REVIEW 4 cited by
Explicit open images for elliptic curves over $\mathbb{Q}$
Not yet reviewed by Pith; the record is open.
This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.
SPECIMEN: schema-true, not a live event
T0 review · schema-true
One-sentence machine reading of the paper's core claim.
pith:XXXXXXXX · record.json · timestamp
abstract
For a non-CM elliptic curve $E$ defined over $\mathbb{Q}$, the Galois action on its torsion points gives rise to a Galois representation $\rho_E: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\widehat{\mathbb{Z}})$ that is unique up to isomorphism. A renowned theorem of Serre says that the image of $\rho_E$ is an open, and hence finite index, subgroup of $GL_2(\widehat{\mathbb{Z}})$. We describe an algorithm that computes the image of $\rho_E$ up to conjugacy in $GL_2(\widehat{\mathbb{Z}})$; this algorithm is practical and has been implemented. Up to a positive answer to a uniformity question of Serre and finding all the rational points on a finite number of explicit modular curves of genus at least $2$, we give a complete classification of the groups $\rho_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))\cap SL_2(\widehat{\mathbb{Z}})$ and the indices $[GL_2(\widehat{\mathbb{Z}}):\rho_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))]$ for non-CM elliptic curves $E/\mathbb{Q}$. Much of the paper is dedicated to the efficient computation of modular curves via modular forms expressed in terms of Eisenstein series.
Forward citations
Cited by 4 Pith papers
-
Non-split Cartan curves and a characterization of fake elliptic curves
An abelian surface over an imaginary quadratic field has quaternionic multiplication exactly when its residual Galois image is non-split Cartan at two or more primes, under a conjecture that is verified for small levels.
-
Degrees of points with rational $j$-invariant on $X_{0}(n)$ and $X_{1}(n)$
The degrees of points with rational j-invariant on X0(n) and X1(n) are classified for all n, unconditionally for infinitely occurring degrees and assuming Zywina's conjecture for finitely occurring ones.
-
Effective bounds for adelic Galois representations attached to elliptic curves over the rationals
For every non-CM elliptic curve over Q, the index of the adelic Galois image is bounded by 10^21(h_F(E)+40)^4.42, and by h_F(E)^{3+o(1)} as the height grows.
-
A uniform bound on the smallest surjective prime of an elliptic curve
For every non-CM elliptic curve over Q, one of its 2-, 3-, or 5-adic Galois representations is surjective unless its j-invariant is one of six listed values, in which case 7 is the minimal surjective prime.
Discussion (0). Continue with ORCID to comment.