arXiv subjects

Jun-Yong Park

Publications and source records attributed to Jun-Yong Park.

15 recordsLinked to original sources

arXiv2026

Exact counts of elliptic curves of bounded height over $\mathbb F_q(t)$ in characteristics $2$ and $3$

Let $p\in\{2,3\}$, let $q=p^r$ with $r\geq1$, and put $K=\mathbb F_q(t)$. We determine the exact weighted and unweighted counts of $K$-isomorphism classes of elliptic curves of bounded Faltings height, equivalently of bounded minimal-discriminant degree. Writing $B$ for the discriminant height bound, the leading term in each count is of order $B^{5/6}$ and has the same coefficient in every characteristic, whereas the lower-order terms depend substantially on $p$ and on the arithmetic of the constant field. In the unweighted count, characteristic two produces a term of order $B^{5/12}$ and, when $r$ is even, a term of order $B^{1/4}$, neither of which occurs for $p>3$. Some characteristic-specific lower-order coefficients are negative. These terms reflect two small-characteristic phenomena. First, the nonsmooth locus of generalized Weierstrass equations contains quasi-elliptic-type strata that are not detected by the rational-singular-section argument in de Jong's count: the unique geometric singular point of the generic fiber may be defined only after a nontrivial purely inseparable extension of $K$. Second, on the $j=0$ locus, the geometric origin-preserving automorphism groups are nonabelian and contain wild elements, while twisting affects which automorphisms descend to $K$. Consequently, the passage from weighted to unweighted counts requires a marked-inertia calculation involving conjugacy classes and centralizer weights. Our formulas show that nonsmooth-locus corrections, extra-automorphism loci, and the removal of minimality defects collectively account for all lower-order terms. Together with the characteristic-greater-than-three formulas of Bejleri-Park-Satriano, this completes the exact weighted and unweighted bounded-height enumerations over $\mathbb F_q(t)$ in every characteristic.

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arXiv2026

Automation Without Understanding

Two developments are unfolding at once: artificial intelligence systems have begun to produce genuine research-level mathematics, and the United States is weakening the pipeline that produces humans capable of understanding what such systems are doing. This essay argues that, taken together, these developments amount to a strategic error. Mathematical capacity, which is the trained ability to verify, interpret, and challenge mathematical reasoning, is not a byproduct of theorem production but a form of infrastructure, built over generations by institutions that cannot be reconstituted on demand. Drawing on the May 2026 AI disproof of a longstanding Erdős conjecture on the planar unit distance problem and on recent disruptions to federal support for the mathematical sciences, the essay makes the case for treating mathematical capacity as a strategic asset on a par with semiconductor capability. It further proposes, among other measures, that AI systems performing consequential reasoning be required to expose their decision-critical claims in formal, machine-checkable form, converting part of AI reasoning from opaque persuasion into auditable structure.

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arXiv2026

Height moduli of elliptic surfaces: Motivic height zeta rationality and Kudla-Millson modularity of Mordell-Weil rank jumps

Let $k$ be a perfect field with $\mathrm{char}(k)\neq 2,3$, set $K=k(t)$, and let $\mathcal{W}_n^{\min}$ be the moduli stack of minimal elliptic curves over $K$ of Faltings height $n$, constructed via the height-moduli framework of Bejleri-Park-Satriano applied to $\overline{\mathcal{M}}_{1,1}\simeq\mathcal{P}(4,6)$. The Shioda-Tate formula $ρ(S)=T(S)+\mathrm{rk}(E/K)$ decomposes the Picard rank of the associated elliptic surface into the trivial lattice rank, which is local (determined by Kodaira fiber types), and the Mordell-Weil rank, which is global. The motivic height zeta function weighted by the trivial lattice rank is rational in $s=t^{1/12}$ in the dimensionally completed Grothendieck ring, via a combination of exact Euler products on the isotrivial loci $j\equiv 0, 1728$ and a motivic discriminant stabilization adapting Vakil-Wood to $Δ=4a_4^3+27a_6^2$; over $k=\mathbb{C}$, this yields bidegree-wise Hodge number stabilization. The Kudla-Millson theta correspondence shows that the distribution of new Mordell-Weil sections by canonical height is governed by a modular form of weight $6n-2$ for $\mathrm{SL}_2(\mathbb{Z})$. Combining Shepherd-Barron's diagonalization of the Gauss-Manin connection with Kodaira-Spencer transversality, we establish unconditionally that at every Faltings height $n\ge 3$ and for every $1 \le r \le \lfloor(10n-2)/(n-1)\rfloor$, there exist infinitely many stable elliptic surfaces with Mordell-Weil rank $\mathrm{rk}(E/K) \ge r$, and that infinitely many canonical heights $\hat{h}(P)=d$ are realized by Mordell-Weil sections.

