$\ell$-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 $X_H$.

Minor corrections; 72 pages; 22 tables

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Cited by in corpus (6)

• Spectral Theory of Isogeny Graphs
• 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 $p=3$
• Near coincidences and nilpotent division fields
• The Statement of Mochizuki's Corollary 3.12: Initial Theta Data