arXiv subjects

Adrian Barquero-Sanchez

Publications and source records attributed to Adrian Barquero-Sanchez.

9 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.

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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.

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arXiv2026

Counting elliptic curves over $\mathbb{Q}$ with bounded naive height

In this paper, we give exact and asymptotic formulas for counting elliptic curves $ E_{A,B} \colon y^2 = x^3 + Ax + B $ with $ A, B \in \mathbb{Z} $, ordered by naive height. We study the family of all such curves and also several natural subfamilies, including those with fixed $ j $-invariant and those with complex multiplication (CM). In particular, we provide formulas for two commonly used normalizations of the naive height appearing in the literature: the calibrated naive height, defined by \[ H^{\mathrm{cal}}(E_{A,B}) := \max\{ 4|A|^3, 27B^2 \}, \] and the uncalibrated naive height, defined by \[ H^{\mathrm{ncal}}(E_{A,B}) := \max\{ |A|^3, B^2 \}. \] In fact, we prove our theorems with respect to the more general naive height $H_{α, β}(E_{A,B}) := \max\{ α|A|^3, βB^2 \}$, defined for arbitrary positive real numbers $α, β\in \mathbb{R}_{> 0}$. As part of our approach, we give a completely explicit parametrization of the set of curves $ E_{A,B} $ with fixed $ j $-invariant and bounded naive height, describing them as twists of the curve $ E_{A_j, B_j} $ of minimal naive height for the given $ j $-invariant. We also include tables comparing and verifying our theoretical predictions with exact counts obtained via exhaustive computer searches, and we compute data for CM elliptic curves of naive height up to $ 10^{30} $. Code in SageMath is provided to compute all exact and asymptotic formulas appearing in the paper.

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arXiv2024

The density and distribution of CM elliptic curves over $\mathbb{Q}$

In this paper we study the density and distribution of CM elliptic curves over $\mathbb{Q}$. In particular, we prove that the natural density of CM elliptic curves over $\mathbb{Q}$, when ordered by naive height, is zero. Furthermore, we analyze the distribution of these curves among the thirteen possible CM orders of class number one. Our results show that asymptotically, $100\%$ of them have complex multiplication by the order $\mathbb{Z}\left[\frac{-1 + \sqrt{-3}}{2} \right]$, that is, have $j$-invariant 0. We conduct this analysis within two different families of representatives for the $\mathbb{Q}$-isomorphism classes of CM elliptic curves: one commonly used in the literature and another constructed using the theory of twists. As part of our proofs, we give asymptotic formulas for the number of elliptic curves with a given $j$-invariant and bounded naive height.

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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.

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arXiv2023

On the embedding of Galois groups into wreath products

In this paper we make explicit an application of the wreath product construction to the Galois groups of field extensions. More precisely, given a tower of fields $F \subseteq K \subseteq L$ with $L/F$ finite and separable, we explicitly construct an embedding of the Galois group $\operatorname{Gal}(L^c/F)$ into the regular wreath product $\operatorname{Gal}(L^c/K^c) \wr_r \operatorname{Gal}(K^c/F)$. Here $L^c$ (resp. $K^c$) denotes the Galois closure of $L/F$ (resp. $K/F$). Similarly, we also construct an explicit embedding of the Galois group $\operatorname{Gal}(L^c/F)$ into the smaller sized wreath product $\operatorname{Gal}(L^c/K) \wr_Ω \operatorname{Gal}(K^c/F)$, where $Ω= \operatorname{Hom}_F(K, K^c)$ is acted on by composition of automorphisms in $\operatorname{Gal}(K^c/F)$. Moreover, when $L/K$ is a Kummer extension we prove a sharper embedding, that is, that $\operatorname{Gal}(L^c/F)$ embeds into the wreath product $\operatorname{Gal}(L/K) \wr_Ω \operatorname{Gal}(K^c/F)$. As corollaries we obtain embedding theorems when $L/K$ is cyclic and when it is quadratic with $\operatorname{char}(F) \neq 2$. We also provide examples of these embeddings and as an illustration of the usefulness of these embedding theorems, we survey some recent applications of these types of results in field theory, arithmetic statistics, number theory and arithmetic geometry.

