Geodesic
$p$-torsion monodromy representations of elliptic curves over geometric function fields
Benjamin Bakker1 ; Jacob Tsimerman2
1 Humboldt-Universität zu Berlin
2 University of Toronto, Toronto, Ontario, Canada
Annals of mathematics, Tome 184 (2016) no. 3, pp. 709-744.

Voir la notice de l'article provenant de la source Annals of Mathematics website

Given a complex quasiprojective curve $B$ and a nonisotrivial family $\mathcal{E}$ of elliptic curves over $B$, the $p$-torsion $\mathcal{E}[p]$ yields a monodromy representation $\rho_\mathcal{E}[p]:\pi_1(B)\rightarrow \mathrm{GL}_2(\mathbb{F}_p)$. We prove that if $\rho_{\mathcal E}[p]\cong \rho_{\mathcal E’}[p]$, then $\mathcal{E}$ and $\mathcal E’$ are isogenous, provided $p$ is larger than a constant depending only on the gonality of $B$. This can be viewed as a function field analog of the Frey–Mazur conjecture, which states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p> 17$. The proof relies on hyperbolic geometry and is therefore only applicable in characteristic 0.
DOI : 10.4007/annals.2016.184.3.2

Benjamin Bakker 1 ; Jacob Tsimerman 2

1 Humboldt-Universität zu Berlin
2 University of Toronto, Toronto, Ontario, Canada 
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Benjamin Bakker; Jacob Tsimerman. $p$-torsion monodromy representations of elliptic curves over geometric function fields. Annals of mathematics, Tome 184 (2016) no. 3, pp. 709-744. doi : 10.4007/annals.2016.184.3.2. http://geodesic.mathdoc.fr/articles/10.4007/annals.2016.184.3.2/

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