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Jackson S. Morrow

Publications and source records attributed to Jackson S. Morrow.

At least 19 recordsLinked to original sources
arXiv2025

Higher dimensional geometry of $p$-jets

In this work, we prove a quantitative version of the prime-to-$p$ Manin--Mumford conjecture for varieties with ample cotangent bundle. More precisely, let $A$ be an abelian variety defined over a number field $F$, and let $X$ be a smooth projective subvariety of $A$ with ample cotangent bundle. We prove that for every prime $p\gg 0$, the intersection of $X(F^{\text{alg}})$ and the geometric prime-to-$p$ torsion of $A$ is finite and explicitly bounded by a summation involving cycle classes in the Chow ring of the reduction of $X$ modulo $p$. This result is a higher dimensional analogue of Buium's quantitative Manin--Mumford for curves. Our proof follows a similar outline to Buium's in that it heavily relies on his theory of arithmetic jet spaces. In this context, we prove that the special fiber of the arithmetic jet space associated to a model of $X$ is affine as a scheme over $\mathbb{F}_p^{\text{alg}}$. As an application of our results, we use a result of Debarre to prove that when $X$ is $\mathbb{Q}^{\text{alg}}$-isomorphic to a complete intersection of $c > \text{dim}(A)/2$ many general hypersurfaces of $A_{\mathbb{Q}^{\text{alg}}}$ of sufficiently large degree, the intersection of $X(F^{\text{alg}})$ and the geometric prime-to-$p$ torsion of $A$ is bounded by a polynomial that depends only on $p$, the dimension of the ambient abelian variety, and intersection numbers of certain products of the hypersurfaces.

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arXiv2025

Global pluripotential theory for adelic line bundles

In this work, we relate recent work of Yuan--Zhang and Song on adelic line bundles over quasi-projective arithmetic varieties to recent advances in pluripotential theory on global Berkovich spaces from Pille-Schneider. In particular, we establish an equivalence between subcategories of adelic line bundles on quasi-projective varieties and line bundles on their Berkovich analytifications equipped with a continuous plurisubharmonic metric. We also provide several applications of this equivalence. For example, we generalize a construction of Pille-Schneider concerning families of Monge--Ampère measures on analytifications of projective arithmetic varieties to the quasi-projective setting. With this construction, we offer a new description of non-degenerate subvarieties which involves Monge--Ampère measures over trivially valued fields. Finally, we define a Monge--Ampère measure on the analytification of a quasi-projective arithmetic variety.

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arXiv2025

Non-Archimedean entire curves in projective varieties dominating an elliptic curve

Let $K$ be an algebraically closed, complete, non-Archimedean valued field of characteristic zero. We prove the non-Archimedean Green--Griffiths--Lang conjecture for projective surfaces of irregularity one. More precisely, we prove that if $X/K$ is a groupless, projective surface that admits a dominant morphism an elliptic curve, then $X$ is $K$-analytically Brody hyperbolic. The main ingredient in our proof is a theorem concerning the algebraic degeneracy of non-Archimedean entire curves in projective, pseudo-groupless varieties admitting a dominant morphism to an elliptic curve.

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arXiv2025

The non-Archimedean Green--Griffiths--Lang--Vojta conjecture for commutative algebraic groups with unipotent rank 1

Let $k$ be algebraically closed field of characteristic zero, let $G$ be a commutative algebraic group over $k$ such that the linear part of $G$ is isomorphic to $\mathbb{G}_a$, and let $X$ be a closed subvariety of $G$. We show that the Kawamata locus of $X$ is equal to a Lang-like exceptional locus of $X$, and furthermore, we identify a condition on $X$ that implies that these loci are proper subschemes of $X$. We also prove the strong form of the non-Archimedean Green--Griffiths--Lang--Vojta conjecture for closed subvarieties of commutative algebraic groups where the linear part is isomorphic to $\mathbb{G}_a \times \mathbb{G}_m^t$.

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arXiv2024

Towards a classification of entanglements of Galois representations attached to elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve, let $\overline{\mathbb{Q}}$ be a fixed algebraic closure of $\mathbb{Q}$, and let $G_{\mathbb{Q}}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. The action of $G_{\mathbb{Q}}$ on the adelic Tate module of $E$ induces the adelic Galois representation $ρ_E\colon G_{\mathbb{Q}} \to \text{GL}(2,\widehat{\mathbb{Z}}).$ The goal of this paper is to explain how the image of $ρ_E$ can be smaller than expected. To this end, we offer a group theoretic categorization of different ways in which an entanglement between division fields can be explained and prove several results on elliptic curves (and more generally, principally polarized abelian varieties) over $\mathbb{Q}$ where the entanglement occurs over an abelian extension.

