperative source for all these wishing to method the frontiers of analysis in mathematics algebraic geometry.

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Proof. Let ξ ∗ = u∈Zn ∩Lg Au x −u ∈ L ∗ (b). We show that (ξ ∗ ) ∈ L ∗ (b/p). , (γ− (x −v ξ ∗ )) ∈ L∗ (b/p). But ξ ∗ ∈ L ∗ (b) implies γ− (x −v ξ ∗ ) ∈ L∗ (b), so we are reduced to proving (L∗ (b)) ⊆ L∗ (b/p). This follows easily from the definitions. To prove part (b), let η = v∈Zn ∩C(g) Bv x v ∈ L(b ) and choose c ∈ R such that ord Bv ≥ b w(v) + c for all v ∈ Zn ∩ C(g). Let ξ ∗ = u∈Zn ∩Lg Au x −u ∈ L ∗ (b) and let t ∈ Zn ∩ Lg . The coefficient of x −t in ηξ ∗ is Ct = Bv Av+t . 3) v∈Zn ∩C(g) Since γ− (x t ξ ∗ ) ∈ L∗ (b), there exists ct ∈ R such that ord Av+t ≥ −bw(v) + ct for all v ∈ Zn ∩ C(g), hence ord Bv Av+t ≥ b w(v) + c − bw(v) + ct = (b − b)w(v) + c + ct .

3) u∈Zn ∩C(g) ∗ , ξ ∈ RK(λ) , we have For ξ ∗ ∈ RK(λ) ∗ (ξ ∗ ), ξ . 4) 23 Exponential sums and generalized hypergeometric functions ∗ ,ξ ∈R We observe for future reference that if ξ ∗ ∈ RK(λ) K(λ) , then (I − γ− )(ξ ∗ ), ξ = 0, where I is the identity map. 5) n i=1 Di,a,λ RK(λ) and ∗ ∗ | γ− Di,a,λ (ξ ∗ ) = 0 for i = 1, . . , n} Ka,λ = {ξ ∗ ∈ RK(λ) ∗ ∗ | Di,a,λ (ξ ∗ ) = 0 for i = 1, . . } Ka,λ = {ξ ∗ ∈ RK(λ) Thus there are identifications Ka,λ = HomK(λ) (Wa,λ , K(λ)) Ka,λ = HomK(λ) (Wa,λ , K(λ)).

We want to describe Wa = RK / ni=1 Di,a RK . 3, namely, approximating Di,a by gr(gi ) will not work. For example, if g = x1 + x2 , then RK /(g1 RK + g2 RK ) = 0, while we shall see that Wa = 0. 5. If g is nondegenerate and a is semi-nonresonant, then the natural map Wa → Wa induced by the inclusion RK → RK is an isomorphism. 16 Alan Adolphson Proof. Let ξ ∈ RK . Then x u ξ ∈ RK for some u ∈ Zn ∩ C(g). 4, there exists η ∈ RK such that xuη ≡ xuξ n (mod Di,a−u RK ). i=1 Dividing this equation by x u gives n η≡ξ (mod Di,a (x −u RK )), i=1 which proves the surjectivity of Wa → Wa .

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