Harris B. Daniels
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Harris B. Daniels has a total of 15 co-authors. Here are the top co-authors by number of shared publications:
- Álvaro Lozano-Robledo
Department of Mathematics, University of Connecticut, Storrs, USA
5 shared publications - Jeffrey Hatley
Department of Mathematics, Union College, Schenectady, USA
4 shared publications - Enrique González-Jiménez
Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain
3 shared publications - Filip Najman
Faculty of Science, Department of Mathematics, University of Zagreb, Zagreb, Croatia
3 shared publications - Jackson Salvatore Morrow
Univ Montreal, Ctr Rech Math, Montreal, PQ H3T 1J4, Canada
3 shared publications - Andrew V. Sutherland
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA
2 shared publications
Latest Publications
- Open Access
- 19 September 2025
- Published by MathDoc/Centre Mersenne in Journal de Théorie des Nombres de Bordeaux
- Vol. 37 (2) , 579-597
Minimal Subgroups of $\mathrm{GL}_2(\mathbb{Z}_{S})$Let E be an elliptic curve over a number field L and for a finite set S of primes, let ρ E , S : Gal ( L ¯ / L ) → GL 2 ( ℤ S ) be the S -adic Galois representation. If L ∩ ℚ ( ζ n ) = ℚ for all positive integers n whose prime factors are in S , then det ρ E , S : Gal ( L ¯ / L ) → ℤ S × is surjective. We say that a finite index subgroup H ⊆ GL 2 ( ℤ S ) is minimal if det : H → ℤ S × is surjective, but det : K → ℤ S × is not surjective for any proper closed subgroup K of H . We show that there are no minimal subgroups of GL 2 ( ℤ S ) unless S = { 2 } , while minimal subgroups of GL 2 ( ℤ 2 ) are plentiful. We give models for all the genus 0 modular curves associated to minimal subgroups of GL 2 ( ℤ 2 ) , and construct an infinite family of elliptic curves over imaginary quadratic fields with bad reduction only at 2 and with minimal 2 -adic image.
- 19 September 2025
- Published by MathDoc/Centre Mersenne in Journal de Théorie des Nombres de Bordeaux
- Vol. 37 (2) , 579-597
Open AccessMinimal Subgroups of $\mathrm{GL}_2(\mathbb{Z}_{S})$Let E be an elliptic curve over a number field L and for a finite set S of primes, let ρ E , S : Gal ( L ¯ / L ) → GL 2 ( ℤ S ) be the S -adic Galois representation. If L ∩ ℚ ( ζ n ) = ℚ for all positive integers n whose prime factors are in S , then det ρ E , S : Gal ( L ¯ / L ) → ℤ S × is surjective. We say that a finite index subgroup H ⊆ GL 2 ( ℤ S ) is minimal if det : H → ℤ S × is surjective, but det : K → ℤ S × is not surjective for any proper closed subgroup K of H . We show that there are no minimal subgroups of GL 2 ( ℤ S ) unless S = { 2 } , while minimal subgroups of GL 2 ( ℤ 2 ) are plentiful. We give models for all the genus 0 modular curves associated to minimal subgroups of GL 2 ( ℤ 2 ) , and construct an infinite family of elliptic curves over imaginary quadratic fields with bad reduction only at 2 and with minimal 2 -adic image.
- 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
Coincidences of division fields
Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $\rho_E\colon {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\rm GL}(2,\widehat{ \mathbb{Z} })$ be the adelic representation associated to the natural action of Galois on the torsion points of $E(\overline{\mathbb{Q}})$. By a theorem of Serre, the image of $\rho_{E}$ is open, but the image is always of index at least $2$ in ${\rm GL}(2,\widehat{\mathbb{Z}})$ due to a certain quadratic entanglement amongst division fields. In this paper, we study other types of abelian entanglements. More concretely, we classify the elliptic curves $E/\mathbb{Q}$, and primes $p$ and $q$ such that $\mathbb{Q}(E[p])\cap \mathbb{Q}(\zeta_{q^k})$ is non-trivial, and determine the degree of the coincidence. As a consequence, we classify all elliptic curves $E/\mathbb{Q}$ and integers $m,n$ such that the $m$-th and $n$-th division fields coincide, i.e., when $\mathbb{Q}(E[n])=\mathbb{Q}(E[m])$, when the division field is abelian.Open AccessCoincidences of division fields
Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $\rho_E\colon {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\rm GL}(2,\widehat{ \mathbb{Z} })$ be the adelic representation associated to the natural action of Galois on the torsion points of $E(\overline{\mathbb{Q}})$. By a theorem of Serre, the image of $\rho_{E}$ is open, but the image is always of index at least $2$ in ${\rm GL}(2,\widehat{\mathbb{Z}})$ due to a certain quadratic entanglement amongst division fields. In this paper, we study other types of abelian entanglements. More concretely, we classify the elliptic curves $E/\mathbb{Q}$, and primes $p$ and $q$ such that $\mathbb{Q}(E[p])\cap \mathbb{Q}(\zeta_{q^k})$ is non-trivial, and determine the degree of the coincidence. As a consequence, we classify all elliptic curves $E/\mathbb{Q}$ and integers $m,n$ such that the $m$-th and $n$-th division fields coincide, i.e., when $\mathbb{Q}(E[n])=\mathbb{Q}(E[m])$, when the division field is abelian. - 1 March 2021
- Published by Elsevier in Journal of Algebra
- Vol. 575, 274-284
