Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.06166

Tower of coverings and polylog-size coverings of rational points

Kenneth Chung Tak Chiu
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that a smooth irreducible proper variety over a number field admits polylogarithmic-size coverings of rational points of bounded height if it admits a tower of finite étale coverings satisfying several conditions. One of the conditions is a height comparison on twists of the finite étale covers. We prove that these conditions hold for abelian varieties.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kenneth Chung Tak Chiu. 2026-10-05. Tower of coverings and polylog-size coverings of rational points. https://arxiv.org/abs/2610.06166

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

New Results for Euler Sums

We present a large number of analytic evaluations of Euler sums, namely sums such as \begin{align} M(m,n_0,n_1,n_2, \ldots, n_t) &= \sum_{k=1}^\infty \frac{H(k)^m}{k^{n_0} (k+1)^{n_1} (k+2)^{n_2} \cdots (k+t)^{n_t}}, \nonumber \end{align} for nonnegative integers $m$ and $(n_i)$, with $m \geq 1$ and $n_0 + n_1 + \cdots + n_t \geq 2$, where $H(k) = \sum_{j=1}^k 1/j$ is the harmonic function. These results were obtained either by algebraic manipulations, or else by very high-precision numerical evaluations combined with an integer relation algorithm to obtain the analytic formulas. We show how many of these results can be derived from a few basic facts, and that these techniques are applicable to Euler sums of even more general forms than the above cases. We then show that these results permit the calculation of constants for Euler sums resembling the Stieltjes $γ$ constants arising in the theory of the Riemann zeta function, and we also present some preliminary results on the asymptotic behavior of these constants. A sign error has been corrected in eq. 98.

math.NT↗
arXiv2026

The moments of split greatest common divisors

Sequences of the form $(\gcd(u_n,v_n))_{n \in \mathbb N}$, with $(u_n)_n$, $(v_n)_n$ sums of $S$-units, have been considered by several authors. The study of $\gcd(n,u_n)$ corresponds, after Silverman, to divisibility sequences arising from the algebraic group $\mathbb G_{\mathrm{a}} \times \mathbb G_{\mathrm{m}}$; in this case, Sanna determined all asymptotic moments of the arithmetic function $\log\,\gcd (n,u_n)$ when $(u_n)_n$ is a Lucas sequence. Here, we characterize the asymptotic behavior of the moments themselves $\sum_{n \leq x}\,\gcd(n,u_n)^λ$, thus solving the moment problem for $\mathbb G_{\mathrm{a}} \times \mathbb G_{\mathrm{m}}$. We give both unconditional and conditional results, the latter only relying on standard conjectures in analytic number theory.

math.NT↗