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The CasselsTate Pairing On Polarized Abelian Varieties
 Ann. of Math
, 1998
"... . Let (A; ) be a principally polarized abelian variety dened over a global eld k, and let X(A) be its Shafarevich{Tate group. Let X(A) nd denote the quotient of X(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing X(A) nd X(A) nd ! Q=Z : If A is an ellip ..."
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. Let (A; ) be a principally polarized abelian variety dened over a global eld k, and let X(A) be its Shafarevich{Tate group. Let X(A) nd denote the quotient of X(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing X(A) nd X(A) nd ! Q=Z : If A is an elliptic curve, then by a result of Cassels' the pairing is alternating. But in general it is only antisymmetric. Using some new but equivalent denitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on X(A) nd . These criteria are expressed in terms of an element c 2 X(A) nd that is canonically associated to the polarization . In the case that A is the Jacobian of some curve, a downtoearth version of the result allows us to determine eectively whether #X(A) (if nite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperell...
Analytic padic cell decomposition and integrals
 Trans. Amer. Math. Soc
"... Abstract. We prove a conjecture of Denef on parameterized padic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), ..."
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Abstract. We prove a conjecture of Denef on parameterized padic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection. 1.
Topology of Diophantine sets: remarks on Mazur’s conjectures. In Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent
 of Contemp. Math
, 1999
"... Abstract. We show that Mazur’s conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers Z in the rational numbers Q, i.e., there is no diophantine set D in some cartesian power Q i such that there exist two binary relations S ..."
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Abstract. We show that Mazur’s conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers Z in the rational numbers Q, i.e., there is no diophantine set D in some cartesian power Q i such that there exist two binary relations S, P on D whose graphs are diophantine in Q 3i (via the inclusion D 3 ⊂ Q 3i), and such that for two specific elements d0, d1 ∈ D the structure (D, S, P, d0, d1) is a model for integer arithmetic (Z,+, ·,0, 1). Using a construction of Pheidas, we give a counterexample to the analogue of Mazur’s conjecture over a global function field, and prove that there is a diophantine model of the polynomial ring over a finite field in the ring of rational functions over a finite field. 1.
OneDimensional PAdic Subanalytic Sets
 J. London Math. Soc
"... Introduction In this article we extend two theorems from [2] on padic subanalytic sets, where p is a fixed prime number, Q p is the field of padic numbers and Z p is the ring of padic integers. One of these theorems, 3.32 in [2], says that each subanalytic subset of Z p is semialgebraic. This is ..."
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Introduction In this article we extend two theorems from [2] on padic subanalytic sets, where p is a fixed prime number, Q p is the field of padic numbers and Z p is the ring of padic integers. One of these theorems, 3.32 in [2], says that each subanalytic subset of Z p is semialgebraic. This is extended here as follows. Theorem A Let S ` Z m+1 p be a subanalytic set. Then there is a semialgebraic set S 0 ` Z m 0 +1 p such that for each x 2 Z m p there is x 0 2 Z m 0 p with S x = S 0 x 0 . Hence the semialgebraic c
Effective Equidistribution of Sintegral points on symmetric varieties
"... Abstract. Let K be a global field of characteristic not 2. Let Z = H\G be a symmetric variety defined over K and S a finite set of places of K. We obtain counting and equidistribution results for the Sintegral points of Z. Our results are effective when K is a number field. 1. ..."
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Abstract. Let K be a global field of characteristic not 2. Let Z = H\G be a symmetric variety defined over K and S a finite set of places of K. We obtain counting and equidistribution results for the Sintegral points of Z. Our results are effective when K is a number field. 1.
On Some Rational Generating Series Occuring in Arithmetic Geometry
"... this paper, by a variety over a ring R, we mean a reduced and separated scheme of finite type over Spec R ..."
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this paper, by a variety over a ring R, we mean a reduced and separated scheme of finite type over Spec R
Classification of SemiAlgebraic PAdic Sets Up to SemiAlgebraic Bijection.
"... We prove that two infinite padic semialgebraic sets are isomorphic (i.e. there exists a semialgebraic bijection between them) if and only if they have the same dimension. The real semialgebraic sets can be classified up to semialgebraic bijection as follows [4]: There exists a real semialg ..."
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We prove that two infinite padic semialgebraic sets are isomorphic (i.e. there exists a semialgebraic bijection between them) if and only if they have the same dimension. The real semialgebraic sets can be classified up to semialgebraic bijection as follows [4]: There exists a real semialgebraic bijection between two real semialgebraic sets if and only if they have the same dimension and Euler characteristic. More generally L. van den Dries [4] gave such a classification for ominimal expansions of the real field, using the dimension and Euler characteristic as defined for ominimal structures. Since this Euler characteristic is in fact the canonical map from the real semialgebraic sets onto the Grothendieck ring of R (which is Z), we see that the isomorphism class of a semialgebraic set only depends on its image in the Grothendieck ring and its dimension. In this paper we treat the padic analogue of this classification. The Grothendieck ring of Q p is recently proved...
Uniform properties of rigid subanalytic sets, submitted
"... Abstract. In the context of rigid analytic spaces over a nonArchimedean valued field, a rigid subanalytic set is a Boolean combination of images of rigid analytic maps. We give an analytic quantifier elimination theorem for (complete) algebraically closed valued fields that is independent of the fi ..."
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Abstract. In the context of rigid analytic spaces over a nonArchimedean valued field, a rigid subanalytic set is a Boolean combination of images of rigid analytic maps. We give an analytic quantifier elimination theorem for (complete) algebraically closed valued fields that is independent of the field; in particular, the analytic quantifier elimination is independent of the valued field’s characteristic, residue field and value group, in close analogy to the algebraic case. This provides uniformity results about rigid subanalytic sets. We obtain uniform versions of smooth stratification for subanalytic sets and the ÃLojasiewicz inequalities, as well as a unfiorm description of the closure of a rigid semianalytic set. 1. Introduction. A