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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 67 (9 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.
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 60 (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
Chow group of 0cycles with modulus and higher dimensional class field theory, preprint
, 2013
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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 4 (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.
Meanperiodicity and zeta functions
, 2008
"... Abstract. This paper establishes new bridges between the class of complex functions, which contains zeta functions of arithmetic schemes and closed with respect to product and quotient, and the class of meanperiodic functions in several spaces of functions on the real line. In particular, the merom ..."
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Abstract. This paper establishes new bridges between the class of complex functions, which contains zeta functions of arithmetic schemes and closed with respect to product and quotient, and the class of meanperiodic functions in several spaces of functions on the real line. In particular, the meromorphic continuation and functional equation of the Hasse zeta function of arithmetic scheme with its expected analytic shape is shown to imply the meanperiodicity of a certain explicitly de ned function associated to the zeta function. Conversely, the meanperiodicity of this function implies the meromorphic continuation and functional equation of the zeta function. This opens a new road to the study of zeta functions via the theory of meanperiodic functions which is a part of modern harmonic analysis. The case of elliptic curves over number elds and their regular models is treated in more details, and many other examples are included as well. Contents
Navigating the motivic world
"... 1. A first look 3 2. Formal statement of the conjectures 8 ..."
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