Quadratic Points Research Articles

Correction: Quadratic points on dynamical modular curves

Correction: Quadratic points on dynamical modular curves

A Positive Proportion of Monic Odd-Degree Hyperelliptic Curves of Genus g ≥ 4 Have no Unexpected Quadratic Points

Abstract Let $\mathcal{F}_{g}$ be the family of monic odd-degree hyperelliptic curves of genus $g$ over ${\mathbb{Q}}$. Poonen and Stoll have shown that for every $g \geq 3$, a positive proportion of curves in $\mathcal{F}_{g}$ have no rational points except the point at infinity. In this note, we prove the analogue for quadratic points: for each $g\geq 4$, a positive proportion of curves in $\mathcal{F}_{g}$ have no points defined over quadratic extensions except those that arise by pulling back rational points from $\mathbb{P}^{1}$.

Quadratic points on dynamical modular curves

Quadratic points on dynamical modular curves

Uniformity of quadratic points

In this paper, we extend a uniformity result of Dimitrov et al. [Uniformity in Mordell-Lang for curves, Ann. of Math. (2) 194(1) (2021) 237–298] to dimension two and use it to get a uniform bound on the cardinality of the set of all quadratic points for non-hyperelliptic non-bielliptic curves which only depend on the Mordell–Weil rank, the genus of the curve and the degree of the number field.

Computing quadratic points on modular curves 𝑋₀(𝑁)

In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves X 0 ( N ) X_0(N) of genus up to 8 8 , and genus up to 10 10 with N N prime, for which they were previously unknown. The values of N N we consider are contained in the set L = { 58 , 68 , 74 , 76 , 80 , 85 , 97 , 98 , 100 , 103 , 107 , 109 , 113 , 121 , 127 } . \begin{equation*} \mathcal {L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}. \end{equation*} We obtain that all the non-cuspidal quadratic points on X 0 ( N ) X_0(N) for N ∈ L N\in \mathcal {L} are complex multiplication (CM) points, except for one pair of Galois conjugate points on X 0 ( 103 ) X_0(103) defined over Q ( 2885 ) \mathbb {Q}(\sqrt {2885}) . We also compute the j j -invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.

Cyclic isogenies of elliptic curves over fixed quadratic fields

Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over Q \mathbb {Q} . Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields Q ( d ) \mathbb {Q}(\sqrt {d}) with | d | > 10 4 |d| > 10^4 we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over 19 19 quadratic fields, including Q ( 213 ) \mathbb {Q}(\sqrt {213}) and Q ( − 2289 ) \mathbb {Q}(\sqrt {-2289}) . To make this procedure work, we determine all of the finitely many quadratic points on the modular curves X 0 ( 125 ) X_0(125) and X 0 ( 169 ) X_0(169) , which may be of independent interest.

Quadratic points on intersections of two quadrics

Quadratic points on intersections of two quadrics

Quadratic points on bielliptic modular curves

Bruin and Najman [LMS J. Comput. Math. 18 (2015), pp. 578–602], Ozman and Siksek [Math. Comp. 88 (2019), pp. 2461–2484], and Box [Math. Comp. 90 (2021), pp. 321–343] described all the quadratic points on the modular curves of genus 2 ≤ g ( X 0 ( n ) ) ≤ 5 2\leq g(X_0(n)) \leq 5 . Since all the hyperelliptic curves X 0 ( n ) X_0(n) are of genus ≤ 5 \leq 5 and as a curve can have infinitely many quadratic points only if it is either of genus ≤ 1 \leq 1 , hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves X 0 ( n ) X_0(n) naturally arises; this question has recently also been posed by Mazur. We answer Mazur’s question completely and describe the quadratic points on all the bielliptic modular curves X 0 ( n ) X_0(n) for which this has not been done already. The values of n n that we deal with are n = 60 n=60 , 62 62 , 69 69 , 79 79 , 83 83 , 89 89 , 92 92 , 94 94 , 95 95 , 101 101 , 119 119 and 131 131 ; the curves X 0 ( n ) X_0(n) are of genus up to 11 11 . We find all the exceptional points on these curves and show that they all correspond to CM elliptic curves. The two main methods we use are Box’s relative symmetric Chabauty method and an application of a moduli description of Q \mathbb {Q} -curves of degree d d with an independent isogeny of degree m m , which reduces the problem to finding the rational points on several quotients of modular curves.

Bielliptic quotient modular curves of 𝑋₀(𝑁)

Let N ≥ 1 N\geq 1 be a non-square free integer and let W N W_N be a nontrivial subgroup of the group of the Atkin-Lehner involutions of X 0 ( N ) X_0(N) such that the modular curve X 0 ( N ) / W N X_0(N)/W_N has genus at least two. We determine all pairs ( N , W N ) (N,W_N) such that X 0 ( N ) / W N X_0(N)/W_N is a bielliptic curve and the pairs ( N , W N ) (N,W_N) such that X 0 ( N ) / W N X_0(N)/W_N has an infinite number of quadratic points over Q \mathbb {Q} .