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arXiv2024

Sporadic Cubic Torsion

Let $K$ be a number field, and let $E/K$ be an elliptic curve over $K$. The Mordell--Weil theorem asserts that the $K$-rational points $E(K)$ of $E$ form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of $E(K)$ for $K$ a cubic number field. To do so, we determine the cubic points on the modular curves $X_1(N)$ for \[N = 21, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 45, 65, 121.\] As part of our analysis, we determine the complete list of $N$ for which $J_0(N)$ (resp., $J_1(N)$, resp., $J_1(2,2N)$) has rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on $J_1(N)(\mathbb{Q})$ is generated by $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-orbits of cusps of $X_1(N)_{\bar{\mathbb{Q}}}$ for $N\leq 55$, $N \neq 54$.

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arXiv2023

Boundedness of hyperbolic varieties

Let $k$ be an algebraically closed field of characteristic zero, and let $X/k$ be a projective variety. The conjectures of Demailly--Green--Griffiths--Lang posit that every integral subvariety of $X$ is of general type if and only if $X$ is algebraically hyperbolic i.e., for any ample line bundle $\mathcal{L}$ on $X$ there is a real number $α(X,\mathcal{L})$, depending only on $X$ and $\mathcal{L}$, such that for every smooth projective curve $C/k$ of genus $g(C)$ and every $k$-morphism $f\colon C\to X$, $\text{deg}_Cf^*\mathcal{L} \leq α(X,\mathcal{L})\cdot g(C) $ holds. In this work, we prove that if $X/k$ is a projective variety such that every integral subvariety is of general type, then for every ample line bundle $\mathcal{L}$ on $X$ and every integer $g\geq 0$, there is an integer $α(X,\mathcal{L},g)$, depending only on $X,\mathcal{L},$ and $g$, such that for every smooth projective curve $C/k$ of genus $g$ and every $k$-morphism $f\colon C\to X$, the inequality $\text{deg}_Cf^*\mathcal{L} \leq α(X,\mathcal{L},g)$ holds, or equivalently, the Hom-scheme $\underline{\text{Hom}}_k(C,X)$ is projective.

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arXiv2023

Integral points on varieties with infinite étale fundamental group

We study integral points on varieties with infinite étale fundamental groups. More precisely, for a number field $F$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $φ\colon Y \to X$ of degree at least $2\dim(X)^2$, there exists an ample line bundle $\mathscr{L}$ on $Y$ such that for a general member $D$ of the complete linear system $|\mathscr{L}|$, $D$ is geometrically irreducible and any set of $φ(D)$-integral points on $X$ is finite. We apply this result to varieties with infinite étale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable.

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arXiv2022

The conjugate uniformization via 1-motives

We use the $p$-divisible group attached to a 1-motive to generalize the conjugate $p$-adic uniformization of Iovita--Morrow--Zaharescu to arbitrary $p$-adic formal semi-abelian schemes or $p$-divisible groups over the ring of integers in a $p$-adic field. This mirrors a mixed Hodge theory construction of the inverse uniformization map for complex semi-abelian varieties. We also highlight the geometric structure of the target of the conjugate uniformization map, which is an étale cover of a negative Banach--Colmez space in the sense of Fargues--Scholze.

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arXiv2022

Ramification of $p$-power torsion points of formal groups

Let $p$ be a rational prime, let $F$ denote a finite, unramified extension of $\mathbb{Q}_p$, let $K$ be the completion of the maximal unramified extension of $\mathbb{Q}_p$, and let $\overline{K}$ be some fixed algebraic closure of $K$. Let $A$ be an abelian variety defined over $F$, with good reduction, let $\mathcal{A}$ denote the Néron model of $A$ over ${\rm Spec}(\mathcal{O}_F)$, and let $\widehat{\mathcal{A}}$ be the formal completion of $\mathcal{A}$ along the identity of its special fiber, i.e. the formal group of $A$. In this work, we prove two results concerning the ramification of $p$-power torsion points on $\widehat{\mathcal{A}}$. One of our main results describes conditions on $\widehat{\mathcal{A}}$, base changed to $\text{Spf}(\mathcal{O}_K) $, for which the field $K(\widehat{\mathcal{A}}[p])/K$ is a tamely ramified extension where $\widehat{\mathcal{A}}[p]$ denotes the group of $p$-torsion points of $\widehat{\mathcal{A}}$ over $\mathcal{O}_{\overline{K}}$. This result generalizes previous work when $A$ is $1$-dimensional and work of Arias-de-Reyna when $A$ is the Jacobian of certain genus 2 hyperelliptic curves.

