Mumford Conjecture Research Articles

INTERSECTING THE TORSION OF ELLIPTIC CURVES

Abstract Bogomolov and Tschinkel [‘Algebraic varieties over small fields’, Diophantine Geometry, U. Zannier (ed.), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73–91] proved that, given two complex elliptic curves $E_1$ and $E_2$ along with even degree- $2$ maps $\pi _j\colon E_j\to \mathbb {P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and $E_2$ under their respective $\pi _j$ is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. The recent proof of the uniform Manin–Mumford conjecture implies a full solution of the Bogomolov–Fu–Tschinkel conjecture. In this paper, we prove a generalisation of the Bogomolov–Fu–Tschinkel conjecture whereby, instead of even degree- $2$ maps, one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna theory with the uniform Manin–Mumford conjecture. With similar techniques, we also prove a result on lower bounds for ranks of elliptic curves over number fields.

Finiteness theorems on elliptical billiards and a variant of the dynamical Mordell–Lang conjecture

AbstractWe offer some theorems, mainly finiteness results, for certain patterns in elliptical billiards, related to periodic trajectories; these seem to be the first finiteness results in this context. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in , there are only finitely many directions 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 (which are shown not to hold for general billiards) have their origin in ‘relative’ cases of the Manin–Mumford conjecture and constitute instances of how arithmetical content may affect chaotic behaviour (in billiards). We shall also interpret the statements through a variant of the dynamical Mordell–Lang conjecture. In turn, this variant embraces cases, which, somewhat surprisingly, sometimes can be treated (only) by completely different methods compared to the former ones; here we shall offer an explicit example related to diophantine equations in algebraic tori.

Moduli spaces of Riemann surfaces as Hurwitz spaces

We consider the moduli space Mg,n of Riemann surfaces of genus g≥0 with n≥1 ordered and directed marked points. For d≥2g+n−1 we show that Mg,n is homotopy equivalent to a component of the simplicial Hurwitz space HurΔ(Sdgeo) associated with the partially multiplicative quandle Sdgeo. As an application, we give a new proof of the Mumford conjecture on the stable rational cohomology of moduli spaces of Riemann surfaces. We also provide a combinatorial model for the infinite loop space Ω∞−2MTSO(2) of Hurwitz flavour.

The Tropical Manin–Mumford Conjecture

Abstract In analogy with the Manin–Mumford conjecture for algebraic curves, one may ask how a metric graph under the Abel–Jacobi embedding intersects torsion points of its Jacobian. We show that the number of torsion points is finite for metric graphs of genus ${g\geq 2}$, which are biconnected and have edge lengths that are “sufficiently irrational” in a precise sense. Under these assumptions, the number of torsion points is bounded by $3g-3$. Next, we study bounds on the number of torsion points in the image of higher-degree Abel–Jacobi embeddings, which send $d$-tuples of points to the Jacobian. This motivates the definition of the “independent girth” of a graph, a number that is a sharp upper bound for $d$ such that the higher-degree Manin–Mumford property holds.

The Manin–Mumford conjecture and the Tate–Voloch conjecture for a product of Siegel moduli spaces

The Manin–Mumford conjecture and the Tate–Voloch conjecture for a product of Siegel moduli spaces

Torsion points of small order on hyperelliptic curves

Let $$\mathscr {C}$$ be a hyperelliptic curve of genus $$g>1$$ over an algebraically closed field K of characteristic zero and $$\mathscr {O}$$ one of the $$(2g{+}2)$$ Weierstrass points in $$\mathscr {C}(K)$$ . Let J be the Jacobian of $$\mathscr {C}$$ , which is a g-dimensional abelian variety over K. Let us consider the canonical embedding of $$\mathscr {C}$$ into J that sends $$\mathscr {O}$$ to the zero of the group law on J. This embedding allows us to identify $$\mathscr {C}(K)$$ with a certain subset of the commutative group J(K). A special case of the famous theorem of Raynaud (Manin–Mumford conjecture) asserts that the set of torsion points in $$\mathscr {C}(K)$$ is finite. It is well known that the points of order 2 in $$\mathscr {C}(K)$$ are exactly the “remaining” $$(2g{+}1)$$ Weierstrass points. One of the authors (Zarhin in Izv Math 83:501–520, 2019) proved that there are no torsion points of order n in $$\mathscr {C}(K)$$ if $$3\leqslant n\leqslant 2g$$ . So, it is natural to study torsion points of order $$2g+1$$ (notice that the number of such points in $$\mathscr {C}(K)$$ is always even). Recently, the authors proved that there are infinitely many (for a given g) mutually non-isomorphic pairs $$(\mathscr {C},\mathscr {O})$$ such that $$\mathscr {C}(K)$$ contains at least four points of order $$2g+1$$ . In the present paper we prove that (for a given g) there are at most finitely many (up to an isomorphism) pairs $$(\mathscr {C},\mathscr {O})$$ such that $$\mathscr {C}(K)$$ contains at least six points of order $$2g+1$$ .

On p-adic Versions of the Manin–Mumford Conjecture

Abstract We establish $p$-adic versions of the Manin–Mumford conjecture, which states that an irreducible subvariety of an abelian variety with dense torsion has to be the translate of a subgroup by a torsion point. We do so in the context of certain rigid analytic spaces and formal groups over a $p$-adic field or its ring of integers, respectively. In particular, we show that the underlying rigidity results for algebraic functions generalize to suitable $p$-adic analytic functions. This leads us to uncover purely $p$-adic Manin–Mumford-type results for formal groups not coming from abelian schemes. Moreover, we observe that a version of the Tate–Voloch conjecture holds: torsion points either lie squarely on a subscheme or are uniformly bounded away from it in the $p$-adic distance.

