Skip to main content Accessibility help

Gonality of abstract modular curves in positive characteristic

  • Anna Cadoret (a1) and Akio Tamagawa (a2)


Let $C$ be a smooth, separated and geometrically connected curve over a finitely generated field $k$ of characteristic $p\geqslant 0$ , $\unicode[STIX]{x1D702}$ the generic point of $C$ and $\unicode[STIX]{x1D70B}_{1}(C)$ its étale fundamental group. Let $f:X\rightarrow C$ be a smooth proper morphism, and $i\geqslant 0$ , $j$ integers. To the family of continuous $\mathbb{F}_{\ell }$ -linear representations $\unicode[STIX]{x1D70B}_{1}(C)\rightarrow \text{GL}(R^{i}f_{\ast }\mathbb{F}_{\ell }(j)_{\overline{\unicode[STIX]{x1D702}}})$ (where $\ell$ runs over primes $\neq p$ ), one can attach families of abstract modular curves $C_{0}(\ell )$ and $C_{1}(\ell )$ , which, in this setting, are the analogues of the usual modular curves $Y_{0}(\ell )$ and $Y_{1}(\ell )$ . If $i\not =2j$ , it is conjectured that the geometric and arithmetic gonalities of these abstract modular curves go to infinity with $\ell$ (for the geometric gonality, under a certain necessary condition). We prove the conjecture for the arithmetic gonality of the abstract modular curves $C_{1}(\ell )$ . We also obtain partial results for the growth of the geometric gonality of $C_{0}(\ell )$ and $C_{1}(\ell )$ . The common strategy underlying these results consists in reducing by specialization theory to the case where the base field $k$ is finite in order to apply techniques of counting rational points.



