On the automorphisms of the nonsplit Cartan modular curves of prime level. (English) Zbl 1365.14034
The main results in the paper under review are the following: if \(p\) is a prime and \(p\geq29\), all the automorphisms of the modular curve \(X_{ns}(p)\) associated to a nonsplit Cartan subgroup of \(\text{GL}_2(\mathbb{F}_p)\), preserve the cusps. Moreover, if there exists an exceptional automorphism, that is, non-modular, defined over \(\mathbb{Q}\) then the modular curve associated to the normalizer of a nonsplit Cartan subgroup \(X^{+}_{ns}(p)\) has a non-CM rational point.
The techniques to prove these results are mainly bases on the work of M. Baker and Y. Hasegawa [J. Number Theory 100, No. 1, 72–87 (2003; Zbl 1088.11049)]; M. A. Kenku and F. Momose [Compos. Math. 65, No. 1, 51–80 (1988; Zbl 0686.14035)]; and A. P. Ogg [Bull. Soc. Math. Fr. 102, 449–462 (1974; Zbl 0314.10018)]. The new ideas brought by the author are bounding by a quadratic field the field of definition of the endomorphisms coming from the CM-part of the Jacobian of \(X_{0}(p^2)\) and proving that there is no CM-point in \(X_{ns}(p)\) defined over \(\mathbb{Q}(\zeta_p)\).
The interest in nonsplit Cartan modular curves comes mainly from Serre’s uniformity conjecture. Which is equivalent to assert that for almost all \(p\) and for every maximal subgroup \(H\) of \(\text{GL}_2(\mathbb{F}_p)\), the modular curve \(X_H\) has no rational points except the cusps and the points associated to elliptic curves with complex multiplication. The only case left to prove concerns modular curves \(X^{+}_{ns}(p)\) associated to the normalizer of a nonsplit Cartan subgroup.
The techniques to prove these results are mainly bases on the work of M. Baker and Y. Hasegawa [J. Number Theory 100, No. 1, 72–87 (2003; Zbl 1088.11049)]; M. A. Kenku and F. Momose [Compos. Math. 65, No. 1, 51–80 (1988; Zbl 0686.14035)]; and A. P. Ogg [Bull. Soc. Math. Fr. 102, 449–462 (1974; Zbl 0314.10018)]. The new ideas brought by the author are bounding by a quadratic field the field of definition of the endomorphisms coming from the CM-part of the Jacobian of \(X_{0}(p^2)\) and proving that there is no CM-point in \(X_{ns}(p)\) defined over \(\mathbb{Q}(\zeta_p)\).
The interest in nonsplit Cartan modular curves comes mainly from Serre’s uniformity conjecture. Which is equivalent to assert that for almost all \(p\) and for every maximal subgroup \(H\) of \(\text{GL}_2(\mathbb{F}_p)\), the modular curve \(X_H\) has no rational points except the cusps and the points associated to elliptic curves with complex multiplication. The only case left to prove concerns modular curves \(X^{+}_{ns}(p)\) associated to the normalizer of a nonsplit Cartan subgroup.
Reviewer: Elisa Lorenzo García (Rennes)
MSC:
| 14G35 | Modular and Shimura varieties |
| 11G05 | Elliptic curves over global fields |
| 11F06 | Structure of modular groups and generalizations; arithmetic groups |
| 11G15 | Complex multiplication and moduli of abelian varieties |
| 14G05 | Rational points |
| 14H37 | Automorphisms of curves |
| 14H52 | Elliptic curves |
Keywords:
modular curves; automorphism groups; nonsplit Cartan subgroups; modular automorphisms; Serre’s uniformity conjectureReferences:
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