We prove a version of the Manin-Mumford conjecture for semiabelian varieties over fields of positive characteristic. The proof presented here contains the details of the proof sketched by the author in the article "Diophantine geometry from model theory," BSL 7 (2001) no. 1, 37 - 57; using the model theory of difference fields along the lines of Hrushovski. Algebraic proofs have been presented by Pink & Roessler and Pillay.
We prove that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture, in the situation where the ambient variety is an abelian variety defined over the function field of a smooth curve over a finite field and the relevant group is a finitely generated group. In particular, in the setting of the last sentence, we provide a proof of the Mordell-Lang conjecture, which does not depend on tools coming from model theory.
We give a short proof of Manin-Mumford in the multiplicative group based on the pigeon-hole principle and the so-called structure theorem for anomalous subvarieties. The arguments appear to be new and perhaps applicable in other situations.
We present a reduction of the function field Mordell-Lang conjecture to the function field Manin-Mumford conjecture, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski structures. In this version 2, the quantifier elimination result in positive characteristic is extended from simple abelian varieties to all abelian varieties, completing the main theorem in the positive characteristic case. In version 3, some corrections are made to the proof of quantifier elimination in positive characteristic, and the paper is substantially reor…
The Manin-Mumford conjecture in characteristic zero was first proved by Raynaud. Later, Hrushovski gave a different proof using model theory. His main result from model theory, when applied to abelian varieties, can be rephrased in terms of algebraic geometry. In this paper we prove that intervening result using classical algebraic geometry alone. Altogether, this yields a new proof of the Manin-Mumford conjecture using only classical algebraic geometry.
We show that Ribet sections are the only obstruction to the validity of the relative Manin-Mumford conjecture for one dimensional families of semi-abelian surfaces. Applications include special cases of the Zilber-Pink conjecture for curves in a mixed Shimura variety of dimension four, as well as the study of polynomial Pell equations with non-separable discriminants.
In the article [PR1] {\it On Hrushovski's proof of the Manin-Mumford conjecture} (Proceedings of the ICM 2002), R. Pink and the author gave a short proof of the Manin-Mumford conjecture, which was inspired by an earlier model-theoretic proof by Hrushovski. The proof given in [PR1] uses a difficult unpublished ramification-theoretic result of Serre. It is the purpose of this note to show how the proof given in [PR1] can be modified so as to circumvent the reference to Serre's result. J. Oesterlé and R. Pink contributed several simplifications and shortcuts to this note.
We give a short proof of the "prime-to-$p$ version" of the Manin-Mumford conjecture for an abelian variety over a number field, when it has supersingular reduction at a prime dividing $p$, by combining the methods of Bogomolov, Hrushovski, and Pink-Roessler. Our proof here is quite simple and short, and neither $p$-adic Hodge theory nor model theory is used. The observation is that a power of a lift of the Frobenius element at a supersingular prime acts on the prime-to-$p$ torsion points via nontrivial homothety.
We prove Zhang's Dynamical Manin-Mumford Conjecture and Dynamical Bogomolov Conjecture for dominant endomorphisms of (P^1)^n. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with an analysis of the symmetries of the Julia set for a rational function.
We give a complete proof for the implication from the Manin-Mumford conjecture to the Mordell-Lang conjecture in positive characteristic, using integral models of semi-abelian varieties over a ring of formal power series, and the machinery of jet schemes.
On the dynamical Manin-Mumford conjecture for plane polynomial maps
Sorbonne Université（索邦大学）；CNRS - Centre de Mathématiques Laurent Schwartz, École Polytechnique（法国国家科学研究中心-洛朗·施瓦茨数学中心，巴黎综合理工学院）；Université de Paris and Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche（巴黎大学和索邦大学，法国国家科学研究中心，朱西厄-左岸巴黎数学研究所）
We present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points as rational points on a complex torus and then compare (i) upper bounds for the number of rational points on a transcendental analytic variety (Bombieri-Pila-Wilkie) and (ii) lower bounds for the degree of a torsion point (Masser), after taking conjugates. In order to be able to deal with (i), we discuss (Thm. 2.1) the semi-algebraic curves contained in an analytic variety supposed invariant for tra…
We prove the Dynamical Bogomolov Conjecture for endomorphisms of P^1\times P^1 defined over a number field. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with a theorem of Levin regarding symmetries of the Julia set. Using a specialization theorem of Yuan and Zhang, we prove the Dynamical Manin-Mumford Conjecture for endomorhisms of P^1\times P^1 defined over the complex numbers.
We use perfectoid spaces associated to abelian varieties and Siegel moduli spaces to study torsion points and ordinary CM points. We reprove the Manin-Mumford conjecture i.e. Raynaud's theorem. We also prove the Tate-Voloch conjecture for a product of Siegel moduli spaces namely ordinary CM points outside a closed subvariety can not be p-adically too close to it.
