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Claudio Stirpe

Publications and source records attributed to Claudio Stirpe.

8 recordsLinked to original sources
arXiv2025

Point counts, automorphisms, and gonalities of Shimura curves

We implement an algorithm to compute the number of points over finite fields for the Shimura curves $X_0^D(N)$ over $\mathbb{Q}$ and their Atkin--Lehner quotients. Our computations identify $116$ such quotients over finite fields (out of $783514$ tested) that attain a number of rational points exceeding that of any previously known curve of the same genus over the same finite field. To illustrate the utility of our point counts algorithm in addressing arithmetic questions, we prove that all automorphisms are Atkin--Lehner for $9288$ of the $10609$ curves $X_0^D(N)$ of genus $g > 2$ with $D$ the discriminant of an indefinite quaternion algebra over $\mathbb{Q}$, $N$ a squarefree positive integer coprime to $D$, and $DN\leq 10000$, and we determine all tetragonal and geometrically tetragonal curves $X_0^D(N)$ up to a small number of possible exceptions.

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arXiv2025

Maximal curves over finite fields and a modular isogeny

We prove the existence of curves of genus $7$ and $12$ over the field with $11^5$ elements, reaching the Hasse-Weil-Serre upper bound. These curves are quotients of modular curves and we give explicit equations. We compute the number of points of many quotient modular curves in the same family without providing equations. For various pairs (genus, finite field) we find new records for the largest known number of points. In other instances we find quotient modular curves that are maximal, matching already known results. To perform these computations, we provide a generalization of Chen's isogeny result.

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arXiv2021

High Order Elements in Finite Fields Arising from Recursive Towers

We provide a recipe to construct towers of fields producing high order elements in $\mathrm{GF}(q,2^n)$, for odd $q$, and in $\mathrm{GF}(2,2 \cdot 3^n)$, for $n \ge 1$. These towers are obtained recursively by $x_{n}^2 + x_{n} = v(x_{n - 1})$, for odd $q$, or $x_{n}^3 + x_{n} = v(x_{n - 1})$, for $q=2$, where $v(x)$ is a polynomial of small degree over the prime field $\mathrm{GF}(q,1)$ and $x_n$ belongs to the finite field extension $\mathrm{GF}(q,2^n)$, for $q$ odd, or to $\mathrm{GF}(2,2\cdot 3^n)$. Several examples are carried out and analysed numerically. The lower bounds of the orders of the groups generated by $x_n$, or by the discriminant $δ_n$ of the polynomial, are similar to the ones obtained in [BCG+09], but we get better numerical results in some cases.

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arXiv2017

Double Covers of Cartan Modular Curves

We present a strategy to obtain explicit equations for the modular double covers associated respectively to both a split and a non-split Cartan subgroup of $\text{GL}_2(\mathbb F_{p})$ with $p$ prime. Then we apply it successfully to the level $13$ case.

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arXiv2016

Modular Curves with many Points over Finite Fields

We describe an algorithm to compute the number of points over finite fields on a broad class of modular curves: we consider quotients $X_H/W$ for $H$ a subgroup of $\GL_2(\mathbb Z/n\mathbb Z)$ such that for each prime $p$ dividing $n$, the subgroup $H$ at $p$ is either a Borel subroup, a Cartan subgroup, or the normalizer of a Cartan subgroup of $\GL_2(\mathbb Z/p^e\mathbb Z)$, and for $W$ any subgroup of the Atkin-Lehner involutions of $X_H$. We applied our algorithm to more than ten thousands curves of genus up to 50, finding more than one hundred record-breaking curves, namely curves $X/\FF_q$ with genus $g$ that improve the previously known lower bound for the maximum number of points over $\FF_q$ of a curve with genus $g$. As a key technical tool for our computations, we prove the generalization of Chen's isogeny to all the Cartan modular curves of composite level.

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