arXiv subjects

Nha Xuan Truong

Publications and source records attributed to Nha Xuan Truong.

4 recordsLinked to original sources

arXiv2026

Local fields, iterated extensions, and Julia Sets

Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. For $α\in K$, let $K_\infty$ be the extension obtained by adjoining all iterated preimages of $α$ under a unicritical polynomial $f_c(z)=z^\ell - c \in K[z]$. We study the extension $K_\infty/K$ and show that its qualitative behavior depends only on the valuation of $c$. This removes the previous restrictions on $\ell$ in work of Anderson--Hamblen--Poonen--Walton and completes the classification for all $\ell \ge 2$. We also relate the ramification to the structure of the Berkovich Julia set of $f_c$.

math.NT↗

arXiv2026

Slopes of modular forms and geometry of eigencurves

Under a stronger genericity condition, we prove the local analogue of ghost conjecture of Bergdall and Pollack. As applications, we deduce in this case (a) a folklore conjecture of Breuil--Buzzard--Emerton on the crystalline slopes of Kisin's crystabelian deformation spaces, (b) Gouvea's $\lfloor\frac{k-1}{p+1}\rfloor$-conjecture on slopes of modular forms, and (c) the finiteness of irreducible components of the eigencurve. In addition, applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce as corollaries in the reducible and strongly generic case, (d) Gouvea--Mazur conjecture, (e) a variant of Gouvea's conjecture on slope distributions, and (f) a refined version of Coleman's spectral halo conjecture.

math.NT↗

arXiv2023

An explicit computation of the Hecke operator and the ghost conjecture

In this paper, we investigate the Hecke operator at p = 5 and show that the upper minors of the matrix have non zero corank and, interestingly, follow the same ghost pattern in the Ghost conjecture of Bergdall and Pollack. Due to this facts, we conjecture that the slope of Hecke action in this case can be computed using an appropriate variant of ghost series. Assume this result, we achieve an upper bound for the slopes that is similar to the Gouvea's (k-1)/(p+1) conjecture.

math.NT↗

arXiv2022

A local analogue of the ghost conjecture of Bergdall-Pollack

We formulate a local analogue of the ghost conjecture of Bergdall and Pollack, which essentially relies purely on the representation theory of GL_2(Q_p). We further study the combinatorial properties of the ghost series as well as its Newton polygon, in particular, giving a characterization of the vertices of the Newton polygon and proving an integrality result of the slopes. In a forthcoming sequel, we will prove this local ghost conjecture under some mild hypothesis and give arithmetic applications.

math.NT↗