We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits of points in a generalised Hecke orbit in terms of this height function, assuming the ‘weakly adelic Mumford–Tate hypothesis’ and prove the generalised André–Pink–Zannier conjecture under this assumption, using Pila–Zannier strategy.

Fertility control strategies became widespread in rural Spain through the twentieth century: a significant number of parents decided to reduce their marital fertility once the advantages of control strategies became widely known. This paper explores the impact of those practices on children through a comparative study of the heights and occupations of grandparents, parents, and children. We analyze more than 1,200 individuals from three different generations born between 1835 and 1959 in 14 rural Spanish villages, studying whether the advantages associated with fertility control were maintained over time favoring a better family status or whether they were diluted in the next generation. The largest increases in height were among children whose parents controlled their fertility by stopping having children before the mother's 36th birthday. However, it does not seem that this increase in biological well-being was accompanied by major episodes of upward social mobility.

In this paper, we investigate the theory of heights in a family of stacky curves following recent work of Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. We count rational points having bounded ESZ-B height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We also show that when the Euler characteristic of stacky curves is non-positive, the ESZ-B height coming from the anti-canonical divisor class fails to have the Northcott property. We prove that a stacky version of a conjecture of Vojta is equivalent to the $abc$-conjecture.

DeMarco, Krieger, and Ye conjectured that there is a uniform bound B, depending only on the degree d, so that any pair of holomorphic maps $f, g :{\mathbb {P}}^1\to {\mathbb {P}}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\mathrm {Rat}_d \times \mathrm {Rat}_d$, for each degree $d\geq 2$. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier and Vigny, Yuan and Zhang, and Mavraki and Schmidt. In addition, we present alternate proofs of the main results of DeMarco, Krieger, and Ye [Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), 949–1001; Common preperiodic points for quadratic polynomials, J. Mod. Dyn. 18 (2022), 363–413] and of Poineau [Dynamique analytique sur $\mathbb {Z}$ II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel, Preprint (2022), arXiv:2207.01574 [math.NT]]. In fact, we prove a generalization of a conjecture of Bogomolov, Fu, and Tschinkel in a mixed setting of dynamical systems and elliptic curves.

Trends in human welfare in Brazil have remained shrouded by a dearth of historical evidence. Although quantitative scholars have revealed the efficacy of the First Republic (1889–1930) in fomenting economic progress, the extent to which Brazil's early economic growth fostered improvements in health remains unclear. This paper fills this void in scholarship by relying on hitherto untapped archival sources with data on human stature—a reliable metric for health and nutritional status. My analysis centres heavily on a large (n ≈ 16,000), geographically-comprehensive series compiled from military inscription files, supplemented by an ancillary dataset drawn from passport records (n ≈ 6,000). I document inferior heights in the North and Northeast that predated the advent of industrialisation. At the national level, my findings reveal an increase in stature of over 2.5 cm between soldiers born in the 1880s and those born in the 1910s. In the South and Southeast, I argue that increased real income and public-health interventions explain the earlier upward trend in heights, while rural sanitary reforms were most important in the North and Northeast, where heights remained stagnant until the 1910 decade and diseases such as hookworm and malaria were most rampant.

Using the coefficient of variations for heights, this paper examines the evolution of net nutrition inequality in Argentina from 1875 to 1950. It uses various samples of recruits and soldiers, previously gathered by the author. Evidence points to two important findings: (a) export-led growth led to stable or declining net-nutrition inequality; while import-substituting industrialisation generated significant net nutrition inequality; and (b) the highest levels of inequality in net nutrition took place during this latter phase in large urban, industrialised areas.

We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over $\mathbb{Q}$ whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems.

We prove a conjecture by Kawaguchi–Silverman on arithmetic and dynamical degrees, for self-morphisms of semi-abelian varieties. Moreover, we determine the set of the arithmetic degrees of orbits and the (first) dynamical degrees of self-morphisms of semi-abelian varieties.

The morphism $f\,:\,{{\mathbb{P}}^{N}}\,\to \,{{\mathbb{P}}^{N}}$ is called post-critically finite $\left( \text{PCF} \right)$ if the forward image of the critical locus, under iteration of $f$ , has algebraic support. In the case $N\,=\,1$ , a result of Thurston implies that there are no algebraic families of PCF morphisms, other than a well-understood exceptional class known as the flexible Lattés maps. A related arithmetic result states that the set of PCF morphisms corresponds to a set of bounded height in the moduli space of univariate rational functions. We prove corresponding results for a certain subclass of the regular polynomial endomorphisms of ${{\mathbb{P}}^{N}}$ for any $N$ .

We prove arithmetic Hilbert–Samuel type theorems for semi-positive singular hermitian line bundles of finite height. This includes the log-singular metrics of Burgos–Kramer–Kühn. The results apply in particular to line bundles of modular forms on some non-compact Shimura varieties. As an example, we treat the case of Hilbert modular surfaces, establishing an arithmetic analogue of the classical result expressing the dimensions of spaces of cusp forms in terms of special values of Dedekind zeta functions.

This paper offers new quantitative evidence on living standards in Bourbon America through a pioneering study of both wages and heights. Wages were not low, nor were heights short, by the international standards of the period. The living standards of Spanish Americans thus compare favourably with those of other regions of the world, including Europe. As in many parts of the West, one can observe a trend towards the deterioration of real wages in Spanish America at the end of the period. Our findings suggest that the ‘Great Divergence’ in living standards between Spanish America and the developed Western countries might have taken place mainly after independence and that currently available GDP per capita estimates might be too low.

