$\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$

Abstract

We discuss the $\ell$-adic case of Mazur's "Program B" over $\mathbb{Q}$, the problem of classifying the possible images of $\ell$-adic Galois representations attached to elliptic curves $E$ over $\mathbb{Q}$, equivalently, classifying the rational points on the corresponding modular curves. The primes $\ell =2$ and $\ell \ge 13$ are addressed by prior work, so we focus on the remaining primes $\ell =3,5,7,11$. For each of these $\ell$, we compute the directed graph of arithmetically maximal $\ell$-power level modular curves $X_H$, compute explicit equations for all but three of them, and classify the rational points on all of them except $X_{\mathrm{ns}}^{+}(N)$, for $N=27,25,49,121$, and two level $49$ curves of genus $9$ whose Jacobians have analytic rank $9$. Aside from the $\ell$-adic images that are known to arise for infinitely many $\bar{\mathbb{Q}}$-isomorphism classes of elliptic curves $E/\mathbb{Q}$, we find only 22 exceptional images that arise for any prime $\ell$ and any $E/\mathbb{Q}$ without complex multiplication; these exceptional images are realized by 20 non-CM rational $j$-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on $X_{\mathrm{ns}}^{+}(\ell )$ with $\ell \ge 19$, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the $\ell$-adic images of Galois for any elliptic curve over $\mathbb{Q}$. In an appendix with John Voight we generalize Ribet's observation that simple abelian varieties attached to newforms on ${\Gamma }_1(N)$ are of ${\mathrm{GL}}_2$-type; this extends Kolyvagin's theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of $X_H$.

arXiv commentMinor corrections; 72 pages; 22 tables

References in corpus (39)

• Computing the endomorphism ring of an ordinary elliptic curve over a finite field
• Computing Hilbert class polynomials with the Chinese Remainder Theorem
• Modular polynomials via isogeny volcanoes
• Twists of X(7) and primitive solutions to x^2+y^3=z^7
• The Mordell-Weil sieve: Proving non-existence of rational points on curves
• Isogeny volcanoes
• Galois theory of iterated endomorphisms
• Sporadic Cubic Torsion
• Computing images of Galois representations attached to elliptic curves
• Growth of torsion groups of elliptic curves upon base change
• Class invariants by the CRT method
• Modular curves of prime-power level with infinitely many rational points
• Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time, II
• Identifying supersingular elliptic curves
• A local-global principle for rational isogenies of prime degree
• Torsion points and Galois representations on CM elliptic curves
• Elliptic curves with 3-adic Galois representation surjective mod 3 but not mod 9
• Torsion subgroups of elliptic curves over quintic and sextic number fields
• Elliptic Curves with abelian division fields
• Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks
• Computing classical modular forms
• Tetrahedral Elliptic Curves and the local-global principle for Isogenies
• On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average
• Torsion subgroups of rational elliptic curves over the compositum of all cubic fields
• A classification of isogeny-torsion graphs of $\mathbb{Q}$-isogeny classes of elliptic curves
• Integral Tate modules and splitting of primes in torsion fields of elliptic curves
• On the torsion of rational elliptic curves over quartic fields
• Entanglement in the family of division fields of elliptic curves with complex multiplication
• Residual Galois representations of elliptic curves with image contained in the normaliser of a non-split Cartan
• Serre's constant of elliptic curves over the rationals
• Modular Curves Of Genus 2
• On the minimal degree of definition of p-primary torsion subgroups of elliptic curves
• Arithmetic invariants from Sato--Tate moments
• Elliptic curves with non-abelian entanglements
• Quadratic Chabauty for modular curves: Algorithms and examples
• A Classification of Genus 0 Modular Curves with Rational Points
• Some uniform bounds for elliptic curves over $\mathbb{Q}$
• Towards a classification of entanglements of Galois representations attached to elliptic curves
• Computing actions on cusp forms

Cited by in corpus (7)

• Spectral Theory of Isogeny Graphs
• Modular Curves with many Points over Finite Fields
• Models of CM elliptic curves with a prescribed $\ell$-adic Galois image
• Towards a classification of isolated $j$-invariants
• Semistable reduction of plane quartics at $p=3$
• The Statement of Mochizuki's Corollary 3.12: Initial Theta Data
• Near coincidences and nilpotent division fields