ON THE AUTOMORPHISMS OF THE NONSPLIT CARTAN MODULAR CURVES OF PRIME LEVEL
ON THE AUTOMORPHISMS OF THE NONSPLIT CARTAN MODULAR CURVES OF PRIME LEVEL
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论素能级不分裂嘉当模曲线的自同构
作者：
V. Dose
我们研究素数级 $p$ 的非分裂嘉当模曲线 $X_{ ext{ns}}(p)$ 的自同构。我们证明，如果 $pgeqslant 29$ 所有自同构都保留尖点。此外，如果 $pequiv 1~ ext{mod}~12$ 和 $p
eq 13$ ，自同构群由 $ ext{GL}_{2}(mathbb{F}_{p})$ 的非分裂嘉坦子群的归一化器给出的模对合生成。我们还证明，对于每个 $pgeqslant 29$，异常有理自同构的存在将在模曲线 $X_{ ext{ns}}^{+}(p)$ 上产生异常有理点，该点与 $ ext{GL}_{2}(mathbb{F}_{p})$ 的非分裂嘉当子群的归一化器相关。
We study the automorphisms of the nonsplit Cartan modular curves $X_{ ext{ns}}(p)$ of prime level $p$ . We prove that if $pgeqslant 29$ all the automorphisms preserve the cusps. Furthermore, if $pequiv 1~ ext{mod}~12$ and $p
eq 13$ , the automorphism group is generated by the modular involution given by the normalizer of a nonsplit Cartan subgroup of $ ext{GL}_{2}(mathbb{F}_{p})$ . We also prove that for every $pgeqslant 29$ the existence of an exceptional rational automorphism would give rise to an exceptional rational point on the modular curve $X_{ ext{ns}}^{+}(p)$ associated to the normalizer of a nonsplit Cartan subgroup of $ ext{GL}_{2}(mathbb{F}_{p})$ .