Searcharxiv⌕ Search

arXiv subjects

Sebastián Herrero

Publications and source records attributed to Sebastián Herrero.

14 recordsLinked to original sources
arXiv2026

p-Adic distribution of CM points and Hecke orbits. II: Linnik equidistribution on the supersingular locus

For every prime number $p$, this paper determines the asymptotic distribution of CM points on the moduli space of elliptic curves over $\mathbb{C}_p$. In stark contrast to the complex case, the $p$-adic setting admits infinitely many distinct accumulation measures. The paper identifies all such measures by translating this problem as a $p$-adic analog of Linnik's classical problem on the asymptotic distribution of integer points on spheres. This translation draws on the relationship between the deformation theories of elliptic curves and formal modules, together with results of Gross and Hopkins. The bounds for Fourier coefficients of cuspidal modular forms of Deligne, Iwaniec, and Duke yield a deviation estimate that solves this $p$-adic Linnik problem. This paper also identifies all the accumulation measures of an arbitrary Hecke orbit.

math.NT↗
arXiv2026

Statistical properties of Hecke correspondences

This paper studies the statistical properties of the dynamical system generated by a Hecke correspondence on the modular curve over $\mathbb{C}$ and, for every prime number $p$, over $\mathbb{C}_p$. Over $\mathbb{C}$, it proves that the equidistribution to the hyperbolic measure established by Clozel and Otal occurs at an exponential rate. Moreover, it determines the sharp rate, assuming an affirmative solution to the Ramanujan-Petersson conjecture. Over $\mathbb{C}_p$, two distinct types of behavior arise. In the first, every orbit converges towards the Gauss point in the Berkovich affine line, and this paper establishes the sharp exponential convergence rate. In the second, a form of unique ergodicity holds on each orbit closure, and this paper establishes a spectral gap property and, as a consequence, a central limit theorem. This complements the central limit theorem proved by Cantat and Le Borgne over $\mathbb{C}$. Finally, the paper extends and strengthens the results of Goren and Kassaei on the associated random walks.

math.NT↗
arXiv2025

Real part of cycle integrals and conjectures of Kaneko

We prove two of Kaneko's conjectures on the "values" $\mathrm{val}(w)$ of the modular $j$ function at real quadratic irrationalities: we prove the lower bound $\mathrm{Re}(\mathrm{val}(w))\geq \mathrm{val}\left(\frac{1+\sqrt{5}}{2}\right)$ for all real quadratics $w$ and the upper bound $\mathrm{Re}(\mathrm{val}(w))\leq \mathrm{val}\left(1+\sqrt{2}\right)$ for all Markov irrationalities $w$. These results generalize to the "values" at quadratic irrationalities of any weakly holomorphic modular function $f$ such that $f(e^{it})$ is real, non-negative and increasing for $t\in [π/3,π/2]$.

math.NT↗
arXiv2025

A Lyapunov exponent attached to modular functions

To each weakly holomorphic modular function $f\not \equiv 0$ for $\mathrm{SL}(2,\mathbb{Z})$, which is non-negative on the geodesic arc $\{e^{it} : π/3\leq t\leq 2π/3\}$, we attach a $\mathrm{GL}(2,\mathbb{Z})$-invariant map $Λ_f:\mathbb{P}^1(\mathbb{R})\to \mathbb{R}$ that generalizes the Lyapunov exponent function introduced by Spalding and Veselov. We prove that it takes every value between $0$ and $Λ_f\left(\frac{1+\sqrt{5}}{2}\right)$ and it gives an increasing convex function on the Markov irrationalities when ordered using their parametrization by Farey fractions in $[0,1/2]$. In the case of quadratic irrationals $w$ with purely periodic continued fraction expansion, the value $Λ_f(w)$ equals the real part of the cycle integral of $f$ along the associated geodesic $C_w$ on the modular surface, normalized with the word length of the associated hyperbolic matrix $A_w$ as a word in the generators $T=\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$ and $V=\left(\begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix}\right)$. These results are related to conjectures of Kaneko who observed several similar behavior for the cycle integrals of the modular $j$ function when normalized by the hyperbolic length of the geodesic $C_w$.

math.NT↗
arXiv2024

Counting rational points on Hirzebruch-Kleinschmidt varieties over global function fields

