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Joan-C. Lario

Publications and source records attributed to Joan-C. Lario.

10 recordsLinked to original sources
arXiv2026

On Galois Embedding Problems Arising from 3-Torsion of Elliptic Curves

We study Galois embedding problems arising from the 3-torsion of elliptic curves defined over $\mathbb{Q}$, extending the correspondence to all possible images of mod 3 Galois representations; namely, $\operatorname{GL}_2(\mathbb{F}_3),SD_{16},D_6,D_4$ and $C_2^2$. In the cyclotomic case, we show that solvability of these embedding problems is equivalent to the existence of infinitely many elliptic curves whose 3-division fields provide the corresponding solutions.

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arXiv2026

Modular forms of CM type mod $\ell$

We say that a normalized modular form is of CM type modulo $\ell$ by an imaginary quadratic field $K$ if its Fourier coefficients $a_p$ are congruent to $0$ modulo a prime $\mathcal L\mid \ell$ for every prime $p$ that is inert in $K$. In this paper, we address the following question. Let $f$ be a weight~$2$ cuspidal Hecke eigenform without complex multiplication which is of CM type modulo $\ell$ by an imaginary quadratic field $K$. Does there exist a congruence modulo $\ell$ between $f$ and a genuine CM modular form of weight~$2$? We conjecture that such a congruence always exists. We prove this conjecture for $\ell>2$ and $\ell\neq 3$ when $K=\mathbb{Q}(\sqrt{-3})$. In this setting, we discuss three situations: (i) modular forms attached to abelian surfaces with quaternionic multiplication, (ii) $\mathbb{Q}$-curves completely defined over an imaginary quadratic field, and (iii) elliptic curves over $\mathbb{Q}$ whose $5$-torsion Galois representation has image the maximal cyclic of order $16$ inside $\operatorname{GL}_2({\mathbb F}_5)$. In all these cases, the modular forms under consideration are of CM type modulo suitable primes~$\ell$, and we show that the associated residual Galois representations are monomial with respect to an imaginary quadratic field $K$ (in some instances, more than one such field). Finally, we present numerical evidence that motivated the conjecture and provides further support for its validity beyond the cases treated in this paper.

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arXiv2025

Faltings elliptic curves in twisted $\mathbb Q$-isogeny classes

Let $G$ be the graph attached to the $\mathbb Q$-isogeny class of an elliptic curve defined over $\mathbb Q$: that is, a vertex for every elliptic curve defined over $\mathbb Q$ in the isogeny class, and edges in correspondence with the prime degree rational isogenies between them. Stevens shows that there is a unique elliptic curve in $G$ with minimal Faltings height. We call this curve the Faltings elliptic curve in $G$. For every square-free integer $d$, we consider the graph~$G^d$ attached to the twisted elliptic curves in $G$ by the quadratic character of $\mathbb Q(\sqrt{d})$. It turns out that $G$ and $G^d$ are canonically isomorphic as abstract graphs (the isomorphism identifies the vertices with equal $j$-invariant). In this paper we determine which vertex is the Faltings elliptic curve in $G^d$. We also obtain the probability of a vertex in $G$ to be the Faltings elliptic curve in~$G^d$. It turns out that this probability depends on the $p$-adic valuations of rational values of certain modular functions.

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arXiv2024

Ruler and compass constructions in the Lemniscate and the 17-gon

We present several ruler and compass practical geometric constructions that can be performed in the lemniscate curve. To be precise, we provide recipes for halving, doubling, adding, subtracting, and transferring lemniscate arcs with ruler and compass. This note complements the instructions for the lemnatomic equilateral triangle and pentagon discussed in \cite{GoLa}, giving the details for the construction of the lemnatomic regular $17$-gon.

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arXiv2022

Periodicity of power Fibonacci sequences modulus a Fibonacci number

Let ${\mathcal F}=(F_i:i\ge 0)$ be the sequence of Fibonacci numbers, and $j$ and $e$ be non negative integers. We study the periodicity of the power Fibonacci sequences ${\mathcal F}^e(F_j)=(F_i^e\pmod{F_j}: i\ge 0)$. It is shown that for every $j,e\ge 1$ the sequence ${\mathcal F}^e(F_j)$ is periodic and its periodicity is computed. The result was previously known for ${\mathcal F}(F_j)$; that is, for $e=1$. For $e\in \{1, 2\}$, the values of the normalized residues $ρ_i\equiv F_i^e\pmod{F_j}$ with $0\le ρ_i<F_j-1$ are obtained.

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arXiv2020

An inverse Jacobian algorithm for Picard curves

We study the inverse Jacobian problem for the case of Picard curves over $\mathbb{C}$. More precisely, we elaborate on an algorithm that, given a small period matrix $Ω\in \mathbb{C}^{3\times 3}$ corresponding to a principally polarized abelian threefold equipped with an automorphism of order $3$, returns a Legendre-Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike-Weng in [Math. Comp., 74(249):499-518, 2005] which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain (numerically) all the isomorphism classes of Picard curves with maximal complex multiplication attached to the sextic CM-fields with class number at most $4$. In particular, we obtain (conjecturally) the complete list of CM Picard curves defined over $\mathbb{Q}$. In the appendix, Vincent gives a correction to the generalization of Takase's formula for the inverse Jacobian problem for hyperelliptic curves given in [Balakrishnan-Ionica-Lauter-Vincent, LMS J. Comput. Math., 19(suppl. A):283-300, 2016].

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arXiv2012

The twisting representation of the $L$-function of a curve

Let C be a smooth projective curve defined over a number field and let C' be a twist of C. In this article we relate the l-adic representations attached to the l-adic Tate modules of the Jacobians of C and C' through an Artin representation. This representation induces global relations between the local factors of the respective Hasse-Weil L-functions. We make these relations explicit in a particularly illustrating situation. For every Qbar-isomorphism class of genus 2 curves defined over Q with automorphism group isomorphic to D_8 or D_{12}, except for a finite number, we choose a representative curve C/Q such that, for every twist C' of C satisfying some mild condition, we are able to determine either the local factor L_p(C'/Q,T) or the product L_p(C'/Q,T)L_p(C'/Q,-T) from the local factor L_p(C/Q,T).

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arXiv2005

Plane quartic twists of X(5,3)

Given an odd representation of the absolute Galois group of Q onto PGL(2,3) and a positive integer N, there exists a twisted modular curve defined over Q whose rational points classify the quadratic Q-curves of degree N realizing the representation. The paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case N=5.

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