arXiv subjects

Stephanie Chan

Publications and source records attributed to Stephanie Chan.

28 records · Page 2Linked to original sources

arXiv2024

Almost all quadratic twists of an elliptic curve have no integral points

For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E_D have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall-Lang conjecture in the case that E has partial 2-torsion. The proof uses a correspondence of Mordell and the reduction theory of binary quartic forms in order to transfer the problem to counting rational points of bounded height on a certain singular cubic surface, together with extensive use of cancellation in character sum estimates, drawn from Heath-Brown's analysis of Selmer group statistics for the congruent number curve.

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arXiv2023

The $3$-isogeny Selmer groups of the elliptic curves $y^2=x^3+n^2$

Consider the family of elliptic curves $E_n:y^2=x^3+n^2$, where $n$ varies over positive cubefree integers. There is a rational $3$-isogeny $ϕ$ from $E_n$ to $\hat{E}_n:y^2=x^3-27n^2$ and a dual isogeny $\hatϕ:\hat{E}_n\rightarrow E_n$. We show that for almost all $n$, the rank of $\mathrm{Sel}_ϕ(E_n)$ is $0$, and the rank of $\mathrm{Sel}_{\hatϕ}(\hat{E}_n)$ is determined by the number of prime factors of $n$ that are congruent to $2\bmod 3$ and the congruence class of $n\bmod 9$.

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arXiv2023

The average number of integral points on the congruent number curves

We show that the total number of non-torsion integral points on the elliptic curves $\mathcal{E}_D:y^2=x^3-D^2x$, where $D$ ranges over positive squarefree integers less than $N$, is $O( N(\log N)^{-1/4+ε})$. The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the $2$-Selmer group of the curves in this family.

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arXiv2023

Integral points on cubic twists of Mordell curves

Fix a non-square integer $k\neq 0$. We show that the number of curves $E_B:y^2=x^3+kB^2$ containing an integral point, where $B$ ranges over positive integers less than $N$, is bounded by $O_k(N(\log N)^{-\frac{1}{2}+ε})$. In particular, this implies that the number of positive integers $B\leq N$ such that $-3kB^2$ is the discriminant of an elliptic curve over $\mathbb{Q}$ is $o(N)$. The proof involves a discriminant-lowering procedure on integral binary cubic forms.

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arXiv2022

Integral points on the congruent number curve

We study integral points on the quadratic twists $\mathcal{E}_D:y^2=x^3-D^2x$ of the congruent number curve. We give upper bounds on the number of integral points in each coset of $2\mathcal{E}_D(\mathbb{Q})$ in $\mathcal{E}_D(\mathbb{Q})$ and show that their total is $\ll (3.8)^{\mathrm{rank} \mathcal{E}_D(\mathbb{Q})}$. We further show that the average number of non-torsion integral points in this family is bounded above by $2$. As an application we also deduce from our upper bounds that the system of simultaneous Pell equations $aX^2-bY^2=d$, $bY^2-cZ^2=d$ for pairwise coprime positive integers $a,b,c,d$, has at most $\ll (3.6)^{ω(abcd)}$ integer solutions.

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arXiv2020

A Density of Ramified Primes

Let $K$ be a cyclic totally real number field of odd degree over $\mathbb{Q}$ with odd class number, such that every totally positive unit is the square of a unit, and such that $2$ is inert in $K/\mathbb{Q}$. We define a family of number fields $\{K(p)\}_p$, depending on $K$ and indexed by the rational primes $p$ that split completely in $K/\mathbb{Q}$, such that $p$ is always ramified in $K(p)$ of degree $2$. Conditional on a standard conjecture on short character sums, the density of such rational primes $p$ that exhibit one of two possible ramified factorizations in $K(p)/\mathbb{Q}$ is strictly between $0$ and $1$ and is given explicitly as a formula in terms of $[K:\mathbb{Q}]$. Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.

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arXiv2019

On the negative Pell equation

Using a recent breakthrough of Smith, we improve the results of Fouvry and Klüners on the solubility of the negative Pell equation. Let $\mathcal{D}$ denote the set of fundamental discriminants having no prime factors congruent to $3$ modulo $4$. Stevenhagen conjectured that the density of $D$ in $\mathcal{D}$ such that the negative Pell equation $x^2-Dy^2=-1$ is solvable with $x,y\in\mathbb{Z}$ is $58.1\%$, to the nearest tenth of a percent. By studying the distribution of the $8$-rank of narrow class groups $\mathrm{CL}^+(D)$ of $\mathbb{Q}(\sqrt{D})$, we prove that the infimum of this density is at least $53.8\%$.

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arXiv2019

Kuroda's formula and arithmetic statistics

Kuroda's formula relates the class number of a multi-quadratic number field $K$ to the class numbers of its quadratic subfields $k_i$. A key component in this formula is the unit group index $Q(K) = [\mathcal{O}_{K}^{\times}: \prod_i\mathcal{O}_{k_i}^{\times}]$. We study how $Q(K)$ behaves on average in certain natural families of totally real biquadratic fields $K$ parametrized by prime numbers.

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arXiv2018

Ranks, $2$-Selmer groups, and Tamagawa numbers of elliptic curves with $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$-torsion

In 2016, Balakrishnan-Ho-Kaplan-Spicer-Stein-Weigandt produced a database of elliptic curves over $\mathbb{Q}$ ordered by height in which they computed the rank, the size of the $2$-Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over $\mathbb{Q}$ whose rational torsion subgroup is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$. Conditional on GRH and BSD, we compute the rank of $92\%$ of the $202461$ curves with parameter height less than $10^3$. We also compute the size of the $2$-Selmer group and the Tamagawa product, and prove that their averages tend to infinity for this family.

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arXiv2017

Rational Right Triangles of a Given Area

Starting from any given rational-sided, right triangle, for example the $(3,4,5)$-triangle with area $6$, we use Euclidean geometry to show that there are infinitely many other rational-sided, right triangles of the same area. We show further that the set of all such triangles of a given area is finitely generated under our geometric construction. Such areas are known as "congruent numbers" and have a rich history in which all the results in this article have been proved and far more. Yet, as far as we can tell, this seems to be the first exploration using this kind of geometric technique.

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