On towers of Isogeny graphs with full level structure
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909308136980480 |
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| author | Lei, Antonio Müller, Katharina |
| author_facet | Lei, Antonio Müller, Katharina |
| contents | 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 Colò-Kohel, enhanced with a level structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_00524 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On towers of Isogeny graphs with full level structure Lei, Antonio Müller, Katharina Number Theory Combinatorics 05C25, 11G20, 11R23, 14G17, 14K02 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 Colò-Kohel, enhanced with a level structure. |
| title | On towers of Isogeny graphs with full level structure |
| topic | Number Theory Combinatorics 05C25, 11G20, 11R23, 14G17, 14K02 |
| url | https://arxiv.org/abs/2309.00524 |