arXiv subjects

Roberto Alvarenga

Publications and source records attributed to Roberto Alvarenga.

13 recordsLinked to original sources

arXiv2026

Moduli of parabolic bundles on an elliptic curve

A lot is known about the moduli space of parabolic bundles over curves of genus $g\geq 2$, but the lower genus cases are notably different. The goal of this article is to study the geometry of the moduli space of semistable parabolic bundles of rank $3$ with trivial determinant and one marked point on an elliptic curve. We show that this moduli space is rational, give an explicit description of its geometry, its group of automorphisms and also prove a Torelli-type result.

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arXiv2026

The Local Bourbaki Degree of a Plane Projective Curve

The Bourbaki degree of a plane projective curve $F$, denoted by $\mathrm{Bour}(F)$, was introduced in \cite{Marcos} by Jardim, Nejad and Simis. It is defined as the degree of $R/I_ε$, where $R = k[x,y,z]$ is the graded polynomial ring, with $k$ algebraically closed, and $I_ε\subseteq R$ is the Bourbaki ideal associated with a minimal generator $ε$ of the module of first syzygies of the Jacobian ideal $J_F$. In this work, we propose the definition of the local Bourbaki degree at a point $P \in \mathbb{P}^2$, denoted by $\mathrm{Bour}_P(F)$, and prove that $\mathrm{Bour}(F) = \sum_{P \in \mathbb{P}^2}\mathrm{Bour}_P(F).$ Furthermore, we present results that follow from this local definition, which are instrumental in determining the Bourbaki degree and in establishing whether a curve is (nearly) free. In addition, we provide examples of computing the Bourbaki degree via the local formula - an approach that is computationally advantageous, as it, generically, demands fewer calculations.

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arXiv2025

Hall algebras and Hecke modifications of vector bundles

In this article, we investigate Hecke modifications of vector bundles on a smooth projective curve $X$ defined over an arbitrary field. We obtain structural results that allow us to reduce the classification problem of Hecke modifications to the case of vector bundles of lower rank. Moreover, when the base field is a finite field and $X$ is the projective line, we apply the Hall algebra of coherent sheaves to provide a full classification of the Hecke modifications, including their multiplicities. These results are applied to study the space of unramified automorphic forms for $\mathrm{PGL}_n$ over the projective line, leading to a proof that the space of unramified toroidal automorphic forms is trivial.

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arXiv2025

Diophantine equations over the generalized Fibonacci sequences: exploring sums of powers

Let (F_n)_{n} be the classical Fibonacci sequence. It is well-known that it satisfies F_{n}^2 + F_{n+1}^2 = F_{2n+1}. In this study, we explore generalizations of this Diophantine equation in several directions. First, we solve the Diophantine equation (F_{n}^{(k)})^2 + (F_{n+d}^{(k)})^2 = F_{m}^{(k)} over the k-generalized Fibonacci numbers for every k \geq 2, generalizing Chaves and Marques. Next, we solve F_{n}^{s} + F_{n+d}^{s} = F_m over the Fibonacci numbers for every s \geq 2, generalizing Luca and Oyono. Finally, we solve the Diophantine equation F_{n}^s + \cdots + F_{n+d}^s = F_m for d+1 < n and s \geq 2.

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arXiv2025

Hecke modifications of vector bundles

Hecke modifications of vector bundles have played a significant role in several areas of mathematics. They appear in subjects ranging from number theory to complex geometry. This article intends to be a friendly introduction to the subject. We give an overview of how Hecke modifications appear in the literature, explain their origin and their importance in number theory and classical algebraic geometry. Moreover, we report the progress made in describing Hecke modifications explicitly and why these explicit descriptions are important. We describe all the Hecke modifications of the trivial rank $2$ vector bundle over a closed point of degree $5$ in the projective line, as well as all the vector bundles over a certain elliptic curve, which admit a rank $2$ and degree $0$ trace bundle as a Hecke modification. This result is not present in existing literature.

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arXiv2025

Fibonacci Numbers as Sums of Consecutive Terms in $k$-Generalized Fibonacci Sequence

Let (F_n^{(k)})_{n\geq -(k-2)} be the k-generalized Fibonacci sequence, defined as the linear recurrence sequence whose first k terms are $0, 0, \ldots, 0, 1$, and whose subsequent terms are determined by the sum of the preceding k terms. This article is devoted to investigating when the sum of consecutive numbers in the k-generalized Fibonacci sequence belongs to the Fibonacci sequence. Namely, given d,k \in \N, with k \geq 3, our main theorem states that there are at most finitely many n \in \N such that F_n^{(k)} + \cdots + F_{n+d}^{(k)} is a Fibonacci number. In particular, the intersection between the Fibonacci sequence and the k-generalized Fibonacci sequence is finite.

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arXiv2024

Hecke eigenspaces for the projective line

In this article we investigate the action of (ramified and unramified) Hecke operators on automorphic forms for the function field of the projective line defined over a finite field and for the group GL_2. We first compute the dimension of the Hecke eigenspaces for every generator of the unramified Hecke algebra. Thus, we consider the ramification in a point of degree one and describe explicitly the action of certain ramified Hecke operators on automorphic forms. Moreover, for those ramified Hecke operators, we also compute the dimensions of its eigenspaces. We finish the article considering more general ramifications, namely, those one attached to a closed point of higher degree.