math.AG↗

arXiv2026

Quantitative rank distribution conjecture over $\mathbb{F}_q(t)$

We combine the exact counting of all elliptic curves over $K = \mathbb{F}_q(t)$ with $\mathrm{char}(K) > 3$ by Bejleri, Satriano and the author, together with the torsion-free nature of most elliptic curves over global function fields proven by Phillips, and the overarching conjecture of Goldfeld and Katz-Sarnak regarding the ``Distribution of Ranks of Elliptic Curves''. Consequently, we arrive at the quantitative statement which naturally renders even finer conjecture regarding the lower order main terms differing for the number of $E/K$ with $|E(K)| = 1$ and $E(K) = \mathbb{Z}$.

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arXiv2025

100% of elliptic curves with a marked point have positive rank

As a consequence of their work on average Selmer ranks of elliptic curves with marked points, Bhargava and Ho proved that $100\%$ of elliptic curves over $\mathbb{Q}$ with an additional marked point have positive rank. In this note we provide an alternate proof which extends the result to global fields of characteristic not two or three.

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arXiv2024

Height moduli on cyclotomic stacks and counting elliptic curves over function fields

For proper stacks, unlike schemes, there is a distinction between rational and integral points. Moreover, rational points have extra automorphism groups. We show that these distinctions exactly account for the lower order main terms appearing in precise counts of elliptic curves over function fields, answering a question of Venkatesh in this case. More generally, using the theory of twisted stable maps and the stacky height functions recently introduced by Ellenberg, Zureick-Brown, and the third author, we construct finite type moduli spaces which parametrize rational points of fixed height on a large class of stacks, so-called cyclotomic stacks. The main tool is a correspondence between rational points, twisted maps and weighted linear series. Along the way, we obtain the Northcott property as well as a generalization of Tate's algorithm for cyclotomic stacks, and compute the exact motives of these moduli spaces for weighted projective stacks.

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arXiv2022

Motivic & Arithmetic probability of a semistable elliptic surface with a Weierstrass torsion section

We prove new sharp asymptotic for counting the semistable elliptic curves with two marked Weierstrass points at $\infty$ and $0$ and also the cases where $0$ is a 2-torsion or a 3-torsion marked Weierstrass point over $\mathbb{F}_q(t)$ by the bounded height of discriminant $Δ(X)$. We consider the motivic probabilities over any basefield $K$ with $\text{char}(K) \neq 2,3$ of picking a nonsingular semistable elliptic surface over $\mathbb{P}^{1}$ with two marked Weierstrass sections at $\infty$ and $0$ such that marked Weierstrass section at $0$ is 2-torsion or 3-torsion. In the end, we formulate an analogous heuristics on $\mathcal{Z}_{\mathbb{Q}}(\mathcal{B})$ for the ratio of the semistable elliptic curves with a marked rational 2-torsion or 3-torsion Weierstrass point at $0$ out of all semistable elliptic curves with a marked rational Weierstrass points at $0$ over $\mathbb{Q}$ by the bounded height of discriminant $Δ$ through the global fields analogy.

