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Ordinary isogeny graphs with level structure
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 27 March 2025, pp. 1-19
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- Article
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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.
