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.


14G05 Rational points
11Y50 Computer solution of Diophantine equations
11G50 Heights
11G18 Arithmetic aspects of modular and Shimura varieties


Full Text: DOI arXiv


[1] Agashe, Amod; Stein, William, Visible evidence for the {B}irch and {S}winnerton-{D}yer conjecture for modular abelian varieties of analytic rank zero, Math. Comp.. Mathematics of Computation, 74, 455-484, (2005) · Zbl 1084.11033
[2] Baran, Burcu, An exceptional isomorphism between modular curves of level 13, J. Number Theory. Journal of Number Theory, 145, 273-300, (2014) · Zbl 1300.11055
[3] Baran, Burcu, An exceptional isomorphism between level 13 modular curves via {T}orelli’s theorem, Math. Res. Lett.. Mathematical Research Letters, 21, 919-936, (2014) · Zbl 1327.14120
[4] Balakrishnan, Jennifer S.; Besser, Amnon, Coleman-{G}ross height pairings and the {\(p\)}-adic sigma function, J. Reine Angew. Math.. Journal f\`“{u}r die Reine und Angewandte Mathematik. [Crelle”s Journal], 698, 89-104, (2015) · Zbl 1348.11091
[5] Balakrishnan, Jennifer S.; Besser, Amnon; M\`“{u}ller, J. Steffen, Quadratic {C}habauty: {\(p\)}-adic heights and integral points on hyperelliptic curves, J. Reine Angew. Math.. Journal f\`‘{u}r die Reine und Angewandte Mathematik. [Crelle”s Journal], 720, 51-79, (2016) · Zbl 1350.11067
[6] Balakrishnan, Jennifer S.; Besser, Amnon; M\"{u}ller, J. Steffen, Computing integral points on hyperelliptic curves using quadratic {C}habauty, Math. Comp.. Mathematics of Computation, 86, 1403-1434, (2017) · Zbl 1376.11053
[7] Bosma, Wieb; Cannon, John; Playoust, Catherine, The {M}agma algebra system. {I}. {T}he user language, J. Symbolic Comput.. Journal of Symbolic Computation, 24, 235-265, (1997) · Zbl 0898.68039
[8] Balakrishnan, Jennifer S.; Dogra, Netan, Quadratic {C}habauty and rational points, {I}: {\(p\)}-adic heights, Duke Math. J.. Duke Mathematical Journal, 167, 1981-2038, (2018) · Zbl 1401.14123
[9] Balakrishnan, Jennifer S.; Dogra, Netan, Quadratic {C}habauty and rational points, {II}: Generalised height functions on {S}elmer varieties, (2017) · Zbl 1401.14123
[10] Balakrishnan, Jennifer S.; Dan-Cohen, Ishai; Kim, Minhyong; Wewers, Stefan, A non-abelian conjecture of {T}ate-{S}hafarevich type for hyperbolic curves, Math. Ann.. Mathematische Annalen, 372, 369-428, (2018) · Zbl 1460.11038
[11] Balakrishnan, Jennifer S.; Dogra, N.; M\"uller, J. S.; Tuitman, J.; Vonk, J., Magma code
[12] P. Berthelot, Cohomologie rigide et cohomologie rigide \`{a} supports propres
[13] Besser, Amnon, Coleman integration using the {T}annakian formalism, Math. Ann.. Mathematische Annalen, 322, 19-48, (2002) · Zbl 1013.11028
[14] Besser, Amnon, The {\(p\)}-adic height pairings of {C}oleman-{G}ross and of {N}ekov\'{a}\v{r}. Number Theory, CRM Proc. Lecture Notes, 36, 13-25, (2004) · Zbl 1153.11316
[15] Betts, L. Alexander, The motivic anabelian geometry of local heights on abelian varieties, (2017)
[16] Bloch, Spencer; Kato, Kazuya, {\(L\)}-functions and {T}amagawa numbers of motives. The {G}rothendieck {F}estschrift, {V}ol. {I}, Progr. Math., 86, 333-400, (1990) · Zbl 0768.14001
[17] Birkenhake, Christina; Lange, Herbert, Complex Abelian Barieties, Grundlehren Math. Wissen., 302, xii+635 pp., (2004) · Zbl 1056.14063
[18] Berthelot, P.; Ogus, A., {\(F\)}-isocrystals and de {R}ham cohomology. {I}, Invent. Math.. Inventiones Mathematicae, 72, 159-199, (1983) · Zbl 0516.14017
[19] Bilu, Yuri; Parent, Pierre, Serre’s uniformity problem in the split {C}artan case, Ann. of Math. (2). Annals of Mathematics. Second Series, 173, 569-584, (2011) · Zbl 1278.11065
[20] Bilu, Yuri; Parent, Pierre; Rebolledo, Marusia, Rational points on {\(X^+_0(p^r)\)}, Ann. Inst. Fourier (Grenoble). Universit\'{e} de Grenoble. Annales de l’Institut Fourier, 63, 957-984, (2013) · Zbl 1307.11075
[21] Bruin, Nils; Poonen, Bjorn; Stoll, Michael, Generalized explicit descent and its application to curves of genus 3, Forum Math. Sigma. Forum of Mathematics. Sigma, 4, 6-80, (2016) · Zbl 1408.11065
[22] Bruin, Nils, Chabauty methods using elliptic curves, J. Reine Angew. Math.. Journal f\`“{u}r die Reine und Angewandte Mathematik. [Crelle”s Journal], 562, 27-49, (2003) · Zbl 1135.11320
[23] Bruin, Nils; Stoll, Michael, Two-cover descent on hyperelliptic curves, Math. Comp.. Mathematics of Computation, 78, 2347-2370, (2009) · Zbl 1208.11078
[24] Balakrishnan, Jennifer S.; Tuitman, Jan, Explicit {C}oleman integration for curves, (2017) · Zbl 1460.30015
[25] Coleman, Robert F.; Gross, Benedict H., {\(p\)}-adic heights on curves. Algebraic Number Theory, Adv. Stud. Pure Math., 17, 73-81, (1989) · Zbl 0758.14009
[26] Chabauty, Claude, Sur les points rationnels des courbes alg\'{e}briques de genre sup\'{e}rieur \`“a l”unit\'{e}, C. R. Acad. Sci. Paris, 212, 882-885, (1941) · Zbl 0025.24902
[27] Chen, Imin, The {J}acobians of non-split {C}artan modular curves, Proc. London Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 77, 1-38, (1998) · Zbl 0903.11019
[28] Coates, John; Kim, Minhyong, Selmer varieties for curves with {CM} {J}acobians, Kyoto J. Math.. Kyoto Journal of Mathematics, 50, 827-852, (2010) · Zbl 1283.11092
[29] Coleman, Robert F., Effective {C}habauty, Duke Math. J.. Duke Mathematical Journal, 52, 765-770, (1985) · Zbl 0588.14015
[30] Chiarellotto, Bruno; Le Stum, Bernard, {\(F\)}-isocristaux unipotents, Compositio Math.. Compositio Mathematica, 116, 81-110, (1999) · Zbl 0936.14017
[31] Chiarellotto, Bruno; Tsuzuki, Nobuo, Cohomological descent of rigid cohomology for \'{e}tale coverings, Rend. Sem. Mat. Univ. Padova. Rendiconti del Seminario Matematico della Universit\`a di Padova. The Mathematical Journal of the University of Padova, 109, 63-215, (2003) · Zbl 1167.14306
[32] Dan-Cohen, Ishai, Mixed {T}ate motives and the unit equation {II}, (2018) · Zbl 1404.11093
[33] Dan-Cohen, Ishai; Wewers, Stefan, Explicit {C}habauty-{K}im theory for the thrice punctured line in depth 2, Proc. Lond. Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 110, 133-171, (2015) · Zbl 1379.11068
[34] Dan-Cohen, Ishai; Wewers, Stefan, Mixed {T}ate motives and the unit equation, Int. Math. Res. Not. IMRN. International Mathematics Research Notices. IMRN, 5291-5354, (2016) · Zbl 1404.11093
[35] Darmon, Henri; Daub, Michael; Lichtenstein, Sam; Rotger, Victor, Algorithms for {C}how-{H}eegner points via iterated integrals, Math. Comp.. Mathematics of Computation, 84, 2505-2547, (2015) · Zbl 1378.11059
[36] Deligne, P., Le groupe fondamental de la droite projective moins trois points. Galois Groups over {\({\bf Q}\)}, Math. Sci. Res. Inst. Publ., 16, 79-297, (1989) · Zbl 0742.14022
[37] Deligne, P., Cat\'{e}gories tannakiennes. The {G}rothendieck {F}estschrift, {V}ol. {II}, Progr. Math., 87, 111-195, (1990)
[38] Darmon, Henri; Rotger, Victor; Sols, Ignacio, Iterated integrals, diagonal cycles and rational points on elliptic curves. Publications Math\'{e}matiques de {B}esan\c{c}on. {A}lg\`“ebre et Th\'”{e}orie des Nombres, 2012/2, Publ. Math. Besan\c{c}on Alg\`“ebre Th\'”{e}orie Nr., 2012/, 19-46, (2012) · Zbl 1332.11054
[39] Edixhoven, Bas, Stable models of modular curves and applications, (1989) · Zbl 0931.11021
[40] Edixhoven, Bas, Minimal resolution and stable reduction of {\(X_0(N)\)}, Ann. Inst. Fourier (Grenoble). Universit\'{e} de Grenoble. Annales de l’Institut Fourier, 40, 31-67, (1990) · Zbl 0679.14009
[41] Ellenberg, Jordan S.; Hast, Daniel Rayor, Rational points on solvable curves over {\( \mathbb{Q} \)} via non-abelian {C}habauty, (2017) · Zbl 1508.14022
[42] Faltings, Gerd, Crystalline cohomology and {\(p\)}-adic {G}alois-representations. Algebraic Analysis, Geometry, and Number Theory, 25-80, (1989) · Zbl 0805.14008
[43] Fontaine, Jean-Marc; Perrin-Riou, Bernadette, Autour des conjectures de {B}loch et {K}ato: cohomologie galoisienne et valeurs de fonctions {\(L\)}. Motives, Proc. Sympos. Pure Math., 55, 599-706, (1994) · Zbl 0821.14013
[44] Flynn, E. Victor; Wetherell, Joseph L., Finding rational points on bielliptic genus 2 curves, Manuscripta Math.. Manuscripta Mathematica, 100, 519-533, (1999) · Zbl 1029.11024
[45] Galbraith, Steven D., Rational points on {\(X^+_0(N)\)} and quadratic {\( \Bbb Q\)}-curves, J. Th\'{e}or. Nombres Bordeaux. Journal de Th\'{e}orie des Nombres de Bordeaux, 14, 205-219, (2002) · Zbl 1035.14008
[46] Grothendieck, A., Groupes de Monodromie en G\'eom\'etrie Alg\'ebrique, 288, (1972)
[47] Gross, Benedict H.; Zagier, Don B., Heegner points and derivatives of {\(L\)}-series, Invent. Math.. Inventiones Mathematicae, 84, 225-320, (1986) · Zbl 0608.14019
[48] Hadian, Majid, Motivic fundamental groups and integral points, Duke Math. J.. Duke Mathematical Journal, 160, 503-565, (2011) · Zbl 1234.14020
[49] Kim, Minhyong, The motivic fundamental group of {\( \bold P^1-\{0,1,\infty\}\)} and the theorem of {S}iegel, Invent. Math.. Inventiones Mathematicae, 161, 629-656, (2005) · Zbl 1090.14006
[50] Kim, Minhyong, The unipotent {A}lbanese map and {S}elmer varieties for curves, Publ. Res. Inst. Math. Sci.. Kyoto University. Research Institute for Mathematical Sciences. Publications, 45, 89-133, (2009) · Zbl 1165.14020
[51] Kolyvagin, V. A.; Logach\"{e}v, D. Yu., Finiteness of the {S}hafarevich-{T}ate group and the group of rational points for some modular abelian varieties, Algebra i Analiz. Algebra i Analiz, 1, 171-196, (1989)
[52] Katz, Nicholas M.; Mazur, Barry, Arithmetic Moduli of Elliptic Curves, Ann. of Math. Stud., 108, xiv+514 pp., (1985) · Zbl 0576.14026
[53] Kim, Minhyong; Tamagawa, Akio, The {\(l\)}-component of the unipotent {A}lbanese map, Math. Ann.. Mathematische Annalen, 340, 223-235, (2008) · Zbl 1126.14035
[54] Le Stum, Bernard, Rigid Cohomology, Cambridge Tracts in Math., 172, xvi+319 pp., (2007) · Zbl 1131.14001
[55] Lawrence, B.; Venkatesh, A., Diophantine problems and \(p\)-adic period mappings, (2018)
[56] Mazur, B., Modular curves and the {E}isenstein ideal, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 33-186, (1977) · Zbl 0394.14008
[57] Mazur, B., Rational isogenies of prime degree (with an appendix by {D}. {G}oldfeld), Invent. Math.. Inventiones Mathematicae, 44, 129-162, (1978) · Zbl 0386.14009
[58] Merel, Lo{\"{i}}c, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math.. Inventiones Mathematicae, 124, 437-449, (1996) · Zbl 0936.11037
[59] Milne, James S., \'{E}tale Cohomology, Princeton Math. Ser., 33, xiii+323 pp., (1980)
[60] Mumford, David, Abelian Varieties, Tata Inst. Fund. Res. Stud. Math., 5, viii+242 pp., (1970) · Zbl 0223.14022
[61] Nekov\'{a}\v{r}, Jan, On {\(p\)}-adic height pairings. S\'{e}minaire de {T}h\'{e}orie des {N}ombres, {P}aris, 1990-91, Progr. Math., 108, 127-202, (1993) · Zbl 1142.90021
[62] Olsson, Martin C., Towards non-abelian {\(p\)}-adic {H}odge theory in the good reduction case, Mem. Amer. Math. Soc.. Memoirs of the American Mathematical Society, 210, vi+157 pp., (2011) · Zbl 1213.14002
[63] Raynaud, Michel, {\(p\)}-groupes et r\'{e}duction semi-stable des courbes. The {G}rothendieck {F}estschrift, {V}ol. {III}, Progr. Math., 88, 179-197, (1990)
[64] Ribet, Kenneth A., Twists of modular forms and endomorphisms of abelian varieties, Math. Ann.. Mathematische Annalen, 253, 43-62, (1980) · Zbl 0421.14008
[65] Schoof, R., The {M}ordell-{W}eil group of a modular curve of level 13, (2012)
[66] Serre, {\relax J-P}, Propri\'{e}t\'{e}s galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math.. Inventiones Mathematicae, 15, 259-331, (1972) · Zbl 0235.14012
[67] Serre, {\relax J-P}, Lectures on the {M}ordell-{W}eil Theorem, Aspects Math., x+218 pp., (1997)
[68] translated from the French by Patrick Ion; revised by the author, Galois Cohomology, Springer Monogr. Math., x+210 pp., (2002)
[69] Shimura, Goro, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2). Annals of Mathematics. Second Series, 91, 144-222, (1970) · Zbl 0237.14009
[70] Siksek, S., Quadratic {C}habauty for modular curves, (2017)
[71] Tuitman, Jan, Counting points on curves using a map to {\( \bold{P}^1\)}, Math. Comp.. Mathematics of Computation, 85, 961-981, (2016) · Zbl 1402.11096
[72] Tuitman, Jan, Counting points on curves using a map to {\( \bold{P}^1\)}, {II}, Finite Fields Appl.. Finite Fields and their Applications, 45, 301-322, (2017) · Zbl 1402.11097
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.