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12
Definable sets, motives and padic integrals
 J. Amer. Math. Soc
, 2001
"... 0.1. Let X be a scheme, reduced and separated, of finite type over Z. For p a prime number, one may consider the set X(Zp) ofitsZprational points. For every n in N, there is a natural map πn: X(Zp) → X(Z/pn+1) assigning to a Zprational point its class modulo pn+1. The image Yn,p of X(Zp) byπn is ..."
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Cited by 44 (10 self)
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0.1. Let X be a scheme, reduced and separated, of finite type over Z. For p a prime number, one may consider the set X(Zp) ofitsZprational points. For every n in N, there is a natural map πn: X(Zp) → X(Z/pn+1) assigning to a Zprational point its class modulo pn+1. The image Yn,p of X(Zp) byπn is exactly the set of
Bertini theorems over finite fields
 Ann. of Math
"... Abstract. Let X be a smooth quasiprojective subscheme of P n of dimension m ≥ 0 over Fq. Then there exist homogeneous polynomials f over Fq for which the intersection of X and the hypersurface f = 0 is smooth. In fact, the set of such f has a positive density, equal to ζX(m + 1) −1, where ζX(s) = Z ..."
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Cited by 32 (8 self)
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Abstract. Let X be a smooth quasiprojective subscheme of P n of dimension m ≥ 0 over Fq. Then there exist homogeneous polynomials f over Fq for which the intersection of X and the hypersurface f = 0 is smooth. In fact, the set of such f has a positive density, equal to ζX(m + 1) −1, where ζX(s) = ZX(q −s) is the zeta function of X. An analogue for regular quasiprojective schemes over Z is proved, assuming the abc conjecture and another conjecture. 1.
INDEPENDENCE OF ℓ OF MONODROMY GROUPS
, 2002
"... Abstract. Let X be a smooth curve over a finite field of characteristic p, let E be a number field, and let {Lλ} be an absolutely irreducible Ecompatible system of lisse sheaves of finite order determinant on the curve X. For each place λ of E not lying over p, the lisse sheaf Lλ has an associated ..."
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Cited by 3 (1 self)
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Abstract. Let X be a smooth curve over a finite field of characteristic p, let E be a number field, and let {Lλ} be an absolutely irreducible Ecompatible system of lisse sheaves of finite order determinant on the curve X. For each place λ of E not lying over p, the lisse sheaf Lλ has an associated arithmetic monodromy group, which is a (possibly nonconnected) semisimple algebraic group over the local field Eλ. We prove an “independence of ℓ ” assertion (5.1) on the isomorphism type of the identity component of these groups: after replacing E by a finite extension, there exists a connected split semisimple algebraic group G0 over the number field E such that for every place λ of E not lying over p, the identity component of the arithmetic monodromy group of Lλ is isomorphic to the group G0 with scalars extended to the local field Eλ. Let X be a smooth curve over a finite field of characteristic p. Let E be a number field, and consider an Ecompatible system {Lλ} of lisse sheaves on X. This means that for every place λ of E not lying over p, we are given a lisse Eλsheaf Lλ on X, and these lisse sheaves are Ecompatible with one another in the sense that for every closed point x of X, the polynomial
ON THE REGULATOR OF FERMAT MOTIVES AND GENERALIZED HYPERGEOMETRIC FUNCTIONS
, 909
"... ABSTRACT. We calculate the Beilinson regulators of motives associated to Fermat curves and express them by special values of generalized hypergeometric functions. As a result, we obtain surjectivity results of the regulator, which support the Beilinson conjecture on special values of Lfunctions. 1. ..."
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ABSTRACT. We calculate the Beilinson regulators of motives associated to Fermat curves and express them by special values of generalized hypergeometric functions. As a result, we obtain surjectivity results of the regulator, which support the Beilinson conjecture on special values of Lfunctions. 1.
ARITHMETIC EQUIVALENCE FOR FUNCTION FIELDS, THE GOSS ZETA FUNCTION AND A GENERALIZATION
, 906
"... ABSTRACT. A theorem of Tate and Turner says that global function fields have the same zeta function if and only if the Jacobians of the corresponding curves are isogenous. In this note, we investigate what happens if we replace the usual (characteristic zero) zeta function by the positive characteri ..."
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ABSTRACT. A theorem of Tate and Turner says that global function fields have the same zeta function if and only if the Jacobians of the corresponding curves are isogenous. In this note, we investigate what happens if we replace the usual (characteristic zero) zeta function by the positive characteristic zeta function introduced by Goss. We prove that for function fields whose characteristic exceeds their degree, equality of the Goss zeta function is the same as Gassmannequivalence (a purely group theoretical property), but this statement fails if the degree exceeds the characteristic. We introduce a ‘Teichmüller lift ’ of the Goss zeta function and show that equality of such is always the same as Gassmann equivalence. 1.
MOTIVES FOR PERFECT PAC FIELDS WITH PROCYCLIC GALOIS GROUP
, 704
"... Abstract. Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudofinite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with procyclic ..."
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Abstract. Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudofinite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with procyclic Galois group. In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of DenefLoeser is not