Hide All
[Abr96] Abramovich, D., A linear lower bound on the gonality of modular curves , Int. Math. Res. Not. IMRN 20 (1996), 10051011.
[ACG11] Arbarello, E., Cornalba, M. and Griffith, P. A., Geometry of algebraic curves: Vol. II, Grundlehren der mathematischen Wissenschaften, vol. 268 (Springer, 2011).
[Ber97] Berthelot, P., Altération des variétés algébriques (d’après A.J. de Jong) , in Séminaire Bourbaki 1995/1996, Vol. 815, Astérisque, vol. 241 (Société Mathématique de France, 1997), 273311.
[Cad12] Cadoret, A., Note on the gonality of abstract modular curves , in The arithmetic of fundamental groups – PIA 2010, MATCH–HGS Contributions in Mathematical and Computational Science, vol. 2, ed. Stix, J. (Springer, 2012), 89106.
[CT09] Cadoret, A. and Tamagawa, A., Torsion of abelian schemes and rational points on moduli spaces , in Algebraic number theory and related topics, RIMS Kôkyûroku Bessatsu, vol. B12, eds Asada, M., Nakamura, H. and Takahashi, H. (RIMS, Kyoto, 2009), 729.
[CT11] Cadoret, A. and Tamagawa, A., On a weak variant of the geometric torsion conjecture , J. Algebra 346 (2011), 227247.
[CT12a] Cadoret, A. and Tamagawa, A., Uniform boundedness of p-primary torsion of abelian schemes , Invent. Math. 188 (2012), 83125.
[CT12b] Cadoret, A. and Tamagawa, A., A uniform open image theorem for -adic representations I , Duke Math. J. 161 (2012), 26052634.
[CT13] Cadoret, A. and Tamagawa, A., A uniform open image theorem for -adic representations II , Duke Math. J. 162 (2013), 23012344.
[CT14a] Cadoret, A. and Tamagawa, A., On the geometric image of $\mathbb{F}_{\ell }$ -linear representations of étale fundamental groups, Preprint (2014).
[CT14b] Cadoret, A. and Tamagawa, A., Genus of abstract modular curves with level- $\ell$ structures, Preprint (2014).
[CKK15] Cornelissen, G., Kato, F. and Kool, J., A combinatorial Li–Yau inequality and rational points on curves , Math. Ann. 361 (2015), 211256.
[Del71] Deligne, P., Théorie de Hodge, II , Publ. Math. Inst. Hautes Études Sci. 40 (1971), 557.
[Del80] Deligne, P., La conjecture de Weil II , Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137252.
[EEHK09] Ellenberg, J., Elsholtz, C., Hall, C. and Kowalski, E., Non-simple abelian varieties in a family: algebraic and analytic approaches , J. Lond. Math. Soc. (2) 80 (2009), 135154.
[EGAIV] Dieudonné, J. and Grothendieck, A., Éléments de géométrie algébrique IV, Étude locale des schémas et des morphismes de schémas – troisième partie , Publ. Math. Inst. Hautes Études Sci. 28 (1966).
[EHK12] Ellenberg, J., Hall, C. and Kowalski, E., Expander graphs, gonality and variation of Galois representations , Duke Math. J. 161 (2012), 12331275.
[Fal91] Faltings, G., Diophantine approximation on abelian varieties , Ann. of Math. (2) 133 (1991), 549576.
[FW84] Faltings, G. and Wüstholz, G., Rational points, Aspects of Mathematics, E6 (Vieweg, 1984).
[Fre94] Frey, G., Curves with infinitely many points of finite degree , Israel J. Math. 85 (1994), 7983.
[Hal08] Hall, C., Big symplectic or orthogonal monodromy groups modulo , Duke Math. J. 141 (2008), 179203.
[Hru96] Hrushovsky, E., The Mordell–Lang conjecture for function fields , J. Amer. Math. Soc. 9 (1996), 667690.
[Kam92] Kamienny, S., Torsion points on elliptic curves over fields of higher degree , Int. Math. Res. Not. IMRN 6 (1992), 129133.
[LN59] Lang, S. and Néron, A., Rational points of abelian varieties over function fields , Amer. J. Math. 81 (1959), 95118.
[Maz77] Mazur, B., Modular curves and the Eisenstein ideal , Publ. Math. Inst. Hautes Études Sci. 47 (1977), 33186.
[Maz78] Mazur, B., Rational isogenies of prime degree (with an appendix by D. Goldfeld) , Invent. Math. 44 (1978), 129162.
[Mer96] Merel, L., Bornes pour la torsion des courbes elliptiques sur les corps de nombres , Invent. Math. 124 (1996), 437449.
[Mil86] Milne, J. S., Jacobian varieties , in Arithmetic geometry, eds Cornell, G. and Silverman, J. H. (Springer, 1986), 167212.
[Mom95] Momose, F., Isogenies of prime degree over number fields , Compositio Math. 97 (1995), 329348.
[Org13] Orgogozo, F., Sur les propriétés d’uniformité des images directes en cohomologie étale, Preprint (2013).
[OV00] Orgogozo, F. and Vidal, I., Le théorème de spécialisation du groupe fondamental , in Courbes semi-stables et groupe fondamental en géométrie algébrique, Progress in Mathematics, vol. 187, eds Bost, J.-B., Loeser, F. and Raynaud, M. (Birkhäuser, 2000), 169184.
[Pin98] Pink, R., l-adic algebraic monodromy groups, cocharacters, and the Mumford–Tate conjecture , J. Reine Angew. Math. 495 (1998), 187237.
[PP07] Poonen, B., Gonality of modular curves in characteristic p , Math. Res. Lett. 14 (2007), 691701.
[RZ82] Rapoport, M. and Zink, T., Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik , Invent. Math. 68 (1982), 21101.
[Sch03] Schweizer, A., On the uniform boundedness conjecture for Drinfeld modules , Math. Z. 244 (2003), 601614.
[Ser68] Serre, J.-P., Corps locaux (Hermann, 1968).
[SGA1] Grothendieck, A. et al. , Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Mathematics, vol. 224 (Springer, 1971).
[SGA3] Grothendieck, A. et al. , Schémas en groupes II (SGA 3 – II), Lecture Notes in Mathematics, vol. 152 (Springer, 1962–1964).
[SGA4] Grothendieck, A. et al. , Théorie des topos et cohomologie étale des schémas (SGA 4), tome 3, Lecture Notes in Mathematics, vol. 305 (Springer, 1973).
[Szp85] L. Szpiro (ed.), Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, Astérisque, vol. 127 (Société Mathématique de France, 1985).
[Tam02] Tamagawa, A., Fundamental groups and geometry of curves in positive characteristic , in Arithmetic fundamental groups and noncommutative algebra, Proceedings of Symposia in Pure Mathematics, vol. 70, eds Fried, M. D. and Ihara, Y. (American Mathematical Society, Providence, RI, 2002), 297333.
[Tat66] Tate, J., Endomorphisms of abelian varieties over finite fields , Invent. Math. 2 (1966), 134144.
[Zar77] Zarhin, Ju. G., Endomorphisms of abelian varieties and points of finite order in characteristic p , Mat. Zametki 21 (1977), 737744 (in Russian).
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

Gonality of abstract modular curves in positive characteristic

  • Anna Cadoret (a1) and Akio Tamagawa (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed