Explicit Chabauty-Kim for the split Cartan modular curve of level 13. (English) Zbl 1469.14050
In his paper [J.-P. Serre, Invent. Math. 15, 259–331 (1972; Zbl 0235.14012)], Serre shows that for a fixed elliptic curve \(E\) over a number field that the representation \(\rho_{E,\ell}\) on the \(\ell\)-torsion points of \(E\) without complex multiplication is surjective for sufficiently large primes \(\ell\). He asks in the same paper whether there is a uniform bound to ensure surjectivity of \(\rho_{E,\ell}\) for all elliptic curves \(E\) over \(\mathbb{Q}\) without complex multiplication.
This question is still open and is equivalent to the determination of rational points on the modular curves classifying non-surjective \(\rho_{E,\ell}\) for sufficiently large \(\ell\). The case of a Borel subgroup was proven by [B. Mazur, Invent. Math. 44, 129–162 (1978; Zbl 0386.14009)] and provided a framework to determine and restrict rational points on modular curves with a rational cusp. The last remaining case, which is what prevents a resolution of Serre’s question, is the normalizer of a non-split Cartan subgroup.
The case of the normalizer of a split Cartan subgroup was recently settled by [Y. Bilu and P. Parent, Ann. Math. (2) 173, No. 1, 569–584 (2011; Zbl 1278.11065)] and [Y. Bilu et al., Ann. Inst. Fourier 63, No. 3, 957–984 (2013; Zbl 1307.11075)], except for the case \(\ell = 13\). Interestingly, it was shown in [B. Baran, J. Number Theory 145, 273–300 (2014; Zbl 1300.11055)] that the normalizer split Cartan and normalizer non-split Cartan modular curve of level \(13\) are isomorphic over \(\mathbb Q\), which provides an explanation for the unsuccessful application of the methods of [Y. Bilu et al., Ann. Inst. Fourier 63, No. 3, 957–984 (2013; Zbl 1307.11075)] for \(\ell = 13\), but also the possibility that the methods in the paper under review may lead to a resolution of Serre’s question.
Mazur’s method in [B. Mazur, Invent. Math. 44, 129–162 (1978; Zbl 0386.14009)] analyzes the rational points on a modular curve by embedding it into its Jacobian and then using precise information about the arithmetic of its Jacobian. A precondition is the need for the Jacobian to have a non-zero rank 0 quotient or for the Jacobian to have a quotient with rank less than its dimension [M. H. Baker, Proc. Am. Math. Soc. 127, No. 10, 2851–2856 (1999; Zbl 0931.11017)]. This fails in the case of the normalizer of a non-split Cartan subgroup, and in particular for \(X_{\text{s}}(13) \simeq X_{\text{ns}}(13)\).
In [M. Kim, Invent. Math. 161, No. 3, 629–656 (2005; Zbl 1090.14006)] and [M. Kim, Publ. Res. Inst. Math. Sci. 45, No. 1, 89–133 (2009; Zbl 1165.14020)], Kim initiated a program to generalize the Chabauty method beyond the natural barrier when the rank is less than the dimension. In this theory, the Jacobian variety is replaced by torsors for the maximal \(n\)-unipotent quotient of the \(\mathbb{Q}_p\)-étale fundamental group of the curve. It has been successfully applied to showing finiteness of the S-unit equation, both theoretically and computationally [I. Dan-Cohen and S. Wewers, Int. Math. Res. Not. 2016, No. 17, 5291–5354 (2016; Zbl 1404.11093)].
A special case of this approach which is more amenable to explicit computation, called quadratic Chabauty, was initiated in [J. S. Balakrishnan et al., Math. Comput. 86, No. 305, 1403–1434 (2017; Zbl 1376.11053)] and [J. S. Balakrishnan and N. Dogra, Duke Math. J. 167, No. 11, 1981–2038 (2018; Zbl 1401.14123)]. This approach is related to the case \(n = 2\) of Kim’s program. It has been shown in [S. Siksek, “Quadratic Chabauty for modular curves”, Prperint, arXiv:1704.00473] that for moduar curves of genus \(\ge 3\), the quadratic Chabauty method satisfies a necessary precondition needed to work beyond the classical Chabauty barrier, namely that the Néron-Severi rank of the Jacobian of the modular curve is \(\ge 2\).
The authors first show how to construct the quadratic Chabauty pairs needed for the quadratic Chabauty method using Nekovář’s theory of \(p\)-adic height functions on Selmer varieties [J. Nekovář, Prog. Math. 108, 127–202 (1993; Zbl 0859.11038)]. An explicit description of this \(p\)-adic height function is then derived using by solving explicit \(p\)-adic differential equations. Finally, to carry out the quadratic Chabauty argument, one uses a number of known rational points to solve for the explicit equations which give a finite \(p\)-adic set containing the rational points of the curve.
The authors successfully apply these methods to determine the rational points on \(X_{\text{s}}(13)\), the modular curve associated to the normalizer of a split Cartan subgroup, thus completing a resolution of Serre’s question in the case of the normalizer of a split Cartan subgroup.
This question is still open and is equivalent to the determination of rational points on the modular curves classifying non-surjective \(\rho_{E,\ell}\) for sufficiently large \(\ell\). The case of a Borel subgroup was proven by [B. Mazur, Invent. Math. 44, 129–162 (1978; Zbl 0386.14009)] and provided a framework to determine and restrict rational points on modular curves with a rational cusp. The last remaining case, which is what prevents a resolution of Serre’s question, is the normalizer of a non-split Cartan subgroup.
The case of the normalizer of a split Cartan subgroup was recently settled by [Y. Bilu and P. Parent, Ann. Math. (2) 173, No. 1, 569–584 (2011; Zbl 1278.11065)] and [Y. Bilu et al., Ann. Inst. Fourier 63, No. 3, 957–984 (2013; Zbl 1307.11075)], except for the case \(\ell = 13\). Interestingly, it was shown in [B. Baran, J. Number Theory 145, 273–300 (2014; Zbl 1300.11055)] that the normalizer split Cartan and normalizer non-split Cartan modular curve of level \(13\) are isomorphic over \(\mathbb Q\), which provides an explanation for the unsuccessful application of the methods of [Y. Bilu et al., Ann. Inst. Fourier 63, No. 3, 957–984 (2013; Zbl 1307.11075)] for \(\ell = 13\), but also the possibility that the methods in the paper under review may lead to a resolution of Serre’s question.
Mazur’s method in [B. Mazur, Invent. Math. 44, 129–162 (1978; Zbl 0386.14009)] analyzes the rational points on a modular curve by embedding it into its Jacobian and then using precise information about the arithmetic of its Jacobian. A precondition is the need for the Jacobian to have a non-zero rank 0 quotient or for the Jacobian to have a quotient with rank less than its dimension [M. H. Baker, Proc. Am. Math. Soc. 127, No. 10, 2851–2856 (1999; Zbl 0931.11017)]. This fails in the case of the normalizer of a non-split Cartan subgroup, and in particular for \(X_{\text{s}}(13) \simeq X_{\text{ns}}(13)\).
In [M. Kim, Invent. Math. 161, No. 3, 629–656 (2005; Zbl 1090.14006)] and [M. Kim, Publ. Res. Inst. Math. Sci. 45, No. 1, 89–133 (2009; Zbl 1165.14020)], Kim initiated a program to generalize the Chabauty method beyond the natural barrier when the rank is less than the dimension. In this theory, the Jacobian variety is replaced by torsors for the maximal \(n\)-unipotent quotient of the \(\mathbb{Q}_p\)-étale fundamental group of the curve. It has been successfully applied to showing finiteness of the S-unit equation, both theoretically and computationally [I. Dan-Cohen and S. Wewers, Int. Math. Res. Not. 2016, No. 17, 5291–5354 (2016; Zbl 1404.11093)].
A special case of this approach which is more amenable to explicit computation, called quadratic Chabauty, was initiated in [J. S. Balakrishnan et al., Math. Comput. 86, No. 305, 1403–1434 (2017; Zbl 1376.11053)] and [J. S. Balakrishnan and N. Dogra, Duke Math. J. 167, No. 11, 1981–2038 (2018; Zbl 1401.14123)]. This approach is related to the case \(n = 2\) of Kim’s program. It has been shown in [S. Siksek, “Quadratic Chabauty for modular curves”, Prperint, arXiv:1704.00473] that for moduar curves of genus \(\ge 3\), the quadratic Chabauty method satisfies a necessary precondition needed to work beyond the classical Chabauty barrier, namely that the Néron-Severi rank of the Jacobian of the modular curve is \(\ge 2\).
The authors first show how to construct the quadratic Chabauty pairs needed for the quadratic Chabauty method using Nekovář’s theory of \(p\)-adic height functions on Selmer varieties [J. Nekovář, Prog. Math. 108, 127–202 (1993; Zbl 0859.11038)]. An explicit description of this \(p\)-adic height function is then derived using by solving explicit \(p\)-adic differential equations. Finally, to carry out the quadratic Chabauty argument, one uses a number of known rational points to solve for the explicit equations which give a finite \(p\)-adic set containing the rational points of the curve.
The authors successfully apply these methods to determine the rational points on \(X_{\text{s}}(13)\), the modular curve associated to the normalizer of a split Cartan subgroup, thus completing a resolution of Serre’s question in the case of the normalizer of a split Cartan subgroup.
Reviewer: Imin Chen (Burnaby)
MSC:
| 14G05 | Rational points |
| 11Y50 | Computer solution of Diophantine equations |
| 11G50 | Heights |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
Citations:
Zbl 0235.14012; Zbl 0386.14009; Zbl 1278.11065; Zbl 1307.11075; Zbl 1300.11055; Zbl 0931.11017; Zbl 1090.14006; Zbl 1165.14020; Zbl 1404.11093; Zbl 1376.11053; Zbl 1401.14123; Zbl 0859.11038Software:
MagmaReferences:
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