R�duction modulop n des sous-ensembles analytiques ferm�s deZ p N

Oesterlé, Joseph

doi:10.1007/bf01389398

Cited by 52 publications

(46 citation statements)

References 5 publications

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“…(a) If X is affine, this is a consequence of the discussion surrounding Théorème 2 of [Oes82], which builds on [Ser81,S3], and the Hahn-Kolmogorov extension theorem. In general, let (X i ) be a finite affine open cover of X.…”

Section: Proofmentioning

confidence: 99%

Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves

Bhargava

Kane

Lenstra

et al. 2015

Cambridge Journal of Mathematics

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Abstract. Using maximal isotropic submodules in a quadratic module over Z p , we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of co-finite type Z p -modules, and then conjecture that as E varies over elliptic curves over a fixed global field k, the distribution of→ 0 is that one. We show that this single conjecture would explain many of the known theorems and conjectures on ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves. We also prove the existence of a discrete probability distribution of the set of isomorphism classes of finite abelian p-groups equipped with a nondegenerate alternating pairing, defined in terms of the cokernel of a random alternating matrix over Z p , and we prove that the two probability distributions are compatible with each other and with Delaunay's predicted distribution for X. Finally, we prove new theorems on the fppf cohomology of elliptic curves in order to give further evidence for our conjecture.

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Section: Proofmentioning

confidence: 99%

Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves

Bhargava

Kane

Lenstra

et al. 2015

Cambridge Journal of Mathematics

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“…The canonical d -dimensional measure on X 0 , cf. [31], [28], is the one induced by the volume form j! 0 j X 0 .…”

Section: 2mentioning

confidence: 99%

Constructible exponential functions, motivic Fourier transform and transfer principle

2010

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We introduce spaces of exponential constructible functions in the motivic setting for which we construct direct image functors in the absolute and relative settings. This allows us to define a motivic Fourier transformation for which we get various inversion statements. We also define spaces of motivic Schwartz-Bruhat functions on which motivic Fourier transformation induces isomorphisms. Our motivic integrals specialize to nonarchimedean integrals. We give a general transfer principle comparing identities between functions defined by exponential integrals over local fields of characteristic zero, resp. of positive characteristic, having the same residue field. We also prove new results about p-adic integrals of exponential functions and stability of this class of functions under p-adic integration.

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“…We obtain in this way a structure for the language L Pas . By a definable subset of [30], [25]), which we shall denote by ν K , for which all definable subsets of X (O K ) are measurable, when K is of characteristic zero (cf. [8], [9], [31]).…”

Section: Byétale -Adic Realization Factorizes Through a Morphisḿmentioning

confidence: 99%

Definable sets, motives and 𝑝-adic integrals

Denef¹,

Loeser²

2000

J. Amer. Math. Soc.

149

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Introduction 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(Z p ) of its Z p -rational points. For every n in N, there is a natural map π n : X(Z p ) → X(Z/p n+1 ) assigning to a Z p -rational point its class modulo p n+1 . The image Y n,p of X(Z p ) by π n is exactly the set of Z/p n+1 -rational points which can be lifted to Z p -rational points. Denote by N n,p the cardinality of the finite set Y n,p . By a result of the first author [7], the Poincaré seriesis a rational function of T . Later Macintyre [24], Pas [26] and Denef [10] proved that the degrees of the numerator and denominator of the rational function P p (T ) are bounded independently of p. One task of the present paper is to prove a much stronger uniformity result by constructing a canonical rational function P ar (T ) which specializes to P p (T ) for almost all p. It follows in particular from our results that there exist, for every n in N, varieties Z n,i and rational numbers r n,i in Q, 1 ≤ i ≤ m n , such that, for almost all p and every n,Hence a natural idea would be to try to construct P ar (T ) as a series with coefficients in K 0 (Sch Q ) ⊗ Q, with K 0 (Sch Q ) the "Grothendieck ring of algebraic varieties over Q", defined in 1.2. However, since different varieties over a number field may have the same L-function, and we want the function P ar (T ) to be canonical, we have to replace the varieties Z n,i by Chow motives and the naive Grothendieck ringin the Grothendieck ring of Chow motives with coefficients inQ, as defined in 1.3.We can now state our uniformity result on the series P p (T ) as follows. In fact, one constructs in 9.2, for X a variety over any field k of characteristic zero, a canonical series P ar (T ) with coefficients in the Grothendieck ring K v 0 (Mot k,Q )⊗Q -defined in 1.3 -which is equal to the former one when k = Q and, furthermore, the series P ar (T ) is rational in a precise sense (Theorem 9.2.1). Now assume X is a subscheme of the affine spaceThe starting point in the proof of the rationality of the serieswhere W p is defined asx ≡ y mod w, and f i (y) = 0, for i = 1, . . . , r . In particular, W p is defined by a formula in the first order language of valued fields independently of the prime p.More generally, let k be a finite extension of Q with ring of integers O andwith N a nonzero multiple of the discriminant. For x a closed point of Spec R, we denote by O x the completion of the localization of R at x, by K x its fraction field, and by F x the residue field at x, a finite field of cardinality q x . Let f (x) be a polynomial in k[x 1 , . . . , x m ] (or more generally a definable function in the first order language of valued fields with variables and values taking place in the valued field and with coefficients in k) and let ϕ be a formula in the language of valued fields with coefficients in k, free variables x 1 , . . . , x m running over the valued field and no other free variables. Now set W x := {y ∈ O m x |ϕ(y) holds in K x } a...

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R�duction modulop n des sous-ensembles analytiques ferm�s deZ p N

Cited by 52 publications

References 5 publications

Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves

Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves

Constructible exponential functions, motivic Fourier transform and transfer principle

Definable sets, motives and 𝑝-adic integrals

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