arXiv subjects

Jit Wu Yap

Publications and source records attributed to Jit Wu Yap.

13 recordsLinked to original sources

arXiv2026

Degree Growth of Iterates of Curves and Likely Intersections

We study the growth of the bidegree of an ample irreducible curve in $\mathbb{P}^1 \times \mathbb{P}^1$ under a product polynomial endomorphism $φ=(f,g)$, where at least one of $f$ and $g$ is non-exceptional. We prove that, if the curve $C$ is not preperiodic under $(f^a,g^b)$ for any $a,b\geq 1$, then the bidegree of $φ^n(C)$ is asymptotic to $(°(g)^n,°(f)^n)$. As an application of this exponential growth, we prove a geometric analogue of Silverman's theorem on the finiteness of $S$-integral points in orbits. Namely if $C$ is not $(f^a,g^b)$-preperiodic and $C'$ is not totally invariant for $φ$, then for any infinite sequence ${n_i}$ of positive integers, the union of the intersections $$ \bigcup_{i \geq 1} \left( φ^{n_i}(C)\cap C' \right) $$ is Zariski dense in $C'$.

math.DS↗

arXiv2026

Finiteness of an Isogeny Class via Equidistribution

In a very recent, beautiful paper, Junyi Xie and Ziquan Yang gave a new proof of Faltings' isogeny theorem over number fields by reducing to Tate's theorem over finite fields. We show that the same ideas can be adapted to prove finiteness of an isogeny class over number fields from the finiteness of an isogeny class over finite fields. Hence we obtain a new proof of the Shafarevich conjecture over number fields. The proof in this paper was obtained through the use of GPT6 Astra Ultra.

math.NT↗

arXiv2026

Non-Archimedean Rigidity and Uniformity for Common Preperiodic Points

Let $k$ be an algebraically closed, complete non-Archimedean field of residue characteristic $0$. Let $f$ be a polynomial of degree at least $2$ over $k$ which does not have potential good reduction. We prove that if $g$ is any other polynomial with the same Julia set, then $f$ and $g$ must be dynamically related. As a consequence, we show that for any two complex polynomials $f,g$ of degree at least $2$, either their sets of preperiodic points coincide, or the number of their common preperiodic points is uniformly bounded above by a constant depending only on the degrees, thereby answering a conjecture of DeMarco--Krieger--Ye for polynomials. We also establish relative results, allowing us to prove special cases of the DeMarco--Mavraki conjecture.

math.DS↗

arXiv2026

Uniform boundedness of small points on abelian varieties over function fields

Let $k$ be a field of characteristic $0$ and let $K = k(B)$ be the function field of a geometrically irreducible projective curve $B$ over $k$. Let $A/K$ be a $g$-dimensional abelian variety with $\mathrm{Tr}_{K/k}(A) = 0$. We prove that any $K$-rational torsion point $x$ of $A$ has order uniformly bounded in terms of $g$ and the gonality of $B$. We also prove a uniform lower bound on the Néron-Tate height $\widehat{h}_{A,L}(x)$ in terms of the stable Faltings height $h_{\mathrm{Fal}}(A)$ for any $K$-rational point $x$ whose forward orbit is Zariski dense, proving the Lang-Silverman conjecture over function fields of characteristic $0$.

math.NT↗

arXiv2026

Uniform Bounds on S-Integral Torsion Points for $\mathbb{G}_m$ and Elliptic Curves

Let $K$ be a number field, $S$ a finite set of places. For $\mathbb{G}_m$ or an elliptic curve $E$ defined over $K$, we establish uniformity results on the number of $S$-integral torsion points relative to a non-torsion point $β$, as $β$ varies over number fields of bounded degree. In particular for $\mathbb{G}_m$, if $D$ is a positive integer, we prove a uniform bound on the degree of a torsion point $ζ$ that is $S$-integral relative to a non-torsion point $β$ with degree $\leq D$.

math.NT↗

arXiv2026

$S$-Integral Points in Orbits on $\mathbb{P}^1$

Let $K$ be a number field and $S$ a finite set of places of $K$ that contains all of the archimedean places. Let $φ: \mathbb{P}^1 \to \mathbb{P}^1$ be a rational map of degree $d \geq 2$ defined over $K$. Given $α\in \mathbb{P}^1(K)$ non-preperiodic and $β\in \mathbb{P}^1(K)$ non-exceptional, we prove an upper bound of the form $O(|S|^{1+ε})$ on the number of points in the forward orbit of $α$ that are $S$-integral relative to $β$, extending results of Hsia--Silverman [HS11]. We also prove uniform bounds when $φ$ is a polynomial, extending resaults of Krieger et al [KLS+15].

math.NT↗

arXiv2026

On the pointwise convergence of the number of abelian varieties over $\mathbb{F}_p$ with fixed trace

Extending Katz-Sarnak heuristics, Ballini-Lombardo-Verzobio [BLV25] conjectures a limiting distribution as $p \to \infty$ for $\# A_g(\mathbb F_p,t)$, the number of $g$-dimensional PPAVs over $\mathbb F_p$ with trace $t$, as a product of natural local factors $v_\ell(t)$ for non-archimedean places $\ell$ and the Sato-Tate measure $\text{ST}_g$ corresponding to $\infty$. We prove that their conjecture is true for all $g$. As a consequence, we obtain analogous results on the distribution of curves of genus $2$ and $3$, answering questions of Bergström-Howe-García-Ritzenthaler [BHLR24] and [BLV25].