Our results concern analytic functions on the open unit $p$-adic poly-disc in $\mathbb{C}^n_p$ centered at the multiplicative unit and we prove that such functions only vanish at finitely many $n$-tuples of roots of unity $(ζ_1-1,\ldots,ζ_n-1)$ unless they vanish along a translate of the formal multiplicative group. For polynomial functions, this follows from the multiplicative Manin-Mumford conjecture. However we allow for a much wider class of analytic functions; in particular we establish a rigidity result for formal tori. Moreover, our methods apply to Lubin-Tate formal groups beyond just …
Let $A$ be a simple abelian surface over an algebraically closed field $k$. Let $S\subset A(k)$ be the set of torsion points $x$ of $A$ such that there exists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in the image of $f$, and $f$ sends a Weierstrass point of $C$ to the origin of $A$. The purpose of this note is to show that if $k$ has characteristic zero, then $S$ is finite -- this is in contrast to the situation where $k$ is the algebraic closure of a finite field, where $S=A(k)$, as shown by Bogomolov and Tschinkel. We deduce that if $k=\bar{\mathbb{Q}}$, the Kummer surfac…
We prove an equidistribution result for torsion points of Drinfeld modules of generic characteristic. We also show that similar equidistribution statements provide proofs for the Manin-Mumford and the Bogomolov conjectures for Drinfeld modules.
In this paper, we propose a new height function for a variety defined over a finitely generated field over Q. For this height function, we will prove Northcott's theorem and Bogomolov's conjecture, so that we can recover the original Raynaud's theorem (Manin-Mumford's conjecture).
We prove a version of Manin's conjecture for a certain family of intrinsic quadrics, the base field being a global field of positive characteristic. We also explain how a very slight variation of the method we use allows to establish the conjecture for a certain generalized del Pezzo surface.
A Mumford curve of genus g (>1) over a non-archimedean valued field k of positive characteristic has at most max{12(g-1), 2 g^(1/2) (g^(1/2)+1)^2} automorphisms. This bound is sharp in the sense that there exist Mumford curves of arbitrary high genus that attain it (they are fibre products of suitable Artin-Schreier curves). The proof provides (via its action on the Bruhat-Tits tree) a classification of discontinuous subgroups of PGL(2,k) that are normalizers of Schottky groups of Mumford curves with more than 12(g-1) automorphisms. As an application, it is shown that all automorphisms of the …
A Mumford group is a discontinuous subgroup $Γ$ of PGL(2,K), where K denotes a non archimedean valued field, such that the quotient by $Γ$ is a curve of genus 0. As abstract group $Γ$ is an amalgam of a finite tree of finite groups. For K of positive characteristic the large collection of amalgams having two or three branch points is classified. Using these data Mumford curves with a large group of automorphisms are discovered. A long combinatorial proof, involving the classification of the finite simple groups, is needed for establishing an upper bound for the order of the group of automorphi…
The generalized Mordell-Lang conjecture (GML) is the statement that the irreducible components of the Zariski closure of a subset of a group of finite rank inside a semi-abelian variety are translates of closed algebraic subgroups. M. McQuillan gave a proof of this statement. We revisit his proof, indicating some simplifications. This text contains a complete elementary proof of the fact that (GML) for groups of torsion points (= generalized Manin-Mumford conjecture), together with (GML) for finitely generated groups imply the full generalized Mordell-Lang conjecture.
We study the rigid analytic geometry of cyclic coverings of the projective line. We determine the defining equation of a cyclic covering of degree $p$ of the projective line by a Mumford curve over a complete discrete valuation field of positive characteristic $p$. Previously, Bradley studied that of any degree over a non-archimedean local field of characteristic zero.
We seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics via characterizing curves in positive characteristics which are reduction of Shimura curve over $\mathbb{C}$. In this paper, we study the liftablity of a curve in the moduli space of principally polarized abelian varieties over $k, \text{char} k=p$. We show that in the generic ordinary case, some tensor decomposition of the isocrystal associated to the family imply that this curve can be lifted to a Shimura curve.
Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups. In this paper, we construct simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a ne…
The goal of this paper is to prove the full geometric Bogomolov conjecture. We first reduce it to the case that the extension of the base fields has transcendence degree 1, and then we prove the later case by intersection theory in algebraic geometry. The proof uses Yamaki's reduction theorem on the geometric Bogomolov conjecture and the Manin--Mumford conjecture proved by Raynaud and Hrushovski.