Laser ablation of graphite materials in the presence of an external magnetic field is studied with the use of the newly developed HEIGHTS-PIC particle-in-cell code and compared with both theoretical and experimental results. Carbon plumes behavior controlled by a strong magnetic field is of interest to evaluate the plume shielding effects to protect the original exposed target from further damage and erosion. Since intense power deposition on plasma facing components is expected during Tokamaks loss of plasma confinement events such as disruptions, vertical displacements event, runaway electrons, or during normal operating conditions such as edge-localized modes, it is critical to better understand the evolving target plasma behavior for more accurate prediction of the potential damage created by these high-energetic dumps which may not be easily mitigated without loss of structural and functional performance of the plasma facing components. Numerical experiments have been performed to provide benchmarking conditions for the HEIGHTS-PIC simulation package originally designed to evaluate the erosion of the Tokamak surfaces, splashing of the melted/ablated-vaporized material, and transport into the bulk plasma with consequent plasma contamination. Evolving target plasma temperature and density are calculated and compared with measured reported values available into literature for similar conditions and show good agreement with the HEIGHTS-PIC package predictions.

Large energy fluxes and energy deposition are often found during off normal events in fusion reactors both magnetic and inertial confinement. Great temperature excursions can be found in the plasma facing components materials resulting in melting, evaporation, and expansion of the target material vapor cloud and successive plasma interaction. A general methodology for the computational solution of this very complex problem is proposed in this paper using a novel approach of the Particle-in-Cell method. This novel approach preserves many of the original Particle-in-Cell methodology proposed by Harlow's but introduces several changes to the method, which makes it particularly suitable in plasma dynamics applications as Tokamaks devices. Benchmarking is done using results from the very well-known and benchmarked HEIGHTS computer package and using available results from the MK200 plasma gun experiments. The general scheme is explained in detail and results and discussion related to the main computational aspects are presented. In particular, the critical importance of the starting computational mesh versus the aimed accuracy is studied and explained. The methodology is different from any other presented in literature because the dynamics of the physical problem dictates that as enough energy is being dumped on the target plate, melting and evaporation of the material takes place. As time progresses, new sample particles are introduced in the system making this boundary a dynamic one along time. Computer time is discussed in relation to the initial sample particle loading, aimed accuracy and importance of the time used for solving the radiation transport at each time step.

We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

We obtain a Bogomolov type of result for the affine space defined over the algebraic closure of a function field of transcendence degree 1 over a finite field.

Let $K$ be a number field, and let $F$ be a symmetric bilinear form in $2N$ variables over $K$. Let $Z$ be a subspace of ${{K}^{N}}$. A classical theorem of Witt states that the bilinear space $\left( Z,\,F \right)$ can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights of $F$ and $Z$. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan-Dieudonné theorem. Namely, we show that every isometry $~\sigma$ of a regular bilinear space $\left( Z,\,F \right)$ can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights of $F$, $Z$, and $~\sigma$.

Nous considérons l’intersection d’une sous-variété $X$ d’une variété abélienne $A$ avec l’union de tous les sous-groupes de $A$ obtenus comme somme d’un sous-groupe donné $\varGamma$ de rang fini et d’un sous-groupe algébrique de dimension donnée. Nous montrons que, quitte à retirer à $X$ un certain ensemble exceptionnel, l’intersection est de hauteur bornée. L’énoncé est optimal pour une courbe $X$. Nous étudions également les épaississements $\varGamma_{\varepsilon}$ introduits par Poonen. La démonstration repose sur une généralisation uniforme de la méthode de Vojta et sur des calculs de nombres d’intersection de cycles réels sur $A$.

We study the intersection of a subvariety $X$ of an abelian variety $A$ over $\bar{\mathbb{Q}}$ with the union of all the subgroups obtained as a sum of a given finite rank subgroup $\varGamma$ and an algebraic subgroup of $A$ of given dimension $d$. Our main result asserts that if we remove a suitable exceptional subset from $X$ then the intersection is a set of bounded height. In some cases, this combines with the output of Part I to yield finiteness. In terms of boundedness of the height, we get an optimal statement for a curve $X$ with $d=2$. We also deal with the fattenings $\varGamma_\varepsilon$ introduced by Poonen. The proof rests on a suitably uniform generalization of the method of Vojta and on computations of intersection numbers of real cycles on $A$.

We prove an almost optimal lower bound for the so called last ‘geometric’ minimum for the height of a subvariety V of a power of the multiplicative group ${\mathbb G}_m^n$. More precisely, one shows that if $\boldsymbol x$ is a point of $V(\overline{\mathbb Q})$ which does not belong to a translate of a sub-torus of ${\mathbb G}_m^n$ lying in V, of height less than a function essentially linear in the inverse of the degree of V, then $\boldsymbol x$ belongs to a finite ‘exceptional’ subset of V. Thus, one proves ‘up to an $\varepsilon$’ the sharpest conjectures that can be formulated on this problem. Previously, the best known result in this direction was due to the second author and Philippon, who provided lower bounds which were inverse monomial in the degree of V (Minorations des hauteurs normalisées des sous-variétés des tores, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 489–543; Errata29 (2000)).

We prove a relation between a generating series for the heights of Heegner cycles on the arithmetic surface associated with a Shimura curve and the second term in the Laurent expansion at s = ½ of an Eisenstein series of weight $\frac32$ for SL(2). On the geometric side, a typical coefficient of the generating series involves the Faltings heights of abelian surfaces isogenous to a product of CM elliptic curves, an archimedean contribution, and contributions from vertical components in the fibers of bad reduction. On the analytic side, these terms arise via the derivatives of local Whittaker functions. It should be noted that s = ½ is not the central point for the functional equation of the Eisenstein series in question. Moreover, the first term of the Laurent expansion at s = ½ coincides with the generating function for the degrees of the Heegner cycles on the generic fiber and, in particular, does not vanish.