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arXiv2022

On $p$-adic uniformization of abelian varieties with good reduction

Let $p$ be a rational prime, let $F$ denote a finite, unramified extension of $\mathbb{Q}_p$, $K$ the maximal unramified extension of $\mathbb{Q}_p$, $\overline{K}$ some fixed algebraic closure of $K$, and $\mathbb{C}_p$ the completion of $\overline{K}$. Let $G_F$ the absolute Galois group of $F$. Let $A$ be an abelian variety defined over $F$, with good reduction. Classically, the Fontaine integral was seen as a Hodge--Tate comparison morphism, i.e. as a map $φ_{A} \otimes 1_{\mathbb{C}_p}\colon T_p(A)\otimes_{\mathbb{Z}_p}\mathbb{C}_p\to \text{Lie}(A)(F)\otimes_F\mathbb{C}_p(1)$, and as such it is surjective and has a large kernel. The present article starts with the observation that if we do not tensor $T_p(A)$ with $\mathbb{C}_p$, then the Fontaine integral is often injective. In particular, it is proved that if $T_p(A)^{G_K} = 0$, then $φ_A$ is injective. As an application, we extend the Fontaine integral to a perfectoid like universal cover of $A$ and show that if $T_p(A)^{G_K} = 0$, then $A(\overline{K})$ has a type of $p$-adic uniformization, which resembles the classical complex uniformization.

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arXiv2022

A group theoretic perspective on entanglements of division fields

In this paper, we initiate a systematic study of entanglements of division fields from a group theoretic perspective. For a positive integer $n$ and a subgroup $G\subseteq \text{GL}_2(\mathbb{Z}/{n}\mathbb{Z})$ with surjective determinant, we provide a definition for $G$ to represent an $(a,b)$-entanglement and give additional criteria for $G$ to represent an explained or unexplained $(a,b)$-entanglement. Using these new definitions, we determine the tuples $((p,q),T)$, with $p<q\in\mathbb{Z}$ distinct primes and $T$ a finite group, such that there are infinitely many non-$\bar{\mathbb{Q}}$-isomorphic elliptic curves over $\mathbb{Q}$ with an unexplained $(p,q)$-entanglement of type $T$. Furthermore, for each possible combination of entanglement level $(p,q)$ and type $T$, we completely classify the elliptic curves defined over $\mathbb{Q}$ with that combination by constructing the corresponding modular curve and $j$-map.

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arXiv2021

A non-Archimedean analogue of Campana's notion of specialness

Let $K$ be an algebraically closed, complete, non-Archimedean valued field of characteristic zero, and let $\mathscr{X}$ be a $K$-analytic space (in the sense of Huber). In this work, we pursue a non-Archimedean characterization of Campana's notion of specialness. We say $\mathscr{X}$ is $K$-analytically special if there exists a connected, finite type algebraic group $G/K$, a dense open subset $\mathscr{U}\subset G^{\text{an}}$ with $\text{codim}(G^{\text{an}}\setminus \mathscr{U}) \geq 2$, and an analytic morphism $\mathscr{U} \to \mathscr{X}$ which is Zariski dense. With this definition, we prove several results which illustrate that this definition correctly captures Campana's notion of specialness in the non-Archimedean setting. These results inspire us to make non-Archimedean counterparts to conjectures of Campana. As preparation for our proofs, we prove auxiliary results concerning the indeterminacy locus of a meromorphic mapping between $K$-analytic spaces, the notion of pseudo-$K$-analytically Brody hyperbolic, and extensions of meromorphic maps from smooth, irreducible $K$-analytic spaces to the analytification of a semi-abelian variety.

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arXiv2019

Irrational points on random hyperelliptic curves

We consider genus $g$ hyperelliptic curves over $\mathbb{Q}$ with a rational Weierstrass point, ordered by height. If $d < g$ is odd, we prove, under an assumption, that there exists $B_d$ such that a positive proportion of these curves have at most $B_d$ points of degree $d$. If $d < g$ is even, we conditionally bound degree $d$ points not pulled back from points of degree $d/2$ on the projective line. Furthermore, we show one may take $B_2=24$ and B_3=114$. Our proofs proceed by refining recent work of Park, which applied tropical geometry to symmetric power Chabauty, and then applying results of Bhargava and Gross on average ranks of Jacobians of hyperelliptic curves.

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arXiv2018

Composite images of Galois for elliptic curves over $\mathbf{Q}$ & Entanglement fields

Let $E$ be an elliptic curve defined over $\mathbf{Q}$ without complex multiplication. For each prime $\ell$, there is a representation $ρ_{E,\ell}\colon \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \text{GL}_2(\mathbf{F}_{\ell})$ that describes the Galois action on the $\ell$-torsion points of $E$. Building on recent work of Rouse--Zureick-Brown and Zywina, we find models for composite level modular curves whose rational points classify elliptic curves over $\mathbf{Q}$ with simultaneously non-surjective, composite image of Galois. We also provably determine the rational points on almost all of these curves. Finally, we give an application of our results to the study of entanglement fields.

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arXiv2018

Chip-firing groups of iterated cones

Let $Γ$ be a finite graph and let $Γ_n$ be the "$n$th cone over $Γ$" (i.e., the join of $Γ$ and the complete graph $K_n$). We study the asymptotic structure of the chip-firing group $\text{Pic}^0(Γ_n)$.

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