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Seokhyun Choi

Publications and source records attributed to Seokhyun Choi.

9 recordsLinked to original sources
arXiv2026

Rank growth of abelian varieties over certain finite Galois extensions

Let $A/K$ be an abelian variety over a number field $K$. We prove that a finite automorphism group $G \subseteq \mathrm{Aut}_K(X)$ of a smooth projective variety $X/K$ such that $X/G \cong \mathbb{P}_K^d$ can force the rank growth of $A$ over infinitely many mutually linearly disjoint $G$-extensions $L_i/K$. The proof is based on Hilbert irreducibility and Néron specialization. We then combine the theorem with finite group representations to obtain explicit lower bounds for rank growth. As applications, we obtain rank growth results for Jacobian varieties and construct explicit examples. We further prove arbitrarily large rank growth of abelian varieties over symmetric extensions. Finally, we study the connection with infinite rank conjectures.

math.NT↗
arXiv2026

Additive Rigidity for Images of Rational Points on Abelian Varieties II: The General Case

We study the interaction between the group law on an abelian variety and the additive structure induced on its image under a morphism to a projective space. Let $A/F$ be an abelian variety, $f:A \rightarrow \mathbb{P}^n$ be a morphism which is finite onto its image, and $Γ\subseteq A(F)$ be a finite-rank subgroup. We show that for any affine chart $\mathbb{A}^n \subseteq \mathbb{P}^n$ and any finite subset $X \subseteq f(Γ) \cap \mathbb{A}^n$, the energy satisfies $E(X) \ll \lvert X \rvert^2$ and the sumset satisfies $\lvert X+X \rvert \gg \lvert X \rvert^2$. Thus images of finite-rank subgroups of abelian varieties cannot have strong additive structure in affine space. This removes the simplicity assumption from the author's previous result. The proof combines the uniform Mordell--Lang conjecture of Gao--Ge--Kühne with a refined use of the Ueno locus, Rémond's boundedness theorem for abelian subvarieties of bounded degree, and induction on the dimension of $A$.

math.NT↗
arXiv2026

Additive Rigidity for Images of Rational Points on Abelian Varieties I: The Simple Case

We study the interaction between the group law on an abelian variety and the additive structure induced on its image under a morphism to projective space. Let $A/F$ be a simple abelian variety, $f:A \rightarrow \mathbb{P}^n$ be a morphism which is finite onto its image, and $Γ\subseteq A(F)$ be a finite-rank subgroup. We show that for any affine chart $\mathbb{A}^n \subseteq \mathbb{P}^n$ and any finite subset $X \subseteq f(Γ) \cap \mathbb{A}^n$, the energy satisfies $E(X) \ll \lvert X \rvert^2$ and the sumset satisfies $\lvert X+X \rvert \gg \lvert X \rvert^2$. We also prove a product version of the main theorem, where the morphism is compatible with the decomposition of the abelian variety into simple factors. The proof uses the uniform Mordell-Lang conjecture proven by Gao--Ge--Kühne.

math.NT↗
arXiv2026

Additive Rigidity for $x$-Coordinates of Rational Points on Elliptic Curves

We study the interaction between the group law on an elliptic curve and the additive structure of $x$-coordinates of rational points on an elliptic curve. Let $E/\mathbb{Q}$ be an elliptic curve of Mordell-Weil rank $r \geq 1$, $d \geq 1$ be an integer, and $0<ρ\leq 1$. We show that if a $d$-dimensional proper generalized arithmetic progression in $\mathbb{Q}$ contains the $x$-coordinates of rational points on $E/\bbq$ with positive proportion $ρ$, then the number of such points is bounded by $A(E,d,ρ)^r$. The proof combines extraction lemmas, gap principles, and the bounds for spherical codes. As an application, we obtain restrictions on sets of rational points whose $x$-coordinates have small sumsets or large additive energy.

math.NT↗
arXiv2026

Analytic rank-one elliptic curves over function fields and their rank over certain ring class fields

Let $E/k$ be a non-isotrivial elliptic curve over a global function field $k$ of characteristic $p>3$, and $G\subset \mathrm{Gal}(k^{\mathrm{sep}}/k)$ be a topologically finitely generated subgroup. We prove that if $E/k$ has analytic rank $1$, then its rank over the fixed subfield $L^G$ is infinite, where $L$ is the infinite ring class extension of some finite separable extension $K/k$. If $E/k$ has analytic rank $0$, then we prove that the same holds provided there exists an imaginary quadratic extension $K/k$ such that $E/K$ has analytic rank $1$ and satisfies the Heegner hypothesis.

math.NT↗
arXiv2026

Number of integral points on quadratic twists of elliptic curves

We study integral points on the quadratic twists $E_D : y^2 = x^3+D^2Ax+D^3B$ of a fixed elliptic curve $E : y^2 = x^3+Ax+B$ over $\overline{Q}$. For sufficiently large squarefree positive integers $D$, we prove that the number of integral points on $E_D$ admits the upper bound $\ll 4^r$, where $r$ denotes the Mordell-Weil rank of $E_D$. The implied constant is absolute and effectively computable. The proof combines gap principles, bounds for spherical codes, and Diophantine approximation. As an application, we prove that the average number of integral points on the quadratic twist family is bounded.

math.NT↗
arXiv2025

Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$

We prove Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$. Specifically, let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$. If $E/\mathbb{Q}$ has analytic rank at most $1$, then we prove that for any topologically finitely generated subgroup $G$ of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the rank of $E$ over the fixed subfield $\overline{\mathbb{Q}}^G$ of $\overline{\mathbb{Q}}$ under $G$ is infinite.

math.NT↗
arXiv2020

S-SGD: Symmetrical Stochastic Gradient Descent with Weight Noise Injection for Reaching Flat Minima

The stochastic gradient descent (SGD) method is most widely used for deep neural network (DNN) training. However, the method does not always converge to a flat minimum of the loss surface that can demonstrate high generalization capability. Weight noise injection has been extensively studied for finding flat minima using the SGD method. We devise a new weight-noise injection-based SGD method that adds symmetrical noises to the DNN weights. The training with symmetrical noise evaluates the loss surface at two adjacent points, by which convergence to sharp minima can be avoided. Fixed-magnitude symmetric noises are added to minimize training instability. The proposed method is compared with the conventional SGD method and previous weight-noise injection algorithms using convolutional neural networks for image classification. Particularly, performance improvements in large batch training are demonstrated. This method shows superior performance compared with conventional SGD and weight-noise injection methods regardless of the batch-size and learning rate scheduling algorithms.

cs.LG↗