P. Mercuri

activity

20142026

most citedAutomorphisms of Cartan modular curves of prime and composite level

6 citations · 17 across the 10 of their papers we have counts for

collaborators

15 papers

math.NT2026

Automorphisms of general modular curves

Valerio Dose, Guido Maria Lido, Pietro Mercuri

We prove that for a geometrically connected modular curve $X_{H}$ of prime level larger than $311$ , all automorphisms are modular, i.e., they arise from the moduli interpretation of…

cs.CR2025

A survey about Hidden Subgroup Problem from a mathematical and cryptographic perspective

Simone Dutto, Pietro Mercuri, Nadir Murru +1

We provide a survey on the Hidden Subgroup Problem (HSP), which plays an important role in studying the security of public-key cryptosystems. We first review the abelian case, wher…

math.NT2025

Point counts, automorphisms, and gonalities of Shimura curves

Pietro Mercuri, Oana Padurariu, Frederick Saia +1

We implement an algorithm to compute the number of points over finite fields for the Shimura curves $X_{0}^{D} (N)$ over $Q$ and their Atkin--Lehner quotients. Our computations…

math.NT2025

Maximal curves over finite fields and a modular isogeny

Valerio Dose, Guido Lido, Pietro Mercuri +1

We prove the existence of curves of genus $7$ and $12$ over the field with $1 1^{5}$ elements, reaching the Hasse-Weil-Serre upper bound. These curves are quotients of modular curves…

math.NT2025

Greenberg's conjecture for real quadratic number fields

Pietro Mercuri, Maurizio Paoluzi, René Schoof

We compute the $3$ -class groups $A_{n}$ of the fields $F_{n}$ in the cyclotomic $Z_{3}$ -extensions of the real quadratic fields of discriminant $f < 100, 000$ . In all cases the ord…

math.NT2023

Intersection matrices for the minimal regular model of $X_{0} (N)$ and applications to the Arakelov canonical sheaf

Paolo Dolce, Pietro Mercuri

Let $N > 1$ be an integer coprime to $6$ such that $N \in / {5, 7, 13}$ and let $g = g (N)$ be the genus of the modular curve $X_{0} (N)$ . We compute the intersection matrices relative to s…

math.NT2022

Parity of 4-regular and 8-regular partition functions

Giacomo Cherubini, Pietro Mercuri

We give a complete characterization of the parity of $b_{8} (n)$ , the number of $8$ -regular partitions of $n$ . Namely, we prove that $b_{8} (n)$ is odd or even depending on whether or no…

math.CO2021

Partition identities associated to Rogers-Ramanujan type identities

Pietro Mercuri

We show that, in many cases, there are infinitely many sets of partitions corresponding to a single analytical Rogers-Ramanujan type identity. This means that a single analytical R…

math.NT2020★ 2 cited

High Order Elements in Finite Fields Arising from Recursive Towers

Valerio Dose, Pietro Mercuri, Ankan Pal +1

We provide a recipe to construct towers of fields producing high order elements in $GF (q, 2^{n})$ , for odd $q$ , and in $GF (2, 2 \cdot 3^{n})$ , for $n \geq 1$ . These towe…

math.NT2020★ 6 cited

Automorphisms of Cartan modular curves of prime and composite level

Valerio Dose, Guido Lido, Pietro Mercuri

We study the automorphisms of modular curves associated to Cartan subgroups of $GL_{2} (Z / n Z)$ and certain subgroups of their normalizers. We prove that if $n…

math.NT2018

Modular forms invariant under non-split Cartan subgroups

Pietro Mercuri, Rene Schoof

In this paper we describe a method for computing a basis for the space of weight $2$ cusp forms invariant under a non-split Cartan subgroup of prime level $p$ . As an application we…

math.NT2017★ 4 cited

Double Covers of Cartan Modular Curves

Valerio Dose, Pietro Mercuri, Claudio Stirpe

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_{…

math.NT2016★ 2 cited

Equations and Rational Points of the Modular Curves $X_{0}^{+} (p)$

Pietro Mercuri

Let $p$ be an odd prime number and let $X_{0}^{+} (p)$ be the quotient of the classical modular curve $X_{0} (p)$ by the action of the Atkin-Lehner operator $w_{p}$ . In this paper we show ho…

math.NT2016★ 3 cited

Modular Curves with many Points over Finite Fields

Valerio Dose, Guido Lido, Pietro Mercuri +1

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…

math.NT2014

Classification of Algebraic Function Fields with Class Number One

Pietro Mercuri, Claudio Stirpe

In this paper we prove that there are exactly eight function fields, up to isomorphism, over finite fields with class number one.

Automorphisms of general modular curves

A survey about Hidden Subgroup Problem from a mathematical and cryptographic perspective

Point counts, automorphisms, and gonalities of Shimura curves

Maximal curves over finite fields and a modular isogeny

Greenberg's conjecture for real quadratic number fields

Intersection matrices for the minimal regular model of X0​(N) and applications to the Arakelov canonical sheaf

Parity of 4-regular and 8-regular partition functions

Partition identities associated to Rogers-Ramanujan type identities

High Order Elements in Finite Fields Arising from Recursive Towers

Automorphisms of Cartan modular curves of prime and composite level

Modular forms invariant under non-split Cartan subgroups

Double Covers of Cartan Modular Curves

Equations and Rational Points of the Modular Curves X0+​(p)

Modular Curves with many Points over Finite Fields

Classification of Algebraic Function Fields with Class Number One

Intersection matrices for the minimal regular model of $X_{0} (N)$ and applications to the Arakelov canonical sheaf

Equations and Rational Points of the Modular Curves $X_{0}^{+} (p)$