Geodesic
Ordinary isogeny graphs with level structure
Canadian mathematical bulletin, Tome 68 (2025) no. 4, pp. 1144-1162

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DOI

We study $\ell $-isogeny graphs of ordinary elliptic curves defined over $\mathbb {F}_q$ with an added level structure. Given an integer N coprime to p and $\ell ,$ we look at the graphs obtained by adding $\Gamma _0(N)$, $\Gamma _1(N),$ and $\Gamma (N)$-level structures to volcanoes. Given an order $\mathcal {O}$ in an imaginary quadratic field $K,$ we look at the action of generalized ideal class groups of $\mathcal {O}$ on the set of elliptic curves whose endomorphism rings are $\mathcal {O}$ along with a given level structure. We show how the structure of the craters of these graphs is determined by the choice of parameters.
DOI : 10.4153/S0008439525000372
Mots-clés : Elliptic curves, level structure, isogeny graphs, class field theory 
Perrin, Derek; Voloch, José Felipe. Ordinary isogeny graphs with level structure. Canadian mathematical bulletin, Tome 68 (2025) no. 4, pp. 1144-1162. doi: 10.4153/S0008439525000372
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     title = {Ordinary isogeny graphs with level structure},
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     year = {2025},
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     doi = {10.4153/S0008439525000372},
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