Serre’s uniformity conjecture for elliptic curves with rational cyclic isogenies

Lemos, Pedro

doi:10.1090/tran/7198

Serre’s uniformity conjecture for elliptic curves with rational cyclic isogenies
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by Pedro Lemos PDF

Trans. Amer. Math. Soc. 371 (2019), 137-146 Request permission

Abstract:

Let $E$ be an elliptic curve over $\mathbb {Q}$ such that $\mathrm {End}_{\bar {\mathbb {Q}}}(E)=\mathbb {Z}$ and admitting a non-trivial cyclic $\mathbb {Q}$-isogeny. We prove that, for $p>37$, the residual mod $p$ Galois representation $\bar {\rho }_{E,p}:G_{\mathbb {Q}}\rightarrow \mathrm {GL}_2(\mathbb {F}_p)$ is surjective.

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