arXiv subjects

Nicolás Sirolli

Publications and source records attributed to Nicolás Sirolli.

12 recordsLinked to original sources

arXiv2026

The distribution of $k$-free ideals in ray class groups

In this paper, we extend the classical problem of studying the distribution of $k$-free integers in arithmetic progressions to the setting of arbitrary number fields. Using the language of ray class groups, we establish asymptotic formulas, together with error terms, for the number of $k$-free ideals of bounded norm lying in a given ray class. In particular, our results show that $k$-free ideals are equidistributed among ray classes. We also obtain improved error estimates in the cases of ideal class groups and narrow class groups by using sharper ideal counting asymptotics due to Landau. Our results recover the classical formulas of Gegenbauer and Cohen--Robinson over $\mathbb{Q}$ and extend previous work of Benkowski, Nymann, and Sittinger to the setting of ray class groups. We also present explicit computational examples that illustrate the asymptotic formulas and the equidistribution of $k$-free ideals among ray classes.

math.NT↗

arXiv2026

On the efficient computation of Fourier coefficients of eta-quotients

The Fourier coefficients of a negative weight eta-quotient, in many particular cases, and after Sussman in general, are known to be expressible by Hardy-Ramanujan-Rademacher type series. We show that the central terms of the coefficients of these series can be efficiently computed, showing that they can be expressed in terms of twisted Kloosterman sums, and that they satisfy multiplicativity relations; this extends the results from Lehmer for the partition function. We also give explicit bounds for the tails of these series, needed for effectively computing the aforementioned Fourier coefficients.

math.NT↗

arXiv2024

Explicit families of congruences for the overpartition function

In this article we exhibit new explicit families of congruences for the overpartition function, making effective the existence results given previously by Treneer. We give infinite families of congruences modulo $m$ for $m = 5, 7, 11$, and finite families for $m = 13, 17, 19$.

math.NT↗

arXiv2023

Efficient computation of the overpartition function and applications

In this paper we develop a method to calculate the overpartition function efficiently using a Hardy-Rademacher-Ramanujan type formula, and we use this method to find many new Ramanujan-style congruences, whose existence is predicted by Treneer and a few of which were first discovered by Ryan, Scherr, Sirolli and Treneer.

math.NT↗

arXiv2023

Congruences satisfied by eta-quotients

The values of the partition function, and more generally the Fourier coefficients of many modular forms, are known to satisfy certain congruences. Results given by Ahlgren and Ono for the partition function and by Treneer for more general Fourier coefficients state the existence of infinitely many families of congruences. In this article we give an algorithm for computing explicit instances of such congruences for eta-quotients. We illustrate our method with a few examples.

math.NT↗

arXiv2022

Effective construction of Hilbert modular forms of half-integral weight

Given a Hilbert cuspidal newform $g$ we construct a family of modular forms of half-integral weight whose Fourier coefficients give the central values of the twisted $L$-series of $g$ by fundamental discriminants. The family is parametrized by quadratic conditions on the primes dividing the level of $g$, where each form has coefficients supported on the discriminants satisfying the conditions. These modular forms are given as generalized theta series and thus their coefficients can be effectively computed. Our construction works over arbitrary totally real number fields, except that in the case of odd degree the square levels are excluded. It includes all discriminants except those divisible by primes whose square divides the level.

math.NT↗

arXiv2020

An explicit Waldspurger formula for Hilbert modular forms II

We describe a construction of preimages for the Shimura map on Hilbert modular forms using generalized theta series, and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted $L$-functions. This formula extends our previous work, allowing to compute these central values when the main central value vanishes.

math.NT↗

arXiv2019

A Gross-Kohnen-Zagier theorem for non-split Cartan curves

Let $p$ be a prime number and let $E/\mathbb{Q}$ be an elliptic curve of conductor $p^2$ and odd analytic rank. We prove that the positions of its special points arising from non-split Cartan curves and imaginary quadratic fields where $p$ is inert are encoded in the Fourier coefficients of a Jacobi form of weight $6$ and lattice index of rank $9$, obtaining a result analogous to that of Gross, Kohnen and Zagier.

math.NT↗

arXiv2018

An explicit Waldspurger formula for Hilbert modular forms

We describe a construction of preimages for the Shimura map on Hilbert modular forms, and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted L-functions. Our construction is inspired by that of Gross and applies to any nontrivial level and arbitrary base field, subject to certain conditions on the Atkin-Lehner eigenvalues and on the weight.

math.NT↗

arXiv2016

Computing Jacobi Forms

We describe an implementation for computing holomorphic and skew-holomorphic Jacobi forms of integral weight and scalar index on the full modular group. This implementation is based on formulas derived by one of the authors which express Jacobi forms in terms of modular symbols of elliptic modular forms. Since this method allows to generate a Jacobi eigenform directly from a given modular eigensymbol without reference to the whole ambient space of Jacobi forms it makes it possible to compute Jacobi Hecke eigenforms of large index. We illustrate our method with several examples.

math.NT↗

arXiv2014

Preimages for the Shimura map on Hilbert modular forms

In this article we give a method to construct preimages for the Shimura correspondence on Hilbert modular forms of odd and square-free level. The method relies in the ideas presented for the rational case by Pacetti and Tornaría, and is such that the Fourier coefficients of the preimages constructed can be computed explicitly.

math.NT↗

arXiv2012

Computing ideal classes representatives in quaternion algebras

Let $K$ be a totally real number field and let $B$ be a totally definite quaternion algebra over $K$. In this article, given a set of representatives for ideal classes for a maximal order in $B$, we show how to construct in an efficient way a set of representatives of ideal classes for any Bass order in $B$. The algorithm does not require any knowledge of class numbers, and improves the equivalence checking process by using a simple calculation with global units. As an application, we compute ideal classes representatives for an order of level 30 in an algebra over the real quadratic field $\Q[\sqrt{5}]$.

math.NT↗