Inspired by Bourqui's work on anticanonical height zeta functions on Hirzebruch surfaces, we study height zeta functions of split toric varieties with Picard rank 2 over global function fields, with respect to height functions associated with big metrized line bundles. We show that these varieties can be naturally decomposed into a finite disjoint union of subvarieties, where precise analytic properties of the corresponding height zeta functions can be given. As application, we obtain asymptotic formulas for the number of rational points of large height on each subvariety, with explicit leading constants and controlled error terms.

math.NT↗
arXiv2024

Counting rational points on Hirzebruch-Kleinschmidt varieties over number fields

We study the asymptotic growth of the number of rational points of bounded height on smooth projective split toric varieties with Picard rank 2 over number fields, with respect to Arakelov height functions associated with big metrized line bundles. We show that these varieties can be naturally decomposed into a finite disjoint union of subvarieties, where explicit asymptotic formulas for the number of rational points of bounded height can be given. Additionally, we present various examples, including the case of Hirzebruch surfaces.

math.NT↗
arXiv2024

Twisted traces of modular functions on hyperbolic $3$-space

We compute analogues of twisted traces of CM values of harmonic modular functions on hyperbolic $3$-space and show that they are essentially given by Fourier coefficients of the $j$-invariant. From this we deduce that the twisted traces of these harmonic modular functions are integers. Additionally, we compute the twisted traces of Eisenstein series on hyperbolic $3$-space in terms of Dirichlet $L$-functions and divisor sums.

math.NT↗
arXiv2024

Special values of Green's functions on hyperbolic $3$-space

Gross, Kohnen and Zagier proved an averaged version of the algebraicity conjecture for special values of higher Green's functions on modular curves. In this work, we study an analogous problem for special values of Green's functions on hyperbolic $3$-space. We prove that their averages can be computed in terms of logarithms of primes and logarithms of units in real quadratic fields. Moreover, we study twisted averages of special values of Green's functions, which yield algebraic numbers instead of logarithms.

math.NT↗
arXiv2023

There are at most finitely many singular moduli that are S-units

We show that for every finite set of prime numbers S, there are at most finitely many singular moduli that are S-units. The key new ingredient is that for every prime number p, singular moduli are p-adically disperse. We prove analogous results for the Weber modular functions, the lambda invariants and the McKay-Thompson series associated to the elements of the monster group. Finally, we also obtain that a modular function that specializes to infinitely many algebraic units at quadratic imaginary numbers must be a weak modular unit.

math.NT↗
arXiv2021

On two notions of expansiveness for continuous semiflows

We study two notions of expansiveness for continuous semiflows: expansiveness in the sense of Alves, Carvalho and Siqueira (2017), and an adaptation of positive expansiveness in the sense of Artigue (2014). We prove that if $X$ is a metric space and $ϕ$ is an expansive semiflow on $X$ according to the first definition, then the semiflow $ϕ$ is trivial and the space $X$ is uniformly discrete. In particular, if $X$ is compact then it is finite. With respect to the second definition, we prove that if $X$ is a compact metric space and $ϕ$ is a positive expansive semiflow on it, then $X$ is a union of at most finitely many closed orbits, unbranched tails and isolated singularities.

math.DS↗
arXiv2020

Solutions of equations involving the modular $j$ function

Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular $j$ function. We show general cases in which these systems have solutions, and then we look at certain situations in which the modular Schanuel conjecture implies that these systems have generic solutions. An unconditional result in this direction is proven for certain polynomial equations on $j$ with algebraic coefficients.

math.NT↗
arXiv2020

p-Adic distribution of CM points and Hecke orbits. I. Convergence towards the Gauss point

We study the asymptotic distribution of CM points on the moduli space of elliptic curves over $\mathbb{C}_p$, as the discriminant of the underlying endomorphism ring varies. In contrast with the complex case, we show that there is no uniform distribution. In this paper we characterize all the sequences of discriminants for which the corresponding CM points converge towards the Gauss point of the Berkovich affine line. We also give an analogous characterization for Hecke orbits. In the companion paper we characterize all the remaining limit measures of CM points and Hecke orbits.

math.NT↗
arXiv2018

Mock modular forms whose shadows are Eisenstein series of integral weight

The purpose of this article is to give a simple and explicit construction of mock modular forms whose shadows are Eisenstein series of arbitrary integral weight, level, and character. As application, we construct forms whose shadows are Hecke's Eisenstein series of weight one associated to imaginary quadratic fields, recovering some results by Kudla, Rapoport and Yang (1999), and Schofer (2009), and forms whose shadows equal $Θ^{2k}(z)$ for $k\in \{1,2,3,4\}$, where $Θ(z)$ denotes Jacobi's theta function.

math.NT↗