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arXiv2022

$\ell$-adic étale cohomology of the moduli of stable elliptic fibrations

We determine the $\ell$-adic étale cohomology and the eigenvalues of the geometric Frobenius for the moduli stack $\mathcal{L}_{1,12n} := \mathrm{Hom}_{n}(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})$ of stable elliptic fibrations over $\mathbb{P}^{1}$ with $12n$ nodal singular fibers and a marked Weierstrass section over $\overline{\mathbb{F}}_q$ with $\mathrm{char}(\overline{\mathbb{F}}_q) \neq 2,3$.

math.AG↗

arXiv2022

Étale cohomological stability of the moduli space of stable elliptic surfaces

We compute the (stable) étale cohomology of $\mathrm{Hom}_{n}(C, \mathcal{P}(\vecλ))$, the moduli stack of degree $n$ morphisms from a smooth projective curve $C$ to the weighted projective stack $\mathcal{P}(\vecλ)$, the latter being a stacky quotient defined by $\mathcal{P}(\vecλ) := \left[\mathbb{A}^N-\{0\}/\mathbb{G}_m\right]$, where $\mathbb{G}_m$ acts by weights $\vecλ = (λ_0, \cdots, λ_N) \in \mathbb{Z}^N_{+}$. Our key ingredient is formulating and proving the étale cohomological descent over the category $ΔS$, the symmetric (semi)simplicial category. An immediate arithmetic consequence is the resolution of the geometric Batyrev--Manin type conjecture for weighted projective stacks over global function fields. Along the way, we also analyze the intersection theory on weighted projectivizations of vector bundles on smooth Deligne-Mumford stacks.

math.AG↗

arXiv2022

Enumerating odd-degree hyperelliptic curves and abelian surfaces over $\mathbb{P}^1$

Given asymptotic counts in number theory, a question of Venkatesh asks what is the topological nature of lower order terms. We consider the arithmetic aspect of the inertia stack of an algebraic stack over finite fields to partially answer this question. Subsequently, we acquire new sharp enumerations on quasi-admissible odd-degree hyperelliptic curves over $\mathbb{F}_q(t)$ ordered by bounded discriminant height.

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arXiv2021

Arithmetic geometry of the moduli stack of Weierstrass fibrations over $\mathbb{P}^1$

Coarse moduli spaces of Weierstrass fibrations over the (unparameterized) projective line were constructed by the classical work of [Miranda] using Geometric Invariant Theory. In our paper, we extend this treatment by using results of [Romagny] regarding group actions on stacks to give an explicit construction of the moduli stack $\mathcal{W}_n$ of Weierstrass fibrations over an unparameterized $\mathbb{P}^{1}$ with discriminant degree $12n$ and a section. We show that it is a smooth algebraic stack and prove that for $n \geq 2$, the open substack $\mathcal{W}_{\mathrm{min},n}$ of minimal Weierstrass fibrations is a separated Deligne-Mumford stack over any base field $K$ with $\mathrm{char}(K) \neq 2,3$ and not dividing $n$. Arithmetically, for the moduli stack $\mathcal{W}_{\mathrm{sf},n}$ of stable Weierstrass fibrations, we determine its motive in the Grothendieck ring of stacks to be $\{\mathcal{W}_{\mathrm{sf},n}\} = \mathbb{L}^{10n - 2}$ in the case that $n$ is odd, which results in its weighted point count to be $\#_q(\mathcal{W}_{\mathrm{sf},n}) = q^{10n - 2}$ over $\mathbb{F}_q$. In the appendix, we show how our methods can be applied similarly to the classical work of [Silverman] on coarse moduli spaces of self-maps of the projective line, allowing us to construct the natural moduli stack and to compute its motive.