math.NT↗

arXiv2025

A quantitative dynamical Zhang fundamental inequality and Bogomolov-type problems

We prove a quantitative version of Zhang's fundamental inequality for heights attached to polarizable endomorphisms. As an application, we obtain a gap principle for the Néron-Tate height on abelian varieties over function fields of arbitrary transcendence degree and characteristic zero, extending the result of Gao-Ge-Kühne. We also establish instances of effective gap principles for regular polynomial endomorphisms of $\mathbb{P}^2$, in the sense that all constants can are explicit. These yield effective instances of uniformity in the dynamical Bogomolov conjecture in both the arithmetic and geometric settings, including examples in prime characteristic.

math.NT↗

arXiv2025

Lower Bounds for Galois Orbits of periodic points for polarized endomorphisms

Let $K$ be a number field, $X$ a smooth projective variety over $K$ and $f: X \to X$ a polarized endomorphism of degree $d \geq 2$. We prove an exponential lower bound on $[K(\Per_n):K]$, where $\Per_n$ is the set of $n$-periodic points, extending results of [Yap24] to higher dimensions. We also prove a quantitative rate of equidistribution for $\Per_n$ to the equilibrium measure.

math.NT↗

arXiv2025

On the Number of Small Points for Rational Maps

Let $K$ be a number field and $f: \mathbb{P}^1 \to \mathbb{P}^1$ a rational map of degree $d \geq 2$ with at most $s$ places of bad reduction, where we include all archimedean places. We prove that there exists constants $c_1,c_2 > 0$, depending only on $d$ and not on $f$ or $K$, such that $$ \# \left\{ x \in \mathbb{P}^1(K) \mid \widehat{h}_f(x) \leq \frac{c_1}{s} h_{\operatorname{rat}_d}(\langle f \rangle) \right\} \leq c_2 s \log(s). $$ Here, $\operatorname{rat}_d$ is the moduli space of rational maps up to conjugacy, $h_{\operatorname{rat}_d}$ is an ample height and $\langle f \rangle$ is the equivalence class associated to $f$. This gives a uniform version of a theorem of Baker as well as generalizing the results of Benedetto and Looper from polynomials to rational maps. The main tool used is the degeneration of sequences of rational maps by Luo which has been recently formalized by Favre-Gong via Berkovich spaces.

math.NT↗

arXiv2024

Quantitative Equidistribution of Small Points for Canonical Heights

Let $X$ be a smooth projective variety defined over a number field $K$ and let $φ: X \to X$ a polarized endomorphism of degree $d \geq 2$. Let $\widehat{h}_φ$ be the canonical height associated to $φ$ on $X(\overline{K})$. Given a generic sequence of points $(x_n)$ with $\widehat{h}_φ(x_n) \to 0$ and a place $v \in M_K$, Yuan [Yua08] has shown that the conjugates of $x_n$ equidistribute to the canonical measure $μ_{φ,v}$. When $v$ is archimedean, we will prove a quantitative version of Yuan's result. We give two applications of our result to polarized endomorphisms $φ$ of smooth projective surfaces that are defined over a number field $K$. The first is an exponential rate of convergence for periodic points of period $n$ to the equilibrium measure and the second is an exponential lower bound on the degree of the extension containing all periodic points of period $n$. When $X$ is an abelian variety, we also give an upper bound on the smallest degree of a hypersurface that contains all points $x \in X(\overline{K})$ satisfying $[K(x):K] \leq D$ and $\widehat{h}_X(x) \leq \frac{c}{D^8}$ for some fixed constant $c > 0$ where $\widehat{h}_X$ is the Neron--Tate height for $X$.

math.NT↗

arXiv2023

Shared Dynamically-Small Points for Polynomials on Average

Given two rational maps $f,g: \mathbb{P}^1 \to \mathbb{P}^1$ of degree $d$ over $\mathbb{C}$, DeMarco-Krieger-Ye [DKY22] has conjectured that there should be a uniform bound $B = B(d) > 0$ such that either they have at most $B$ common preperiodic points or they have the same set of preperiodic points. We study their conjecture from a statistical perspective and prove that the average number of shared preperiodic points is zero for monic polynomials of degree $d \geq 6$ with rational coefficients. We also investigate the quantity $\liminf_{x \in \overline{\mathbb{Q}}} \left(\widehat{h}_f(x) + \widehat{h}_g(x) \right)$ for a generic pair of polynomials and prove both lower and upper bounds for it.

math.DS↗

arXiv2019

Nilpotent orbits of orthogonal groups over $p$-adic fields, and the DeBacker parametrization

For local non-archimedean fields $k$ of sufficiently large residual characteristic, we explicitly parametrize and count the rational nilpotent adjoint orbits in each algebraic orbit of orthogonal and special orthogonal groups. We separately give an explicit algorithmic construction for representatives of each orbit. We then, in the general setting of groups $\mathrm{GL}_n(D)$, $\mathrm{SL}_n(D)$ (where $D$ is a central division algebra over $k$) or classical groups, give a new characterisation of the "building set" (defined by DeBacker) of an $\mathfrak{sl}_2(k)$-triple in terms of the building of its centralizer. Using this, we prove our construction realizes DeBacker's parametrization of rational nilpotent orbits via elements of the Bruhat-Tits building.

math.GR↗