Jackson Salvatore Morrow
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Jackson Salvatore Morrow has a total of 12 co-authors. Here are the top co-authors by number of shared publications:
- Harris B. Daniels
Department of Mathematics and Statistics, Amherst College, Amherst, MA 01002, USA
3 shared publications - David Zureick-Brown
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322 USA
2 shared publications - Davide Lombardo
Dipartimento di Matematica Università di Pisa Pisa Italy
2 shared publications - Morgan V. Brown
University of Miami
2 shared publications Sonny Arora
The Pennsylvania State University
2 shared publicationsAaron Landesman
Stanford University
1 shared publication
Latest Publications
- 1 May 2023
- Published by Foundation Compositio Mathematica in Algebraic Geometry
- Vol. 10 (3) , 262-297
A non-Archimedean analogue of Campana's notion of specialness
- 1 May 2023
- Published by Foundation Compositio Mathematica in Algebraic Geometry
- Vol. 10 (3) , 262-297
A non-Archimedean analogue of Campana's notion of specialness
- Open Access
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 $\rho_E\colon G_{\mathbb{Q}} \to \text{GL}(2,\widehat{\mathbb{Z}}).$ The goal of this paper is to explain how the image of $\rho_E$ 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.Open AccessTowards 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 $\rho_E\colon G_{\mathbb{Q}} \to \text{GL}(2,\widehat{\mathbb{Z}}).$ The goal of this paper is to explain how the image of $\rho_E$ 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. - Open Access
- 13 October 2022
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society, Series B
- Vol. 9 (27) , 827-858
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 n and a subgroup G ⊆ G L 2 ( Z / n Z ) G\subseteq GL_2( \mathbb {Z}/n\mathbb {Z}) with surjective determinant, we provide a definition for G G to represent an ( a , b ) (a,b) -entanglement and give additional criteria for G G to represent an explained or unexplained ( a , b ) (a,b) -entanglement. Using these new definitions, we determine the tuples ( ( p , q ) , T ) ((p,q),T) , with p > q ∈ Z p>q\in \mathbb {Z} distinct primes and T T a finite group, such that there are infinitely many non- Q ¯ \overline {\mathbb {Q}} -isomorphic elliptic curves over Q \mathbb {Q} with an unexplained ( p , q ) (p,q) -entanglement of type T T . Furthermore, for each possible combination of entanglement level ( p , q ) (p,q) and type T T , we completely classify the elliptic curves defined over Q \mathbb {Q} with that combination by constructing the corresponding modular curve and j j -map.
- 13 October 2022
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society, Series B
- Vol. 9 (27) , 827-858
Open AccessA 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 n and a subgroup G ⊆ G L 2 ( Z / n Z ) G\subseteq GL_2( \mathbb {Z}/n\mathbb {Z}) with surjective determinant, we provide a definition for G G to represent an ( a , b ) (a,b) -entanglement and give additional criteria for G G to represent an explained or unexplained ( a , b ) (a,b) -entanglement. Using these new definitions, we determine the tuples ( ( p , q ) , T ) ((p,q),T) , with p > q ∈ Z p>q\in \mathbb {Z} distinct primes and T T a finite group, such that there are infinitely many non- Q ¯ \overline {\mathbb {Q}} -isomorphic elliptic curves over Q \mathbb {Q} with an unexplained ( p , q ) (p,q) -entanglement of type T T . Furthermore, for each possible combination of entanglement level ( p , q ) (p,q) and type T T , we completely classify the elliptic curves defined over Q \mathbb {Q} with that combination by constructing the corresponding modular curve and j j -map.
- Open Access
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.Open AccessA 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. - Open Access
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.Open AccessNon-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. - 5 September 2020
- Published by Springer Nature in Mathematische Annalen
- Vol. 379 (3-4) , 1003-1010
Non-archimedean entire curves in closed subvarieties of semi-abelian varieties
- 5 September 2020
- Published by Springer Nature in Mathematische Annalen
- Vol. 379 (3-4) , 1003-1010
Non-archimedean entire curves in closed subvarieties of semi-abelian varieties
- Open Access
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.Open AccessIrrational 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. - Open AccessComposite images of Galois for elliptic curves over $\mathbf{Q}$ & Entanglement fieldsLet $E$ be an elliptic curve defined over $\mathbf{Q}$ without complex multiplication. For each prime $\ell$, there is a representation $\rho_{E,\ell}\colon \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \text{GL}_2(\mathbf{F}_{\ell})$ 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.Open AccessComposite images of Galois for elliptic curves over $\mathbf{Q}$ & Entanglement fieldsLet $E$ be an elliptic curve defined over $\mathbf{Q}$ without complex multiplication. For each prime $\ell$, there is a representation $\rho_{E,\ell}\colon \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \text{GL}_2(\mathbf{F}_{\ell})$ 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.
- 1 November 2018
- Published by Elsevier in Linear Algebra and its Applications
- Vol. 556, 46-54
Chip-firing groups of iterated cones
- 1 November 2018
- Published by Elsevier in Linear Algebra and its Applications
- Vol. 556, 46-54
Chip-firing groups of iterated cones
- Open Access
Chip-firing groups of iterated cones
Let $\Gamma$ be a finite graph and let $\Gamma_n$ be the "$n$th cone over $\Gamma$" (i.e., the join of $\Gamma$ and the complete graph $K_n$). We study the asymptotic structure of the chip-firing group $\text{Pic}^0(\Gamma_n)$.Open AccessChip-firing groups of iterated cones
Let $\Gamma$ be a finite graph and let $\Gamma_n$ be the "$n$th cone over $\Gamma$" (i.e., the join of $\Gamma$ and the complete graph $K_n$). We study the asymptotic structure of the chip-firing group $\text{Pic}^0(\Gamma_n)$.