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arXiv2021

Motive of the moduli stack of rational curves on a weighted projective stack

We show the compactly supported motive of the moduli stack of degree $n$ rational curves on the weighted projective stack $\mathcal{P}(a,b)$ is of mixed Tate type over any base field $K$ with $\text{char}(K) \nmid a,b$ and has class $\mathbb{L}^{(a+b)n+1}-\mathbb{L}^{(a+b)n-1}$ in the Grothendieck ring of stacks. In particular, this improves upon the result of [HP] regarding the arithmetic invariant of the moduli stack $\mathcal{L}_{1,12n} := \mathrm{Hom}_{n}(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})$ of stable elliptic fibrations over $\mathbb{P}^{1}$ with $12n$ nodal singular fibers and a marked Weierstrass section.

math.AG↗

arXiv2019

Arithmetic of the moduli of semistable elliptic surfaces

We prove a new sharp asymptotic with the lower order term of zeroth order on $\mathcal{Z}_{\mathbb{F}_q(t)}(\mathcal{B})$ for counting the semistable elliptic curves over $\mathbb{F}_q(t)$ by the bounded height of discriminant $Δ(X)$. The precise count is acquired by considering the moduli of nonsingular semistable elliptic fibrations over $\mathbb{P}^{1}$, also known as semistable elliptic surfaces, with $12n$ nodal singular fibers and a distinguished section. We establish a bijection of $K$-points between the moduli functor of semistable elliptic surfaces and the stack of morphisms $\mathcal{L}_{1,12n} \cong \mathrm{Hom}_n(\mathbb{P}^{1}, \overline{\mathcal{M}}_{1,1})$ where $\overline{\mathcal{M}}_{1,1}$ is the Deligne-Mumford stack of stable elliptic curves and $K$ is any field of characteristic $\neq 2,3$. For $\mathrm{char}(K)=0$, we show that the class of $\mathrm{Hom}_n(\mathbb{P}^1,\mathcal{P}(a,b))$ in the Grothendieck ring of $K$-stacks, where $\mathcal{P}(a,b)$ is a 1-dimensional $(a,b)$ weighted projective stack, is equal to $\mathbb{L}^{(a+b)n+1}-\mathbb{L}^{(a+b)n-1}$. Consequently, we find that the motive of the moduli $\mathcal{L}_{1,12n}$ is $\mathbb{L}^{10n + 1}-\mathbb{L}^{10n - 1}$ and the cardinality of the set of weighted $\mathbb{F}_q$-points to be $\#_q(\mathcal{L}_{1,12n}) = q^{10n + 1}-q^{10n - 1}$. In the end, we formulate an analogous heuristic on $\mathcal{Z}_{\mathbb{Q}}(\mathcal{B})$ for counting the semistable elliptic curves over $\mathbb{Q}$ by the bounded height of discriminant $Δ$ through the global fields analogy.

math.AG↗

arXiv2015

Unique fiber sum decomposability of genus 2 Lefschetz fibrations

By applying the lantern relation substitutions to the positive relation of the genus two Lefschetz fibration over $\mathbb{S}^{2}$. We show that $K3\#2 \overline{\mathbb{CP}}{}^{2}$ can be rationally blown down along seven disjoint copies of the configuration $C_2$. We compute the Seiberg-Witten invariant of the resulting symplectic 4-manifolds, and show that they are symplectically minimal. We also investigate how these exotic smooth 4-manifolds constructed via lantern relation substitution method are fiber sum decomposable. Furthermore by considering all the possible decompositions for each of our decomposable exotic examples, we will find out that there is a uniquely decomposing genus 2 Lefschetz fibration which is not a self sum of the same fibration up to diffeomorphism on the indecomposable summands.

math.GT↗

arXiv2014

Lantern substitution and new symplectic 4-manifolds with ${b_{2}}^{+} = 3$

Motivated by the construction of H. Endo and Y. Gurtas, changing a positive relator in Dehn twist generators of the mapping class group by using lantern substitutions, we show that 4-manifold $K3#2\CPb$ equipped with the genus two Lefschetz fibration can be rationally blown down along six disjoint copies of the configuration $C_2$. We compute the Seiberg-Witten invariants of the resulting symplectic 4-manifold, and show that it is symplectically minimal. Using our example, we also construct an infinite family of pairwise non-diffeomorphic irreducible symplectic and non-symplectic 4-manifolds homeomorphic to $M = 3\CP# (19-k)\CPb$ for $1 \leq k \leq 4$.

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