We study the automorphisms of the nonsplit Cartan modular curves Xns(p) of prime level p. We prove that if p ≥ 29 all the automorphisms preserve the cusps. Furthermore, if p ≡ 1 mod 12 and p ≠13 , the automorphism group is generated by the modular involution given by the normalizer of a nonsplit Cartan subgroup of GL2(Fp). We also prove that for every p ≥ 29 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 nonsplit Cartan subgroup of GL2(Fp).
On the automorphisms of the nonsplit cartan modular curves of prime level / Dose, V.. - In: NAGOYA MATHEMATICAL JOURNAL. - ISSN 0027-7630. - 224:1(2016), pp. 74-92. [10.1017/nmj.2016.32]
On the automorphisms of the nonsplit cartan modular curves of prime level
Dose V.
2016
Abstract
We study the automorphisms of the nonsplit Cartan modular curves Xns(p) of prime level p. We prove that if p ≥ 29 all the automorphisms preserve the cusps. Furthermore, if p ≡ 1 mod 12 and p ≠13 , the automorphism group is generated by the modular involution given by the normalizer of a nonsplit Cartan subgroup of GL2(Fp). We also prove that for every p ≥ 29 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 nonsplit Cartan subgroup of GL2(Fp).File | Dimensione | Formato | |
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