We introduce $\ell$-Galois special subvarieties as an $\ell$-adic analog of the Hodge-theoretic notion of a special subvariety. The Mumford-Tate conjecture predicts that both notions are equivalent. We study some properties of these subvarieties and prove this equivalence for subvarieties satisfying a simple monodromy condition. As applications, we show that the $\ell$-Galois exceptional locus is a countable union of algebraic subvarieties and, if the derived group of the generic Mumford-Tate group of a family is simple, its part of positive period dimension coincides with the Hodge locus of p…
We study bounds for the Castelnuovo-Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular our aim is to give a positive answer to a question posed by Bayer and Mumford, by showing that the known upper bound in characteristic zero holds true also in positive characteristic. We first analyze Giusti's proof, which provides the result in characteristic 0, giving some insight on the combinatorial properties needed in that context. For the general case we provide a new argument which employs Bayer and Stillma…
We compute the dimension of the tangent space to, and the Krull dimension of the pro-representable hull of two deformation functors. The first one is the ``algebraic'' deformation functor of an ordinary curve X over a field of positive charateristic with prescribed action of a finite group G, and the data are computed in terms of the ramification behaviour of X -> G\X. The second one is the ``analytic'' deformation functor of a fixed embedding of a finitely generated discrete group N in PGL(2,K) over a non-archimedean valued field K, and the data are computed in terms of the Bass-Serre represe…
Pour tout $t\in\mN$ nous définissons un certain entier positif $\N_t$ et nous conjecturons: si $H$ est un fibré de Gauss-Manin d'une fibration semi-stable alors la $t$-ème classe de Chern de $H$ est annulée par $\N_t$. Nous démontrons diverses conséquences de cette conjecture. For any $t\in\mN$, we define a certain positive integer $\N_t$ and we conjecture: if $H$ is a Gauss-Manin bundle of a semi-stable fibration then the $t$-th Chern class of $H$ is kiled by $\N_t$. We prove various consequences of this conjecture.
We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes p we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic 0. We also investigate the principles of Geometric Manin's Conjecture for weak del Pezzo surfaces. In the course of this investigation, we give examples of weak del Pezzo surfaces defined over $\mathbb{F}_{2}(t)$ or $\mathbb{F}_{3}(t)$ such that the exceptional sets in Manin's Conjecture are Zariski dense.
We study the algbraic dynamics for endomorphisms of projective spaces with coefficients in a p-adic field whose reduction in positive characteritic is the Frobenius. In particular, we prove a version of the dynamical Manin-Mumford conjecture and the dynamical Mordell-Lang conjecture for the coherent backward orbits for such endomorphisms. We also give a new proof of a dynamical version of the Tate-Voloch conjecture in this case. Our method is based on the theory of perfectoid spaces introduced by P. Scholze. In the appendix, we prove that under some technical condition on the field of definiti…
We introduce and study a class of algebraic stacks with finite inertia in positive and mixed characteristic, which we call tame algebraic stacks. They include tame Deligne-Mumford stacks, and are arguably better behaved than general Deligne-Mumford stacks. We also give a complete characterization of finite flat linearly reductive schemes over an arbitrary base. Our main result is that tame algebraic stacks are étale locally quotient by actions of linearly reductive finite group schemes. In a subsequent paper we will show that tame algebraic stacks admit a good theory of stable maps.
Stratifications and iterative differential equations are analogues in positive characteristic of complex linear differential equations. There are few explicit examples of stratifications. The main goal of this paper is to construct stratifications on projective or affine curves in positive characteristic and to determine the possibilities for their differential Galois groups. For the related "differential Abhyankar conjecture" we present partial answers, supplementing the literature. The tools for the construction of regular singular stratifications and the study of their differential Galois g…
We prove a version of Manin's conjecture for the projective plane blown up in three collinear points, the base field being a global field of positive characteristic.
This paper investigates the Castelnuovo-Mumford regularity of the generic hyperplane section of projective curves in positive characteristic case, and yields an application to a sharp bound on the regularity for nondegenerate projective varieties.
If $M$ is a projective manifold in $P^N$, then one can associate to each one parameter subgroup $H$ of $SL(N+1)$ the Mumford $μ$ invariant. The manifold $M$ is Chow-Mumford stable if $μ$ is positive for all $H$. Tian has defined the notion of K-stability, and has shown it to be intimately related to the existence of Kähler-Einstein metrics. The manifold $M$ is K-stable if $μ'$ is positive for all $H$, where $μ'$ is an invariant which is defined in terms of the Mabuchi K-energy. In this paper we derive an explicit formula for $μ'$ in the case where $M$ is a curve. The formula is similar to Mumf…
We offer some theorems, mainly of finiteness, for certain patterns in elliptical billiards, related to periodic trajectories. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in $(0,π)$, there are only finitely many cases for both trajectories being periodic. Another instance is the finiteness of the billiard shots which send a given ball into another one so that this falls eventually in a hole. These results have their origin in `relative' cases of the Manin-Mumford conjecture, and constitute instances of how arithmetical content may affect…
Let $K$ be a global field of positive characteristic. We prove that the Brauer-Manin obstructions to the Hasse principle, to weak approximation and to strong approximation are the only ones for homogeneous spaces of reductive groups with reductive stabilizers. The methods involve abelianization techniques and arithmetic duality theorems for complexes of tori over K.
