The Geometric Bogomolov Conjecture for Small Genus Curves
X. W. C. Faber
备注：
47 pages total (31 pages in body of article, 8 pages of Mathematica code, 8 pages of Mathematica notebooks); ramification typo corrected in formulas on pp.13-14, this is the final version
The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4 defined over a function field of characteristic zero. We recover the known result for genus 2 curves and in many cases improve upon the known bound for genus 3 curves. For many curves of genus 4 with bad reduction, the conjecture was previously unproved.
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 "Uniformity in Mordell-Lang for curves".
We formulate a conjecture for semiabelian varieties A over number fields that includes both the Mordell-Lang conjecture (now proven) and the Bogomolov conjecture. We prove the "Mordellic" (finitely generated) part of the conjecture when A is isogenous to the product of an abelian variety and a torus. The proof makes use of the Mordell-Lang conjecture, the Bogomolov conjecture, and an equidistribution theorem.
In "Mordell-Lang plus Bogomolov", we formulated a conjecture combining the Mordell-Lang conjecture (now proven) and the Bogomolov conjecture, and proved the Mordellic part when A is almost split (i.e., isogenous to the product of an abelian variety and a torus). Here we prove that the full conjecture would follow from the Bogomolov conjecture and an equidistribution result for general semiabelian varieties. These assumptions have been proved when A is almost split, so the conjecture is completely proven in that case.
We prove a dynamical version of the Bogomolov conjecture in the special case of lines in affine space A^m under the action of a map (f_1,...,f_m) where each f_i is a polynomial in Q-bar[X] of the same degree.
We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang's Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of genus at most 4 and a few other special cases. We also either verify or improve the previous results. We relate the invariants involved in Zhang's Conjecture to the tau constant of metrized graphs. Then we use and extend our previous results on the tau constant. By proving another Conjecture of Zhang, we o…
We prove the Bogomolov conjecture for an abelian variety A over a function field which is totally degenerate at a place v. We adapt Zhang's proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place. As an application, we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v.
The Bogomolov conjecture claims that a closed subvariety containing a dense subset of small points is a special kind of subvarieties. 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 field, 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 …
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 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 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 $>1$ in families of abelian varieties with trivial trace.
Let K be a finitely generated field over Q, and A an abelian variety over K. Let <, > : A(K^a) x A(K^a) --> R be an arithmetic height pairing on A, where K^a is the algebric closure of K. For x_1,..., x_l \in A(K^a), we denote det(<x_i, x_j>) by d(x_1,..., x_l). Let G be a subgroup of finite rank in A(K^a), and X a subvariety of A_{K^a}. Fix a basis {g_1,..., g_n} of G_Q. In this note, we prove a generalization of Poonen's theorem: If the set {x \in X(K^a) | d(g_1,..., g_n, x) <= e} is Zariski dense in X for every positive number e, then X is a translation of an abelian subvariety by an elemen…
This is an introduction to the topics of the title, from the 2017 Grenoble Summer school on Arakelov geometry and arithmetic applications. We review Arithmetic intersection numbers, explain the definition of the height of a variety and its properties, both in the framework of classical Arakelov geometry and of Zhang's adelic formalism. We then discuss arithmetic ampleness and its application to the equidistribution theorem of Szpiro-Ullmo-Zhang-Yuan, both at complex and nonarchimedean places. We conclude with Ullmo-Zhang's proof of the Bogomolov conjecture over number fields.
In this paper we study the degeneration behavior of the norm of the Riemann $θ$-function in a family of principally polarized abelian varieties over the punctured complex unit disc in terms of the associated polarized real torus. As an application, we obtain the degeneration behavior of the Zhang--Kawazumi invariant $φ(M_t)$ of a family of Riemann surfaces $M_t$ in terms of Zhang's invariant $φ(Γ)$ of the associated metrized reduction graph $Γ$. This allows us to deduce a uniform lower bound for the essential minimum of the Néron-Tate height on the tautological cycles of any Jacobian variety o…
This is an enhanced version of the author's 1998 Harvard Ph.D. thesis, as published by IMRN in 2005. We propose an extension of Bogomolov's conjecture about commutator subgroups of Galois groups to arbitrary fields. A somewhat weaker conjecture is then shown to follow, together with the famous Milnor-Bloch-Kato conjecture, from a certain hypothesis about homological properties of the Milnor K-theory algebras.
