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arXiv · 2309.00524

On towers of Isogeny graphs with full level structure

Antonio Lei·Katharina Müller
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Abstract

Let $p,q,l$ be three distinct prime numbers and let $N$ be a positive integer coprime to $pql$. For an integer $n\ge 0$, we define the directed graph $X_l^q(p^nN)$ whose vertices are given by isomorphism classes of elliptic curves over a finite field of characteristic $q$ equipped with a level $p^nN$ structure. The edges of $X_l^q(p^nN)$ are given by $l$-isogenies. We are interested in when the connected components of $X_l^q(p^nN)$ give rise to a tower of Galois covers as $n$ varies. We show that only in the supersingular case we do get a tower of Galois covers. We also study similar towers of isogeny graphs given by oriented supersingular curves, as introduced by Col\`o-Kohel, enhanced with a level structure.

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Antonio Lei, Katharina Müller. 2023-09-01. On towers of Isogeny graphs with full level structure. https://arxiv.org/abs/2309.00524

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