Enlarging the frequency dynamic range of DMZI sensors based on the switching control feedback system.

The feedback loop in dual Mach-Zehnder interferometer (DMZI) sensors stabilizes the system operating at the quadratic point for the highest sensitivity but requires the minimum measurable vibration frequency out of the feedback bandwidth, resulting in a limited dynamic range. In this paper, we point out that the feedback operation is unnecessary while vibration is occurring and propose a strategy to adaptively enable/disable the feedback phase compensation depending on the vibration state, lowering the minimum measurable vibration frequency tenfold. Moreover, the state variable employed enables direct extraction of vibration-related data, with no need of complicated postprocessing algorithms.

Possible realization of optical Dirac points in woodpile photonic crystals.

The simulation of fermionic relativistic physics, e.g., Dirac and Weyl physics, has led to the discovery of many unprecedented phenomena in photonics, of which the optical-frequency realization is, however, still challenging. Here, surprisingly, we discover that the woodpile photonic crystals commonly used for optical frequency applications host exotic fermion-like relativistic degeneracies: a Dirac nodal line and a fourfold quadratic point, as protected by the nonsymmorphic crystalline symmetry. Deforming the woodpile photonic crystal leads to the emergence of type-II Dirac points from the fourfold quadratic point. Such type-II Dirac points can be detected by its anomalous refraction property which is manifested as a giant birefringence in a slab setup. Our findings provide a promising route towards 3D optical Dirac physics in all-dielectric photonic crystals.

Splitting of primes in number fields generated by points on some modular curves

We study the splitting of primes in number fields generated by points on modular curves. Momose (Nagoya Math J 96:139–165, 1984) was the first to notice that quadratic points on $$X_1(N)$$ generate quadratic fields over which certain primes split in a particular way and his results were later expanded upon by Krumm (Quadratic Points on Modular Curves. PhD thesis, University of Georgia, 2013). We prove results about the splitting behaviour of primes in quadratic fields generated by points on the modular curves $$X_0(N)$$ which are hyperelliptic (except for $$N=37$$ ) and in cubic fields generated by points on $$X_1(2,14)$$ .

Ideal Quadratic Nodal Point with Large Surface Arc States in Half-Heusler Material Scbipt

Ideal Quadratic Nodal Point with Large Surface Arc States in Half-Heusler Material Scbipt

Fermat’s Last Theorem and modular curves over real quadratic fields

In this paper we study the Fermat equation $x^n+y^n=z^n$ over quadratic fields $\mathbb{Q}(\sqrt{d})$ for squarefree $d$ with $26 \leq d \leq 97$. By studying quadratic points on the modular curves $X_0(N)$, $d$-regular primes, and working with Hecke operators on spaces of Hilbert newforms, we extend work of Freitas and Siksek to show that for most squarefree $d$ in this range there are no non-trivial solutions to this equation for $n \geq 4$.

Optimum Processing Conditions for Bakasang Using the Response Surface Methodology with Central Composite Design (CCD).

<b>Background and Objective:</b> Bakasang is a typical food of North Sulawesi and its surroundings, made from fermented fish viscera. This food is made by fermentation so that it is rich in amino acids and polypeptides. This study aimed to determine the optimum processing conditions for Bakasang with Peroxide Value (PV) parameters based on Response Surface Methodology (RSM) with Central Composite Design (CCD). <b>Materials and Methods:</b> Viscera from fresh Skipjack tuna consisting of intestines, liver, heart and eggs were mixed with salt and left in a fermenter for varying times and temperatures to obtain Bakasang. The fermentation temperature, salt content, and fermentation time, determined based on the Central Composite Design (CCD), were variables in this study. Thus, 20 experiments consisted of eight quadratic points, six centre points and six axial points. <b>Results:</b> The results showed that the correlation test between temperature, salinity and fermentation time variables on PV were 0.521, 0.305 and 0.591. The regression test resulted in an R<sup>2</sup> value of 0.988. The model equation obtained was y = 4.35194 - 0.11363x<sub>1 </sub>- 0.07459x<sub>2 </sub>- 0.25300x<sub>3 </sub>+ 0.00157x<sub>12 </sub>+ 0.00310x<sub>22</sub> + 0.01571x<sub>32</sub> - 0.00064x<sub>1</sub>x<sub>2 </sub>+ 0.00172x<sub>1</sub>x<sub>3</sub> + 0.00340x<sub>2</sub>x<sub>3</sub>. The contour plot graph obtained from the experiment using CCD showed that the optimum processing conditions for Bakasang were at a temperature of 32.9324°C, salinity 10.2631% and fermentation time of 4.7793 days, which will produce Bakasang with an optimum PV of 1.51256 meq kg<sup>1</sup> sample. <b>Conclusion:</b> According to the findings of this study, the optimum temperature, salinity and fermentation time for producing Bakasang with a PV of 1.51256 meq kg<sup>1</sup> sample were 32.932°C, 10.263% and 4.779 days, respectively.

