Elliptic curves over finite fields and Ramanujan graphs on two vertices. (English) Zbl 07852306
Summary: Let \(p\) be an odd prime and let \(q = p^{\alpha}\), where \(a = 1\) or \(a = 2\). We show that there is a one-to-one correspondence between \(\mathbb{F}_q\)-isogeny classes of elliptic curves \(C\) over \(\mathbb{F}_q\) satisfying \(|C(\mathbb{F}_q )| \in 4\mathbb{Z}\) and \((q + 1)\)-regular Ramanujan graphs on two vertices. The correspondence is obtained by matching the Hasse-Weil zeta function of the elliptic curves with the Ihara zeta function of the Ramanujan graphs using the Honda-Tate theorem.
MSC:
| 11G20 | Curves over finite and local fields |
| 11G07 | Elliptic curves over local fields |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
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