On an abelian scheme over a smooth curve over $\overline{\mathbb Q}$ a symmetric relatively ample line bundle defines a fiberwise Néon-Tate height. If the base curve is inside a projective space, we also have a height on its $\overline{\mathbb Q}$-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 …
We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of inequivalent height functions on $\mathbb{P}^1(\overline{\mathbb{Q}}).$ We apply this strategy to prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on the number of common torsion points of elliptic curves in the case of two Legendre curves over $\mathbb{C}$. As a consequence, we obtain two uniform bounds for a two-dimensional family of genus 2 curves: a uniform Manin-Mumford bound for the family over $\mathbb{C}$, and a uniform Bogom…
We finish the proof of the conjecture of F. Bogomolov and F. Pop: Let $F_{1}$ and $F_{2}$ be fields finitely-generated and of transcendence degree $\geq 2$ over $k_{1}$ and $k_{2}$, respectively, where $k_{1}$ is either $\bar{\mathbb{Q}}$ or $\bar{\mathbb{F}}_{p}$, and $k_{2}$ is algebraically closed. We denote by $G_{F_1}$ and $G_{F_2}$ their respective absolute Galois groups. Then the canonical map $φ_{F_{1}, F_{2}}: \Isom^i(F_1, F_2)\rightarrow \Isom^{\Out}_{\cont}(G_{F_2}, G_{F_1})$ from the isomorphisms, up to Frobenius twists, of the inseparable closures of $F_1$ and $F_2$ to continuous …
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 $ε$ for all polarized metrized graphs of genus $3$. This improves the lower bound given for Effective Bogomolov Conjecture for such curves.
Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca（米兰比可卡大学数学与应用系）；Femto-St, Université Marie et Louis Pasteur（玛丽和路易·巴斯德大学 Femto-St）
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…
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).
In this note, we show how the classical Hodge index theorem implies the Hodge index conjecture of Beilinson for height pairing of homologically trivial codimension two cycles over function field of characteristic 0. Such an index conjecture has been used in our paper on Gross-Schoen cycles to deduce the Bogomolov conjecture and a lower bound for Hodge class (or Faltings height) from some conjectures about metrized graphs which have just been recently proved by Zubeyir Cinkir.
We extend to the case of semi-abelian varieties the statements of various variants of the conjecture alla Bogomolov about the non-density of small points of small height in abelian varieties. Inspired by recent work of Ullmo, Zhang and Bilu, we then prove these conjectures when the semi-abelian variety is almost split. The proof uses Arakelov geometry and equirepartition arguments.
We study the Selmer varieties of smooth projective curves of genus at least two defined over $\mathbb{Q}$ which geometrically dominate a curve with CM Jacobian. We extend a result of Coates and Kim to show that Kim's non-abelian Chabauty method applies to such a curve. By combining this with results of Bogomolov-Tschinkel and Poonen on unramified correspondences, we deduce that any cover of $\mathbf{P}^1$ with solvable Galois group, and in particular any superelliptic curve over $\mathbb{Q}$, has only finitely many rational points over $\mathbb{Q}$.
The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multiplication---to their $p$-torsion Galois representations is one-to-one over function fields of small genus complex curves for sufficiently large $p$ relative to the genus.
We conjecture a Fibonacci-like property on the number of numerical semigroups of a given genus. Moreover we conjecture that the associated quotient sequence approaches the golden ratio. The conjecture is motivated by the results on the number of semigroups of genus at most 50. The Wilf conjecture has also been checked for all numerical semigroups with genus in the same range.
The Birch and Swinnerton-Dyer conjecture has been numerically verified for the Jacobians of 32 modular hyperelliptic curves of genus 2 by Flynn, Leprévost, Schaefer, Stein, Stoll and Wetherell, using modular methods. In the calculation of the real period, there is a slight inaccuracy, which might give problems for curves with non-reduced components in the special fibre of their Néron model. In this present paper we explain how the real period can be computed, and how the verification has been extended to many more hyperelliptic curves, some of genus 3, 4 and 5, without using modular methods.
We consider some string invariants at genus two that appear in the analysis of the $D^8\mathcal{R}^4$ and $D^6\mathcal{R}^5$ interactions in type II string theory. We conjecture a Poisson equation involving them and the Kawazumi--Zhang invariant based on their asymptotic expansions around the non--separating node in the moduli space of genus two Riemann surfaces.
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…
We suspect that the ``genus part'' of the class number of a number field K may be an obstruction for an ``easy proof'' of the classical p-rank epsilon-conjecture for p-class groups and, a fortiori, for a proof of the ``strong epsilon-conjecture'': \# (Cl\_K \otimes \Z\_p) <<\_(d,p,epsilon) ($\sqrt$D\_K)^epsilon for all K of degree d. We analyze the weight of genus theory in this inequality by means of an infinite family of degree p cyclic fields with many ramified primes, then we prove the p-rank epsilon-conjecture: \# (Cl\_K \otimes \F\_p) <<\_(d,p,epsilon) ($\sqrt$D\_K)^epsilon, for d=p and …
Shimura's conjectire (1963) concerns the rationality of the generating series for Hecke operators for the symplectic group of genus g. This conjecture wes proved by Andrianov for arbitrary genus g. For genus g=4, we explicify the rational fraction in this conjecture. Using formulas for images of double cosets, we first compute the sum of the generating series under the Satake spherical map, which is a rational fraction with polynomial coefficients. Then we recover the coefficients of this fraction as elements of the Hecke algebra using polynomial representation of basic Hecke operators under s…
