On the automorphisms of the non-split Cartan modular curves of prime level

TL;DR

This paper investigates the automorphism groups of non-split Cartan modular curves of prime level, establishing conditions under which automorphisms preserve cusps and identifying generators of the automorphism group for certain primes.

Contribution

It proves that for primes p ≥ 37, all automorphisms preserve cusps, and for p ≡ 1 mod 12 (p ≠ 13), the automorphism group is generated by a specific modular involution.

Findings

Automorphisms preserve cusps for p ≥ 37.

Automorphism group generated by modular involution when p ≡ 1 mod 12, p ≠ 13.

Existence of exceptional rational automorphisms linked to rational points on related curves.

Abstract

We study the automorphisms of the non-split Cartan modular curves $X_{ns}(p)$ of prime level $p$. We prove that if $p\geq 37$ all the automorphisms preserve the cusps. Furthermore, if $p\equiv 1\text{ mod }12$ and $p\neq 13$, the automorphism group is generated by the modular involution given by the normalizer of a non-split Cartan subgroup of $\text{GL}_2(\mathbb F_p)$. We also prove that for every $p\geq 37$ such that $X_{ns}(p)$ has a CM rational point, the existence of an exceptional rational automorphism would give rise to an exceptional rational point on the modular curve $X_{ns}^+(p)$ associated to the normalizer of a non-split Cartan subgroup of $\text{GL}_2(\mathbb F_p)$.