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arXiv2019

Theta series and number fields: theorems and experiments

We construct certain $θ$-series associated to number fields and prove that for number fields of degree less than equal to 4, these $θ$-series are number field invariants. We also investigate whether or not the collection of $θ$-series associated to number fields of the same degree and discriminant are linearly independent. This is known to be true if the degree of the number field is less than or equal to 3. We do not prove in this paper that they are linearly independent in general but we do give computational and heuristic evidence that we would expect them to be.

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arXiv2019

The distribution of $G$-Weyl CM fields and the Colmez conjecture

Let $G$ be a transitive subgroup of $S_d$ and $E$ be a CM field of degree $2d$ with a maximal totally real $G$-field. If the Galois group of the Galois closure of $E$ is isomorphic to the wreath product of $C_2$ and $G$, then we say that $E$ is a $G$-Weyl CM field. Let $N_{2d}^{\textrm{Weyl}}(X,G)$ count the $G$-Weyl CM fields $E$ of degree $2d$ with discriminant $|d_E| \leq X$ and define \begin{align*} N_{2d}^{\textrm{Weyl}}(X):=\sum_{G \leq S_d}N_{2d}^{\textrm{Weyl}}(X,G). \end{align*} Further, let $N_{2d}^{\textrm{cm}}(X)$ count the CM fields $E$ of degree $2d$ with discriminant $|d_E| \leq X$. Assuming a weak form of the upper bound in Malle's conjecture which is known to be true in many cases, we build upon an approach of Klüners to prove that \begin{align*} \frac{N_{2d}^{\textrm{Weyl}}(X,G)}{N_{2d}^{\textrm{cm}}(X)} = C(d, G) + O(X^{-α(d,G)}) \end{align*} and \begin{align} \frac{N_{2d}^{\textrm{Weyl}}(X)}{N_{2d}^{\textrm{cm}}(X)} = 1 + O(X^{-β(d)}) \qquad \qquad (0.1) \end{align} for some explicit positive constants $C(d,G), α(d,G)$, and $β(d)$. We then apply these distribution results to study the Colmez conjecture. Using the recently proved averaged Colmez conjecture, we deduce that the Colmez conjecture is true for $G$-Weyl CM fields. Combined with (0.1), we conclude that the Colmez conjecture is true for an asymptotic density of 100% of CM fields of degree $2d$; in other words, the Colmez conjecture is true for a random CM field.

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arXiv2016

On the Colmez conjecture for non-abelian CM fields

The Colmez conjecture relates the Faltings height of an abelian variety with complex multiplication by the ring of integers of a CM field $E$ to logarithmic derivatives of certain Artin $L$--functions at $s=0$. In this paper, we prove that if $F$ is any fixed totally real number field of degree $[F:\mathbb{Q}] \geq 3$, then there are infinitely many CM extensions $E/F$ such that $E/\mathbb{Q}$ is $\textit{non-abelian}$ and the Colmez conjecture is true for $E$. Moreover, these CM extensions are explicitly constructed to be ramified at "arbitrary" prescribed sets of prime ideals of $F$. We also prove that the Colmez conjecture is true for a generic class of non-abelian CM fields called Weyl CM fields, and use this to develop an arithmetic statistics approach to the Colmez conjecture based on counting CM fields of fixed degree and bounded discriminant. We illustrate these results by evaluating the Faltings height of the Jacobian of a genus 2 hyperelliptic curve with complex multiplication by a non-abelian quartic CM field in terms of the Barnes double Gamma function at algebraic arguments. This can be seen as an explicit non-abelian Chowla-Selberg formula. A crucial input to the proofs is an averaged version of the Colmez conjecture which was recently proved independently by Andreatta-Goren-Howard-Madapusi Pera and Yuan-Zhang.

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