Under rank-zero conditions on elliptic curves E1 and E2 over K, the K-quadratic points on the Fermat quartic are finite and computable, coinciding with Q-quadratic points for odd-degree K.
On the possible images of the mod ell representations associated to elliptic curves over Q
6 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
Consider a non-CM elliptic curve $E$ defined over $\mathbb{Q}$. For each prime $\ell$, there is a representation $\rho_{E,\ell}: G \to GL_2(\mathbb{F}_\ell)$ that describes the Galois action on the $\ell$-torsion points of $E$, where $G$ is the absolute Galois group of $\mathbb{Q}$. A famous theorem of Serre says that $\rho_{E,\ell}$ is surjective for all large enough $\ell$. We will describe all known, and conjecturally all, pairs $(E,\ell)$ such that $\rho_{E,\ell}$ is not surjective. Together with another paper, this produces an algorithm that given an elliptic curve $E/\mathbb{Q}$, outputs the set of such exceptional primes $\ell$ and describes all the groups $\rho_{E,\ell}(G)$ up to conjugacy. Much of the paper is dedicated to computing various modular curves of genus $0$ with their morphisms to the $j$-line.
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UNVERDICTED 6representative citing papers
Arithmetic intersection numbers of CM divisors on X_ns^+(p) are determined at all finite primes when p is inert, via a new moduli interpretation of the smooth locus in the regular integral model.
Solvability of Galois embedding problems arising from 3-torsion of elliptic curves over Q is equivalent to the existence of infinitely many elliptic curves with the corresponding 3-division fields, for all possible images of the mod 3 representation.
Introduces minimal torsion curves in isogeny classes of elliptic curves and gives complete characterizations for N a prime power (odd degree) and for CM classes.
For non-CM elliptic curves over Q, the Cojocaru constant C_E,j is positive except when there are coincidences of division fields, which the authors classify in several abelian families.
Arithmetic progressions of five or six squares over quadratic extensions of a number field K are shown to take specific forms, with none of length greater than six existing under stated assumptions on K.
citing papers explorer
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Quadratic points on the Fermat quartic over number fields
Under rank-zero conditions on elliptic curves E1 and E2 over K, the K-quadratic points on the Fermat quartic are finite and computable, coinciding with Q-quadratic points for odd-degree K.
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Arithmetic intersections on non-split Cartan modular curves
Arithmetic intersection numbers of CM divisors on X_ns^+(p) are determined at all finite primes when p is inert, via a new moduli interpretation of the smooth locus in the regular integral model.
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On Galois Embedding Problems Arising from 3-Torsion of Elliptic Curves
Solvability of Galois embedding problems arising from 3-torsion of elliptic curves over Q is equivalent to the existence of infinitely many elliptic curves with the corresponding 3-division fields, for all possible images of the mod 3 representation.
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Minimal torsion curves in geometric isogeny classes
Introduces minimal torsion curves in isogeny classes of elliptic curves and gives complete characterizations for N a prime power (odd degree) and for CM classes.
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The Smallest Invariant Factor of Elliptic Curves, and Coincidences
For non-CM elliptic curves over Q, the Cojocaru constant C_E,j is positive except when there are coincidences of division fields, which the authors classify in several abelian families.
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Squares in arithmetic progression over quadratic extensions of number fields
Arithmetic progressions of five or six squares over quadratic extensions of a number field K are shown to take specific forms, with none of length greater than six existing under stated assumptions on K.