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

Abstract

We study the automorphisms of the non-split Cartan modular curves $X_{ns}(p)$ of prime level $p$. We prove that if $p\ge 37$ all the automorphisms preserve the cusps. Furthermore, if $p\equiv 1\text{ mod }12$ and $p\ne 13$, 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 $p\ge 37$ 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 ${\text{GL}}_2({\mathbb{F}}_p)$.

References in corpus (1)

• The automorphism group of the non-split Cartan modular curve of level 11

Cited by in corpus (4)

• Automorphisms of Cartan modular curves of prime and composite level
• Double Covers of Cartan Modular Curves
• Quadratic Points on Non-Split Cartan Modular Curves
• Modular Curves with many Points over Finite Fields