arXiv subjects

Francesco Naccarato

Publications and source records attributed to Francesco Naccarato.

6 recordsLinked to original sources

arXiv2026

The arithmetic of critical values II: critical elliptic curves

In this second chapter of the $\textit{Arithmetic of critical values}$ series (ACV), we study certain double covers $E_f\to\mathbb{P}^1$ whose branch locus coincides with that of a quartic polynomial $f$. We give a direct proof of the fact, already shown non-constructively in ACV I, that the elliptic curves $E_f$ admit a $3$-isogeny. Our methods are Galois-theoretic, and lead us to a thorough analysis of the Galois closure of $f:\mathbb{P}^1\to\mathbb{P}^1$. We exploit its rich geometry to prove a Selmer companionship theorem for the family $E_f$, allowing us to exhibit elements in certain Tate-Shafarevich groups which are visible in an abelian surface. We also give some dynamical and Diophantine applications of our constructions, as well as new examples of Jacobians isogenous to a power of an elliptic curve.

math.NT↗

arXiv2026

The arithmetic of critical values I: equicritical quartic polynomials

A polynomial $f$ of degree $d$ and coefficients in an algebraically closed field $k$ defines a morphism $f:\mathbb{P}^1_k\longrightarrow\mathbb{P}^1_k$ which, if char$(k)\nmid d$, is unramified outside a finite set of points in the image: the critical values of $f$. In this work we establish a rigorous framework for the study of their arithmetic, which we carry out for $d=4$ and $k=\overline{\mathbb{Q}}$, uncovering a connection to the arithmetic of elliptic curves. Recent progress in the theory of Weyl sums has sparked some interest in finding pairs of polynomials having the same critical values for "nontrivial" reasons: building on our analysis, we provide a complete classification of such pairs in the case of quartics over number fields.

math.NT↗

arXiv2026

Finding the complement of an elliptic curve inside a Jacobian

This note gives a simple algorithm for the following effectivity problem: given a genus $2$ curve $X$ together with a nonconstant map $π:X\to E$ to an elliptic curve, determine an elliptic curve $E'$ and a map $π':X\to E'$ independent of $π$. Equivalently, we compute the complementary elliptic factor in the decomposition of $\operatorname{Jac}(X)$ up to isogeny. While the problem has been studied extensively, and more general ones have been solved by deep and powerful techniques, we are not aware of a reference for the simple explicit procedure described here.

math.NT↗

arXiv2022

SnP$_2$S$_6$: A Promising Infrared Nonlinear Optical Crystal with Strong Non-Resonant Second Harmonic Generation and Phase-matchability

High-power infrared laser systems with broadband tunability are of great importance due to their wide range of applications in spectroscopy and free-space communications. These systems require nonlinear optical (NLO) crystals for wavelength up/down conversion using sum/difference frequency generation, respectively. NLO crystals need to satisfy many competing criteria, including large nonlinear optical susceptibility, large laser induced damage threshold (LIDT), wide transparency range and phase-matchability. Here, we report bulk single crystals of SnP_2S_6 with a large non-resonant SHG coefficient of d33= 53 pm/V at 1550nm and a large LIDT of 350 GW/cm^2 for femtosecond laser pulses. It also exhibits a broad transparency range from 0.54 μm to 8.5μm (bandgap of ~2.3 eV) and can be both Type I and Type II phase-matched. The complete linear and SHG tensors are measured as well as predicted by first principles calculations, and they are in excellent agreement. A proximate double-resonance condition in the electronic band structure for both the fundamental and the SHG light is shown to enhance the non-resonant SHG response. Therefore, SnP2S6 is an outstanding candidate for infrared laser applications.

physics.optics↗

arXiv2021

Counting rational points on elliptic curves with a rational 2-torsion point

Let $E/\mathbb{Q}$ be an elliptic curve over the rational numbers. It is known, by the work of Bombieri and Zannier, that if $E$ has full rational $2$-torsion, the number $N_E(B)$ of rational points with Weil height bounded by $B$ is $\exp\left(O\left(\frac{\log B}{\sqrt{\log\log B}}\right)\right)$. In this paper we exploit the method of descent via $2$-isogeny to extend this result to elliptic curves with just one nontrivial rational $2$-torsion point. Moreover, we make use of a result of Petsche to derive the stronger upper bound $N_{E}(B) = \exp\left(O\left(\frac{\log B}{\log\log B}\right)\right)$ for these curves and to remove a deep transcendence theory ingredient from the proof.

math.NT↗

arXiv2019

Searching for Materials with High Refractive Index and Wide Band Gap: A First-Principles High-Throughput Study

Materials combining both a high refractive index and a wide band gap are of great interest for optoelectronic and sensor applications. However, these two properties are typically described by an inverse correlation with high refractive index appearing in small gap materials and vice-versa. Here, we conduct a first-principles high-throughput study on more than 4000 semiconductors (with a special focus on oxides). Our data confirm the general inverse trend between refractive index and band gap but interesting outliers are also identified. The data are then analyzed through a simple model involving two main descriptors: the average optical gap and the effective frequency. The former can be determined directly from the electronic structure of the compounds, but the latter cannot. This calls for further analysis in order to obtain a predictive model. Nonetheless, it turns out that the negative effect of a large band gap on the refractive index can counterbalanced in two ways: (i) by limiting the difference between the direct band gap and the average optical gap which can be realized by a narrow distribution in energy of the optical transitions and (ii) by increasing the effective frequency which can be achieved through either a high number of transitions from the top of the valence band to the bottom of the conduction or a high average probability for these transitions. Focusing on oxides, we use our data to investigate how the chemistry influences this inverse relationship and rationalize why certain classes of materials would perform better. Our findings can be used to search for new compounds in many optical applications both in the linear and non-linear regime (waveguides, optical modulators, laser, frequency converter, etc.).

cond-mat.mtrl-sci↗