On the torsion values for sections of an elliptic scheme

We shall consider sections of a complex elliptic scheme ℰ {{{\mathcal{E}}}} over an affine base curve B, and study the points of B where the section takes a torsion value. In particular, we shall relate the distribution in B of these points with the canonical height of the section, proving an integral formula involving a measure on B coming from the so-called Betti map of the section. We shall show that this measure is the same one which appears in dynamical issues related to the section. This analysis will also involve the multiplicity with which a torsion value is attained, which is an independent problem. We shall prove finiteness theorems for the points where the multiplicity is higher than expected. Such multiplicity has also a relation with Diophantine Approximation and quasi-integral points on ℰ {{{\mathcal{E}}}} (over the affine ring of B), and in Sections 5 and 6 of the paper we shall exploit this viewpoint, proving an effective result in the spirit of Siegel’s theorem on integral points.

A Genomics Resource for Genetics, Physiology, and Breeding of West African Sorghum

Local landrace and breeding germplasm is a useful source of genetic diversity for regional and global crop improvement initiatives. Sorghum (Sorghum bicolor L. Moench) in West Africa has diversified across a mosaic of cultures and end-uses, and along steep precipitation and photoperiod gradients. To facilitate germplasm utilization, a West African sorghum association panel (WASAP) of 756 accessions from national breeding programs of Niger, Mali, Senegal, and Togo was assembled and characterized. Genotyping-by-sequencing was used to generate 159,101 high-quality biallelic SNPs, with 43% in intergenic regions and 13% in genic regions. High genetic diversity was observed within the WASAP (π = 0.00045), only slightly less than in a global diversity panel (π = 0.00055). Linkage disequilibrium decayed to background level (r2 < 0.1) by ~50 kb in the WASAP. Genome-wide diversity was structured both by botanical type, and by populations within botanical type, with eight ancestral populations identified. Most populations were distributed across multiple countries, suggesting several potential common gene pools across the national programs. Genome-wide association studies of days to flowering and plant height revealed eight and three significant quantitative trait loci (QTL), respectively, with major height QTL at canonical height loci Dw3 and SbHT7.1. Colocalization of two of eight major flowering time QTL with flowering genes previously described in US germplasm (Ma6 and SbCN8) suggests that photoperiodic flowering in WA sorghum is conditioned by both known and novel genes. This genomic resource provides a foundation for genomics-enabled breeding of climate-resilient varieties in West Africa.

Canonical Heights on Hyper-Kähler Varieties and the Kawaguchi–Silverman Conjecture

The Kawaguchi–Silverman conjecture predicts that if $f: X \dashrightarrow X$ is a dominant rational-self map of a projective variety over $\overline{{\mathbb{Q}}}$, and $P$ is a $\overline{{\mathbb{Q}}}$-point of $X$ with a Zariski dense orbit, then the dynamical and arithmetic degrees of $f$ coincide: $\lambda _1(f) = \alpha _f(P)$. We prove this conjecture in several higher-dimensional settings, including all endomorphisms of non-uniruled smooth projective threefolds with degree larger than $1$, and all endomorphisms of hyper-Kähler manifolds in any dimension. In the latter case, we construct a canonical height function associated with any automorphism $f: X \to X$ of a hyper-Kähler manifold defined over $\overline{{\mathbb{Q}}}$. We additionally obtain results on the periodic subvarieties of automorphisms for which the dynamical degrees are as large as possible subject to log concavity.

Rationality of dynamical canonical height

We present a dynamical proof of the well-known fact that the Néron–Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field $k=\mathbb{C}(X)$, where $X$ is a curve. More generally, we investigate the mechanism by which the local canonical height for a map $f:\mathbb{P}^{1}\rightarrow \mathbb{P}^{1}$ defined over a function field $k$ can take irrational values (at points in a local completion of $k$), providing examples in all degrees $\deg f\geq 2$. Building on Kiwi’s classification of non-archimedean Julia sets for quadratic maps [Puiseux series dynamics of quadratic rational maps. Israel J. Math.201 (2014), 631–700], we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application we prove that if the heights $\widehat{h}_{f}(a),\widehat{h}_{g}(b)$ are rational and positive, for maps $f$ and $g$ of multiplicatively independent degrees and points $a,b\in \mathbb{P}^{1}(\bar{k})$, then the orbits $\{f^{n}(a)\}_{n\geq 0}$ and $\{g^{m}(b)\}_{m\geq 0}$ intersect in at most finitely many points, complementing the results of Ghioca et al [Intersections of polynomials orbits, and a dynamical Mordell–Lang conjecture. Invent. Math.171 (2) (2008), 463–483].

Canonical Heights and Preperiodic Points for Certain Weighted Homogeneous Families of Polynomials

A family f of polynomials over a number field K will be called weighted homogeneous if and only if ft(z) = F(ze, t) for some binary homogeneous form F(X, Y) and some integer e ≥ 2. For example, the family zd + t is weighted homogeneous. We prove a lower bound on the canonical height, of the form \begin{align*} \hat{h}_{f_{t}}(z)\geq \varepsilon \max\!\left\{h_{\mathsf{M}_{d}}(f_{t}), \log|\operatorname{Norm}\mathfrak{R}_{f_{t}}|\right\},\end{align*} for values z ∈ K which are not preperiodic for ft. Here ε depends only on the number field K, the family f, and the number of places at which ft has bad reduction. For suitably generic morphisms $\varphi :\mathbb {P}^{1}\to \mathbb {P}^{1}$, we also prove an absolute bound of this form for t in the image of φ over K (assuming the abc Conjecture), as well as uniform bounds on the number of preperiodic points (unconditionally).

On the complex dynamics of birational surface maps defined over number fields

We show that any birational selfmap of a complex projective surface that has dynamical degree greater than one and is defined over a number field automatically satisfies the Bedford–Diller energy condition after a suitable birational conjugacy. As a consequence, the complex dynamics of the map is well behaved. We also show that there is a well-defined canonical height function.