Corrigendum to “Torsion subgroups of rational elliptic curves over the compositum of all D4 extensions of the rational numbers” [J. Algebra 509 (2018) 535–565]
- 1 March 2021
- Published by Elsevier in Journal of Algebra
- Vol. 575, 274-284
Corrigendum to “Torsion subgroups of rational elliptic curves over the compositum of all D4 extensions of the rational numbers” [J. Algebra 509 (2018) 535–565]
- Open Access
Bounds of the rank of the Mordell-Weil group of jacobians of hyperelliptic curves
In this article we extend work of Shanks and Washington on cyclic extensions, and elliptic curves associated to the simplest cubic fields. In particular, we give families of examples of hyperelliptic curves $C: y^2=f(x)$ defined over $\mathbb{Q}$, with $f(x)$ of degree $p$, where $p$ is a Sophie Germain prime, such that the rank of the Mordell--Weil group of the jacobian $J/\mathbb{Q}$ of $C$ is bounded by the genus of $C$ and the $2$-rank of the class group of the (cyclic) field defined by $f(x)$, and exhibit examples where this bound is sharp.Open AccessBounds of the rank of the Mordell-Weil group of jacobians of hyperelliptic curves
In this article we extend work of Shanks and Washington on cyclic extensions, and elliptic curves associated to the simplest cubic fields. In particular, we give families of examples of hyperelliptic curves $C: y^2=f(x)$ defined over $\mathbb{Q}$, with $f(x)$ of degree $p$, where $p$ is a Sophie Germain prime, such that the rank of the Mordell--Weil group of the jacobian $J/\mathbb{Q}$ of $C$ is bounded by the genus of $C$ and the $2$-rank of the class group of the (cyclic) field defined by $f(x)$, and exhibit examples where this bound is sharp. - 2 December 2019
- Published by Taylor & Francis in Experimental Mathematics
- Vol. 31 (2) , 518-536
Serre’s Constant of Elliptic Curves Over the Rationals
Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer A(E), that we call the Serre’s constant associated to E, that gives necessary conditions to conclude that the mod m Galois representation associated to E, is non-surjective. In particular, if there exists a prime factor p of m satisfying then is non-surjective. Conditionally under Serre’s Uniformity Conjecture, we determine all the Serre’s constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod p Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre’s constants.- 2 December 2019
- Published by Taylor & Francis in Experimental Mathematics
- Vol. 31 (2) , 518-536
Serre’s Constant of Elliptic Curves Over the Rationals
Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer A(E), that we call the Serre’s constant associated to E, that gives necessary conditions to conclude that the mod m Galois representation associated to E, is non-surjective. In particular, if there exists a prime factor p of m satisfying then is non-surjective. Conditionally under Serre’s Uniformity Conjecture, we determine all the Serre’s constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod p Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre’s constants.- Open Access
Serre's constant of elliptic curves over the rationals
Let $E$ be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer $A(E)$, that we call the {\it Serre's constant associated to $E$}, that gives necessary conditions to conclude that $\rho_{E,m}$, the mod m Galois representation associated to $E$, is non-surjective. In particular, if there exists a prime factor $p$ of $m$ satisfying ${\rm val}_p(m) > {\rm val}_p(A(E))>0$ then $\rho_{E,m}$ is non-surjective. {Conditionally under Serre's Uniformity Conjecture, w}e determine all the Serre's constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod $p$ Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over $\mathbb{Q}$, and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of ${\rm GL}_2(\mathbb{Z}_p)$. Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre's constant.Open AccessSerre's constant of elliptic curves over the rationals
Let $E$ be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer $A(E)$, that we call the {\it Serre's constant associated to $E$}, that gives necessary conditions to conclude that $\rho_{E,m}$, the mod m Galois representation associated to $E$, is non-surjective. In particular, if there exists a prime factor $p$ of $m$ satisfying ${\rm val}_p(m) > {\rm val}_p(A(E))>0$ then $\rho_{E,m}$ is non-surjective. {Conditionally under Serre's Uniformity Conjecture, w}e determine all the Serre's constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod $p$ Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over $\mathbb{Q}$, and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of ${\rm GL}_2(\mathbb{Z}_p)$. Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre's constant. - Open Access
- 14 June 2019
- Published by Wiley in Transactions of the London Mathematical Society
- Vol. 6 (1) , 22-52
Groups of generalized G‐type and applications to torsion subgroups of rational elliptic curves over infinite extensions of Q
- 14 June 2019
- Published by Wiley in Transactions of the London Mathematical Society
- Vol. 6 (1) , 22-52
Open AccessGroups of generalized G‐type and applications to torsion subgroups of rational elliptic curves over infinite extensions of Q