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arXiv2023

On the number of elements with prescribed norm and trace

Let F_q be the finite field with cardinality q, where q is a prime power. Given a finite field extension F_q^n over F_q and a,b in (F_q)^{*}, we investigate in this article the number N_n(a,b) of elements in F_q^n whose norm equals a and trace equals b. Our approach to probe N_n(a,b) is to connect it with the number of rational points on certain Artin-Schreier curve. After establish an improvement of the Hasse-Weil bound for that Artin-Schreier curve, we improve the known estimates for N_n(a,b) when (roughly speaking) n \geq \sqrt{q}-1. Moreover, we use this approach to improve the bound given by Moisio and Wan for the number of rational points on the toric Calabi-Yau variety studied by Rojas-Leon and Wan in 2011. We finish the paper with explicit calculations of N_n(a,b) and an application to the number of irreducible monic polynomials in an arithmetic progression.

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arXiv2023

On unramified automorphic forms over the projective line

Let $q$ be a prime power and $\mathbb{F}_q$ be the finite field with $q$ elements. In this article we investigate the space of unramified automorphic forms for $\mathrm{PGL}_n$ over the rational function field defined over $\mathbb{F}_q$ (i.e.\ for $\mathbb{P}^1$ defined over $\mathbb{F}_q$). In particular, we prove that the space of unramified cusp form is trivial and (for $n=3$) that the space of eigenforms is one dimensional. Moreover, we show that there are no nontrivial unramified toroidal forms for $\mathrm{PGL}_3$ over $\mathbb{P}^1$ and conjecture that the space of all toroidal automorphic forms is trivial.

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arXiv2021

Automorphic forms for PGL(3) over elliptic function fields. Part 1: Graphs of Hecke operators

This is a first part of a series of papers in which we develop explicit computational methods for automorphic forms for GL(3) and PGL(3) over elliptic function fields. In this first part, we determine explicit formulas for the action of the Hecke operators on automorphic forms on GL(2) and GL(3) in terms of their graphs. Our primary result consists in a complete description of the graphs of degree 1 Hecke operators for GL(3). As complementary results, we describe the 'even component' of the graphs of degree 2 Hecke operators for GL(2) and the 'neighborhood of the identity' of the graphs of degree 2 Hecke operators for GL(3). In addition, we establish two dualities for Hecke operators for GL(n) and PGL(n), which hold for all n, all degrees and all function fields.

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arXiv2020

Hall algebras and graphs of Hecke operators for elliptic curves

The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms over a global function field. These graphs were introduced by Lorscheid in his PhD thesis for $\text{PGL}_{2}$ and we generalized to $\text{GL}_{n}$ in the paper "On graphs of Hecke operators". After reviewing some general properties, we explain the connection to the Hall algebra of the function field. In the case of an elliptic function field, we can use structure results of Burban-Schiffmann and Fratila to develop an algorithm which explicitly calculate these graphs. We apply this algorithm to determine some structure constants and provide explicitly the rank two case in the last section.

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arXiv2019

$p$-adic Wan-Riemann Hypothesis for $\mathbb{Z}_p$-towers of curves

Our goal in this paper is to investigate four conjectures proposed by Daqing Wan about the stable behavior of a geometric $\mathbb{Z}_p$-tower of curves $X_{\infty}/X$. Let $h_n$ be the class number of the $n$-th layer in $X_{\infty}/X$. It is known from Iwasawa theory that there are integers $μ(X_{\infty}/X), λ(X_{\infty}/X)$ and $ ν(X_{\infty}/X)$ such that the $p$-adic valuation $v_p(h_n)$ equals to $μ(X_{\infty}/X) p^n + λ(X_{\infty}/X) n+ ν(X_{\infty}/X)$ for $n$ sufficiently large. Let $\mathbb{Q}_{p,n}$ be the splitting field (over $\mathbb{Q}_p$) of the zeta-function of $n$-th layer in $X_{\infty}/X$. The $p$-adic Wan-Riemann Hypothesis conjectures that the extension degree $[\mathbb{Q}_{p,n}:\mathbb{Q}_p]$ goes to infinity as $n$ goes to infinity. After motivating and introducing the conjectures, we prove the $p$-adic Wan-Riemann Hypothesis when $λ(X_{\infty}/X)$ is nonzero.

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arXiv2019

On graphs of Hecke operators

The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms. Let $X$ be a curve over $\mathbb{F}_q$, $F$ its function field and $\mathbb{A}$ the adele ring of $F$. In this paper we will exhibit the first properties for the graph of Hecke operators for $\mathrm{GL}_n(\mathbb{A}),$ for every $n \geq 1.$ This includes a description of the graph in terms of coherent sheaves on $X.$ We provide a numerical condition for two vertices to be connected by an edge. Moreover, we describe how to calculate these graphs in the case of the projective line $X = \mathbb{P}^1(\mathbb{F}_q).$

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