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Matias Alvarado

Publications and source records attributed to Matias Alvarado.

7 recordsLinked to original sources
arXiv2026

On some Diophantine and Ergodic properties of the Schneider map over function fields

We introduce and study continued fractions defined by Schneider-like maps over polynomial rings, where the maps are associated with a fixed polynomial of arbitrary degree. In particular, we prove the existence and uniqueness of the continued fraction expansion for every element of the field of Laurent series. We then establish precise Diophantine approximation properties of the corresponding convergents. We also study the dynamical aspects of the underlying map, proving that the Haar measure is invariant and ergodic. As an arithmetic consequence, we determine the asymptotic sum of the digits of the expansion for almost every element. Finally, we identify the set of elements that are worst approximable in this framework and compute its Hausdorff dimension, showing that it is a fractal set of positive dimension.

math.NT↗
arXiv2026

Asymptotic distribution of CM points on the reduction of the Drinfeld modular curve

We study a distribution problem over global function fields. More precisely, we describe the asymptotic distribution of rank $2$ CM Drinfeld modules among the irreducible components of the analytic reduction of the Drinfeld modular curve. Our approach relies on the properties of the building map and the spectral decomposition of the adjacency operator on a quotient of the Bruhat-Tits tree.

math.NT↗
arXiv2026

Multifractal analysis of power means for the Schneider map on $p\mathbb{Z}_p$

We study the asymptotic power means of the coefficients associated with the Schneider continued fraction map on $p\mathbb{Z}_p$. Using tools from thermodynamic formalism, we compute the Hausdorff dimension of the corresponding level sets and obtain explicit formulas for the associated multifractal spectra. The locally constant nature of the geometric potential enables a precise description in terms of polylogarithm functions, in sharp contrast with the classical real setting.

math.DS↗
arXiv2026

The Lyapunov spectrum for Schneider map on $p\mathbb{Z}_p$

We study the thermodynamic formalism associated with the Schneider map on the p-adic integers $p\mathbb{Z}_p$ . By introducing a geometric potential that captures the expansion of cylinder sets generated by the map, we define a Lyapunov exponent adapted to this non-Archimedean setting. We investigate the corresponding Lyapunov spectrum and show that it is real analytic on its natural domain. Moreover, we obtain an explicit closed formula for the spectrum. As a consequence, we recover and refine known results on the Hausdorff dimension of sets defined by a prescribed asymptotic arithmetic mean of the continued fraction digits. Finally, we relate the Lyapunov exponent to the exponential rate of convergence of rational approximations arising from truncations of the Schneider continued fraction expansion. This provides a $p$-adic analogue of classical results from Diophantine approximation and yielding precise dimension formulas for the associated level sets.

math.DS↗
arXiv2025

Counting algebraic points of bounded degree on curves

Let $X$ be a smooth projective curve over a number field $k$. Let $f\colon X \to \mathbb{P}^1$ be a non-constant morphism that realizes the gonality of $X$. In this article we study the growth rate of $\left\{P\in X\left(\overline{k} \right)\left| [k(x):k]=ν, k(x)=k(f(x)), h(x)\leq T \right.\right\}.$

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arXiv2024

Equidistribution of Hecke Orbits on the Picard group of definite Shimura curves

We prove an equidistribution result about Hecke orbits on the Picard group of Shimura curves coming from definite quaternion algebras over function fields. In particular, we show the equidistribution of Hecke orbits of supersingular Drinfeld modules of rank 2. Our approach is via the automorphic method, using bounds for coefficients of cuspidal automorphic forms of Drinfeld type as the main tool.

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