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Improved bounds for Serre’s open image theorem

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  10 September 2025

Imin Chen  and
Joshua Swidinsky
Imin Chen*
Affiliation:
Department of Mathematics, https://ror.org/0213rcc28 Simon Fraser University , Burnaby, BC V5A 1S6, Canada e-mail: joshua_swidinsky@sfu.ca
Joshua Swidinsky
Affiliation:
Department of Mathematics, https://ror.org/0213rcc28 Simon Fraser University , Burnaby, BC V5A 1S6, Canada e-mail: joshua_swidinsky@sfu.ca
*
e-mail: ichen@sfu.ca
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Abstract

Let E be an elliptic curve over the rationals which does not have complex multiplication. Serre showed that the adelic representation attached to $E/\mathbb {Q}$ has open image, and in particular, there is a minimal natural number $C_E$ such that the mod $\ell $ representation ${\bar {\rho }}_{E,\ell }$ is surjective for any prime $\ell> C_E$. Assuming the Generalized Riemann Hypothesis, Mayle–Wang gave explicit bounds for $C_E$ which are logarithmic in the conductor of E and have explicit constants. The method is based on using effective forms of the Chebotarev Density Theorem together with the Faltings–Serre method, in particular, using the “deviation group” of the $2$-adic representations attached to two elliptic curves.

By considering quotients of the deviation group and a characterization of the images of the $2$-adic representation $\rho _{E,2}$ by Rouse and Zureick–Brown, we show in this article how to further reduce the constants in Mayle–Wang’s results. Another result of independent interest are improved effective isogeny theorems for elliptic curves over the rationals.

MSC classification

Primary: 11G05: Elliptic curves over global fields 11F80: Galois representations

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Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 19
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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