A consequence of the relative Bogomolov conjecture

We propose a formulation of the relative Bogomolov conjecture and show that it gives an affirmative answer to a question of Mazur's concerning the uniformity of the Mordell–Lang conjecture for curves. In particular we show that the relative Bogomolov conjecture implies the uniform Manin–Mumford conjecture for curves. The proof is built up on our previous work [DGH21].

Generic rank of Betti map and unlikely intersections

Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$, the real torus of dimension $2g$, by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

Cyclotomic torsion points in elliptic schemes

An elliptic curve defined over a number field possesses only a finite number of torsion points defined over the cyclotomic closure of its field of definition. In analogy to the relative version of the Manin–Mumford conjecture stated by Masser and Zannier, we propose a family version of the above statement and prove it under a suitable integrality condition.

A REFORMULATION OF THE DYNAMICAL MANIN–MUMFORD CONJECTURE

We advance a new conjecture in the spirit of the dynamical Manin–Mumford conjecture. We show that our conjecture holds for all polarisable endomorphisms of abelian varieties and for all polarisable endomorphisms of $(\mathbb{P}^{1})^{N}$. Furthermore, we show various examples which highlight the restrictions one would need to consider in formulating any general conclusion in the dynamical Manin–Mumford conjecture.

The Dynamical Manin–Mumford Conjecture and the Dynamical Bogomolov Conjecture for split rational maps

We prove the Dynamical Bogomolov Conjecture for endomorphisms \Phi:\mathbb P^1\times \mathbb P^1\to \mathbb P^1\times \mathbb P^1 , where \Phi(x,y):=(f(x), g(y)) for any rational functions f and g defined over \bar {\mathbb Q} . 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 can prove the Dynamical Manin–Mumford Conjecture for endomorhisms \Phi=(f,g) of \mathbb P^1\times \mathbb P^1 , where f and g are rational functions defined over an arbitrary field of characteristic 0.

An application of the value distribution theory for semi-abelian varieties to problems of Ax–Lindemann and Manin–Mumford types

The aim of this paper is to prove a theorem of Ax–Lindemann type for complex semi-abelian varieties as an application of a big Picard theorem proved by the author in 1981, and then apply it to prove a theorem of classical Manin–Mumford Conjecture for semi-abelian varieties, which was proved by M. Raynaud 1983, M. Hindry 1988, . . . , and Pila–Zannier 2008 by a different method from others, which is most relevant to ours. The present result might be a first instance of a direct connection at the proof level between the value distribution theory of holomorphic maps and the arithmetic (Diophantine) theory over algebraic number fields, while there have been many analogies between them.

The dynamical Manin–Mumford conjecture and the dynamical Bogomolov conjecture for endomorphisms of

We prove Zhang’s dynamical Manin–Mumford conjecture and dynamical Bogomolov conjecture for dominant endomorphisms$\unicode[STIX]{x1D6F7}$of$(\mathbb{P}^{1})^{n}$. 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.

On the arithmetic dynamics of monomial maps

We prove several results for the arithmetic dynamics of monomial maps, including Silverman’s conjectures on height growth, dynamical Mordell–Lang conjecture, and dynamical Manin–Mumford conjecture. These results were originally known for monomial maps on algebraic tori. We extend them to arbitrary toric varieties.

An infinitesimal p-adic multiplicative Manin–Mumford Conjecture

Our results concern certain analytic functions on the open unit poly-disc in ℂ p n centered at the multiplicative unit and we show such functions only vanish at finitely many n-tuples of roots of unity (ζ 1 -1,...,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 formal 𝔾 m and we extend the results to this setting.

Uniform bounds for the number of rational points on curves of small Mordell–Weil rank

Let X be a curve of genus g≥2 over a number field F of degree d=[F:Q]. The conjectural existence of a uniform bound N(g,d) on the number #X(F) of F-rational points of X is an outstanding open problem in arithmetic geometry, known by the work of Caporaso, Harris, and Mazur to follow from the Bombieri–Lang conjecture. A related conjecture posits the existence of a uniform bound Ntors,†(g,d) on the number of geometric torsion points of the Jacobian J of X which lie on the image of X under an Abel–Jacobi map. For fixed X, the finiteness of this quantity is the Manin–Mumford conjecture, which was proved by Raynaud. We give an explicit uniform bound on #X(F) when X has Mordell–Weil rank r≤g−3. This generalizes recent work of Stoll on uniform bounds for hyperelliptic curves of small rank to arbitrary curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of F-rational torsion points of J lying on the image of X under an Abel–Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of J lying on X when the reduction type of X is highly degenerate. Our methods combine Chabauty–Coleman’s p-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs.

On function field Mordell–Lang and Manin–Mumford

We give a reduction of the function field Mordell–Lang conjecture to the function field Manin–Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the “Theorem of the Kernel”, and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: “quantifier-elimination” for the corresponding [Formula: see text] where [Formula: see text] is a saturated separably closed field.

The Weak b-principle: Mumford Conjecture

AbstractIn this note we introduce and study a new class of maps called oriented colored broken submersions. This is the simplest class of maps that satisfies a version of the b–principle and in dimension 2 approximates the class of oriented submersions well in the sense that every oriented colored broken submersion of dimension 2 to a closed simply connected manifold is bordant to a submersion. We show that the Madsen–Weiss theorem (the standard Mumford Conjecture) fits a general setting of the b–principle, namely, a version of the b–principle for oriented colored broken submersions together with the Harer stability theorem and Miller–Morita theorem implies the Madsen–Weiss theorem.

Relative Manin–Mumford for Semi-Abelian Surfaces

AbstractWe 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 4, as well as the study of polynomial Pell equations with non-separable discriminants.