Bogomolov Conjecture Research Articles

Height coincidences in products of the projective line

We consider hypersurfaces in $${\mathbb {P}}_1^n$$ that contain a generic sequence of small dynamical height with respect to a split map and project onto $$n-1$$ coordinates. We show that these hypersurfaces satisfy strong coincidence relations between their points with zero height coordinates. More precisely, it holds that in a Zariski-open dense subset of such a hypersurface $$n-1$$ coordinates have height zero if and only if all coordinates have height zero. This is a key step in the resolution of the dynamical Bogomolov conjecture for split maps.

The relative Bogomolov conjecture for fibered products of elliptic curves

Abstract We deduce an analogue of the Bogomolov conjecture for non-degenerate subvarieties in fibered products of families of elliptic curves from the author’s recent theorem on equidistribution in families of abelian varieties. This generalizes results of DeMarco and Mavraki and improves certain results of Manin–Mumford type proven by Masser and Zannier to results of Bogomolov type, yielding the first results of this type for subvarieties of relative dimension > 1 {>1} in families of abelian varieties with trivial trace.

Geometric Bogomolov conjecture in arbitrary characteristics

We give a proof of the full geometric Bogomolov conjecture.

Points of small height on semiabelian varieties

The equidistribution conjecture is proved for general semiabelian varieties over number fields. Previously, this conjecture was only known in the special case of almost split semiabelian varieties through work of Chambert-Loir. The general case has remained intractable so far because the height of a semiabelian variety is negative unless it is almost split. In fact, this places the conjecture outside the scope of Yuan’s equidistribution theorem on algebraic dynamical systems. To overcome this, an asymptotic adaption of the equidistribution technique invented by Szpiro, Ullmo, and Zhang is used here. It also allows a new proof of the Bogomolov conjecture and hence a self-contained proof of the strong equidistribution conjecture in the same general setting.

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].

Semi-stable models of modular curves X0(p2) and some arithmetic applications

In this paper, we compute the semi-stable models of modular curves $X_0(p^2)$ for odd primes $p > 3$ and compute the Arakelov self-intersection numbers of the relative dualising sheaves for these models. We give two arithmetic applications of our computations. In particular, we give an effective version of the Bogomolov conjecture following the strategy outlined by Zhang and find the stable Faltings heights of the arithmetic surfaces corresponding to these modular curves.

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.

The geometric Bogomolov conjecture

We prove the geometric Bogomolov conjecture over a function field of characteristic zero.

Heights in families of abelian varieties and the Geometric Bogomolov Conjecture

On an abelian scheme over a smooth curve over $\bar{\mathbb{Q}}$ a symmetric relatively ample line bundle defines a fiberwise Neron–Tate height. If the base curve is inside a projective space, we also have a height on its $\bar{\mathbb{Q}}$-points that serves as a measure of each fiber, an abelian variety. Silverman proved an asymptotic equality between these two heights on a curve in the abelian scheme. In this paper we prove an inequality between these heights on a subvariety of any dimension of the abelian scheme. As an application we prove the Geometric Bogomolov Conjecture for the function field of a curve defined over $\bar{\mathbb{Q}}$. Using Moriwaki's height we sketch how to extend our result when the base field of the curve has characteristic 0.

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.

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.

Survey on the geometric Bogomolov conjecture

This is a survey paper of the developments on the geometric Bogomolov conjecture. We explain the recent results by the author as well as previous works concerning the conjecture. This paper also includes an introduction to the height theory over function fields and a quick review on basic notions on non-archimedean analytic geometry.

Uniform bound for the effective Bogomolov conjecture

We obtain a uniform bound for the effective Bogomolov conjecture, which depends only on the genus g of the curve. The bound grows as O(g−3) as g tends to infinity.

Geometric Bogomolov conjecture for nowhere degenerate abelian varieties of dimension $5$ with trivial trace

We prove that the geometric Bogomolov conjecture holds for nowhere degenerate abelian varieties of dimension $5$ with trivial trace. By this result together with our previous work, we see that the conjecture holds for an abelian variety such that the difference between the dimension of its maximal nowhere degenerate abelian subvariety and that of its trace equals $5$.

