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 might and July of 2001 to commemorate Professor Bernard Dwork (1923 - 1998). It offers a wide-ranging review of a few of the main lively components of up to date examine in mathematics algebraic geometry, with unique emphasis at the geometric functions of thep-adic analytic recommendations originating in Dwork's paintings, their connection to varied fresh cohomology theories and to modular types. the 2 volumes comprise either very important new study and illuminating survey articles written via top specialists within the box. The ebook willprovide an quintessential source for all these wishing to technique the frontiers of study in mathematics algebraic geometry.

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Then a Ng is also nonresonant so we get a basis {ξi,a ,λp }i=1 of Ka ,λp . We define γ (a, a , λ) to ∗ be the transpose of the matrix of the isomorphism αa,a ,λ relative to these bases. Since N g a − u is also nonresonant for all u ∈ Zn ∩ L(g), we also have bases {ξi,a−u,λ }i=1 for the Ka−u,λ . We recall that M(a, a − u, λ) is the transpose of the matrix of x u : Ka−u,λ → Ka,λ relative to these bases. 12. Assume B ⊆ Z[λ]. For fixed µ ∈ Zn , γ (a, (a + µ)/p, λ) extends to a function of (a, λ) meromorphic in the region inf(1, ord (a + µ)) > eλ + 1 1 + p p−1 with polar factor det M(a , a − u0 , λp ), where a = (a + µ)/p and u0 is any element of Zn ∩ C(g) such that Zn ∩ u0 − µ + C(g) ⊆ C(g).

8) 21 Exponential sums and generalized hypergeometric functions We conclude this section with a result on the b-invariant that will be used in section 8. M(d +1) bm/M (λ(0) ) = 0. 6. Suppose λ(0) ∈ K¯ N satisfies m=0g (0) g(λ , x) is nondegenerate relative to (g) and dimK(λ(0) ) Wa,λ(0) = Ng . In particN g is a basis for Wa,λ(0) . ular, {x uj }j =1 Proof. 3 implies that dimK(λ(0) ) Wa,λ(0) = Ng . 3 that if M(dg +1) bm/M (λ(0) ) m=0 N g = 0, then {x uj }j =1 spans Wa,λ(0) , hence it is a ba- M(d +1) sis for Wa,λ(0) .

Let gτ = ν∈τ ∩ gν (λ(0) )x ν , a homogeneous element of degree 1 in the graded ring gr(RK(λ(0) ) )τ . 2] implies that g(λ(0) , x) is nondegenerate. 3, the nonvanishing of m=0g bm/M (λ(0) ) implies that N g {x uj }j =1 spans gr(RK(λ(0) ) )/ ni=1 gr(gi (λ(0) , x))gr(RK(λ(0) ) ). 9) by showing that the subset {x uj }uj ∈C(τ ) spans the quotient n gr(RK(λ(0) ) )τ i=1 gτ,i gr(RK(λ(0) ) )τ . Let ξ be a homogeneous element of degree m/M in gr(RK(λ(0) ) )τ . 10) i=1 where ci ∈ K(λ(0) ), ηi ∈ gr(RK(λ(0) ) )(m/M−1) , and the subscript τ appearing on the right-hand side means that we select only those terms from the product gr(gi (λ(0) , x))ηi that lie in gr(RK(λ(0) ) )τ .

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