Obvious Surface States Connecting to the Projected Triple Points in NaCl's Phonon Dispersion.

With the development of computer technology and theoretical chemistry, the speed and accuracy of first-principles calculations have significantly improved. Using first-principles calculations to predict new topological materials is a hot research topic in theoretical and computational chemistry. In this work, we focus on a well-known material, sodium chloride (NaCl), and propose that the triple point (TP), quadratic contact triple point (QCTP), linear and quadratic nodal lines can be found in the phonon dispersion of NaCl with Fm m type structure. More importantly, we propose that the clear surface states connected to the projected TP and QCTP are visible on the (001) surface. It is hoped that further experimental investigation and verification for these properties as mentioned above.

Quadratic points on non-split Cartan modular curves

In this paper, we study quadratic points on the non-split Cartan modular curves [Formula: see text], for [Formula: see text] and [Formula: see text]. Recently, Siksek proved that all quadratic points on [Formula: see text] arise as pullbacks of rational points on [Formula: see text]. Using similar techniques for [Formula: see text], and employing a version of Chabauty for symmetric powers of curves for [Formula: see text], we show that the same holds for [Formula: see text] and [Formula: see text]. As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular.

On conjectures of Samart

In this work, we verify all the conjectural formulas for the Mahler measure of the Laurent polynomial $$\begin{aligned} \left( X+\frac{1}{X}\right) ^{2}\left( Y+\frac{1}{Y}\right) ^{2}(1+Z)^{3}Z^{-2}-s \end{aligned}$$ parametrized by s posed by Samart using properties of spherical theta functions, and show that when s is induced by a CM point, these Mahler measures are all expressible in terms of special values of modular L-functions. In addition, we derive all new Samart-type formulas attached to a family of particular s as byproducts of this work. We remark that our method may also be used to verify all Samart’s remaining conjectural formulas associated to the Laurent polynomials $$\begin{aligned} \left( X+\frac{1}{X}\right) \left( Y+\frac{1}{Y}\right) \left( Z+\frac{1}{Z}\right) +s^{1/2}\quad \text{ and }\quad X^{4}+Y^{4}+Z^{4}+1+s^{1/4}XYZ, \end{aligned}$$ validating his hypothesis that $$n_{2}(s)$$ must be a linear combination of modular L-values at the s induced by the modularity of the associated K3 surface. At the end, we also affirm a conjecture of Samart on elliptic trilogarithms related to $$n_{2}(s)$$ by showing that the value of the elliptic trilogarithm associated to an elliptic curve E induced by an imaginary quadratic point at some 4-torsion point of E can be written as a linear combination of special values of Dirichlet L-series and modular L-functions.

Bielliptic smooth plane curves and quadratic points

Let [Formula: see text] be a smooth plane curve of degree [Formula: see text] defined over a global field [Formula: see text] of characteristic [Formula: see text] or [Formula: see text] (up to an extra condition on [Formula: see text]). Unless the curve is bielliptic of degree four, we observe that it always admits finitely many quadratic points. We further show that there are only finitely many quadratic extensions [Formula: see text] when [Formula: see text] is a number field, in which we may have more points of [Formula: see text] than these over [Formula: see text]. In particular, we have this asymptotic phenomenon valid for Fermat’s and Klein’s equations. Second, we conjecture that there are two infinite sets [Formula: see text] and [Formula: see text] of isomorphism classes of smooth projective plane quartic curves over [Formula: see text] with a prescribed automorphism group, such that all members of [Formula: see text] (respectively [Formula: see text]) are bielliptic and have finitely (respectively infinitely) many quadratic points over a number field [Formula: see text]. We verify the conjecture over [Formula: see text] for [Formula: see text] and [Formula: see text]. The analog of the conjecture over global fields with [Formula: see text] is also considered.

Diagonal restrictions of p-adic Eisenstein families

We compute the diagonal restriction of the first derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series attached to a general (odd) character of the narrow class group of a real quadratic field, and express the Fourier coefficients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of Darmon and Vonk (Duke Math J, to appear, 2020) at appropriate real quadratic points of Drinfeld’s p-adic upper half-plane. This can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier (J Reine Angew Math 355:191–220, 1985, §7) which arose in their “analytic proof” of the factorisation of differences of singular moduli, and whose inspiration can be traced to Siegel’s proof of the rationality of the values at negative integers of the Dedekind zeta function of a totally real field. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic fields based on RM values of rigid meromorphic cocycles, and leads to an expression for the p-adic logarithms of Gross–Stark units and Stark–Heegner points in terms of the first derivatives of certain twisted Rankin triple product p-adic L-functions.