Non-density of small points on divisors on Abelian varieties and the Bogomolov conjecture

The Bogomolov conjecture for a curve claims finiteness of algebraic points on the curve which are small with respect to the canonical height. Ullmo has proved that this conjecture holds over number fields, and Moriwaki generalized it to the assertion over finitely generated fields over Q \mathbb {Q} with respect to arithmetic heights. As for the case of function fields with respect to the geometric heights, Cinkir has proved the conjecture over function fields of characteristic 0 0 and of transcendence degree 1 1 . However, the conjecture has been open over other function fields. In this paper, we prove that the Bogomolov conjecture for curves holds over any function field. In fact, we show that any non-special closed subvariety of dimension 1 1 in an abelian variety over function fields has only a finite number of small points. This result is a consequence of the investigation of non-density of small points of closed subvarieties of abelian varieties of codimension 1 1 . As a by-product, we show that the geometric Bogomolov conjecture, which is a generalization of the Bogomolov conjecture for curves over function fields, holds for any abelian variety of dimension at most 3 3 . Combining this result with our previous works, we see that the geometric Bogomolov conjecture holds for all abelian varieties for which the difference between its nowhere degeneracy rank and the dimension of its trace is not greater than 3 3 .

Trace of abelian varieties over function fields and the geometric Bogomolov conjecture

Abstract We prove that the geometric Bogomolov conjecture for any abelian varieties is reduced to that for nowhere degenerate abelian varieties with trivial trace. In particular, the geometric Bogomolov conjecture holds for abelian varieties whose maximal nowhere degenerate abelian subvariety is isogenous to a constant abelian variety. To prove the results, we investigate closed subvarieties of abelian schemes over constant varieties, where constant varieties are varieties over a function field which can be defined over the constant field of the function field.

Strict supports of canonical measures and applications to the geometric Bogomolov conjecture

The Bogomolov conjecture claims that a closed subvariety containing a dense subset of small points is a special kind of subvariety. In the arithmetic setting over number fields, the Bogomolov conjecture for abelian varieties has already been established as a theorem of Ullmo and Zhang, but in the geometric setting over function fields, it has not yet been solved completely. There are only some partial results known such as the totally degenerate case due to Gubler and our recent work generalizing Gubler’s result. The key in establishing the previous results on the Bogomolov conjecture is the equidistribution method due to Szpiro, Ullmo and Zhang with respect to the canonical measures. In this paper we exhibit the limits of this method, making an important contribution to the geometric version of the conjecture. In fact, by the crucial investigation of the support of the canonical measure on a subvariety, we show that the conjecture in full generality holds if the conjecture holds for abelian varieties which have anywhere good reduction. As a consequence, we establish a partial answer that generalizes our previous result.

Admissible invariants of genus 3 curves

Several invariants of polarized metrized graphs and their applications in Arithmetic Geometry are studied recently. In this paper, we explicitly calculated these admissible invariants for all curves of genus 3. We find the sharp lower bound for the invariants φ, λ and e for all polarized metrized graphs of genus 3. This improves the lower bound given for Effective Bogomolov Conjecture for such curves.

Self-intersection of the relative dualizing sheaf on modular curves X_1(N)

Let $N$ be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than 4. Our main theorem is an asymptotic formula solely in terms of $N$ for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves $X_1(N)/ \mathbb{Q}$. From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian $J_1(N) / \mathbb{Q}$ of $X_1(N)/ \mathbb{Q}$, and, for sufficiently large N, an effective version of Bogomolov's conjecture for $X_1(N) / \mathbb{Q}$.

Special points on fibered powers of elliptic surfaces

Consider a fibered power of an elliptic surface. We characterize its subvarieties that contain a Zariski dense set of points that are torsion points in fibers with complex multiplication. This result can be viewed as a mix of the Manin–Mumford and André–Oort Conjecture and is related to a conjecture of Pink. The main technical tool is a new height inequality. We also use it to give another proof of a case of Gubler's result on the Bogomolov Conjecture over function fields.