Hostname: page-component-6bf8c574d5-pdxrj Total loading time: 0 Render date: 2025-03-09T02:19:08.707Z Has data issue: false hasContentIssue false
  • English
  • Français

The Manin constant in the semistable case

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  13 August 2018

Kęstutis Česnavičius
Kęstutis Česnavičius*
Affiliation:
Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France email kestutis@math.u-psud.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

For an optimal modular parametrization $J_{0}(n){\twoheadrightarrow}E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, Manin conjectured the agreement of two natural $\mathbb{Z}$-lattices in the $\mathbb{Q}$-vector space $H^{0}(E,\unicode[STIX]{x1D6FA}^{1})$. Multiple authors generalized his conjecture to higher-dimensional newform quotients. We prove the Manin conjecture for semistable $E$, give counterexamples to all the proposed generalizations, and prove several semistable special cases of these generalizations. The proofs establish general relations between the integral $p$-adic étale and de Rham cohomologies of abelian varieties over $p$-adic fields and exhibit a new exactness result for Néron models.

Keywords

elliptic curveHecke algebraManin constantmodular curveNéron modelnewform

MSC classification

Primary: 11G05: Elliptic curves over global fields 11G10: Abelian varieties of dimension $> 1$
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties 11F11: Holomorphic modular forms of integral weight 14G35: Modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 9 , September 2018 , pp. 1889 - 1920
Copyright
© The Author 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbes, A. and Ullmo, E., A propos de la conjecture de Manin pour les courbes elliptiques modulaires , Compositio Math. 103 (1996), 269–286; MR 1414591.Google Scholar
Agashe, A., Ribet, K. and Stein, W. A., The Manin constant , Pure Appl. Math. Q. 2 (2006), 617–636, doi:10.4310/PAMQ.2006.v2.n2.a11; MR 2251484.Google Scholar
Agashe, A., Ribet, K. A. and Stein, W. A., The modular degree, congruence primes, and multiplicity one , in Number theory, analysis and geometry (Springer, New York, NY, 2012), 19–49; MR 2867910.Google Scholar
Bégueri, L., Dualité sur un corps local à corps résiduel algébriquement clos , Mém. Soc. Math. Fr. (N.S.) 4 (1980), 121 pp; MR 615883 (82k:12019).Google Scholar
Bhatt, B., Morrow, M. and Scholze, P., Integral  $p$ -adic Hodge theory, Preprint (2016),arXiv:1602.03148.Google Scholar
Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21 (Springer, Berlin, 1990); MR 1045822 (91i:14034).Google Scholar
Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system. I. The user language , J. Symbolic Comput. 24 (1997), 235–265, doi:10.1006/jsco.1996.0125.Google Scholar
Česnavičius, K., The Manin–Stevens constant in the semistable case, Preprint (2016),arXiv:1604.02165.Google Scholar
Česnavičius, K., A modular description of X0(n) , Algebra Number Theory 11 (2017), 2001–2089; MR 3735461.Google Scholar
Cojocaru, A. C. and Kani, E., The modular degree and the congruence number of a weight 2 cusp form , Acta Arith. 114 (2004), 159–167, doi:10.4064/aa114-2-5; MR 2068855.Google Scholar
Conrad, B., Grothendieck duality and base change, Lecture Notes in Mathematics, vol. 1750 (Springer, Berlin, 2000); MR 1804902.Google Scholar
Conrad, B., Chow’s K/k-image and K/k-trace, and the Lang–Néron theorem , Enseign. Math. (2) 52 (2006), 37–108; MR 2255529.Google Scholar
Conrad, B., Edixhoven, B. and Stein, W., J 1(p) has connected fibers , Doc. Math. 8 (2003), 331–408; MR 2029169.Google Scholar
Cremona, J., Manin constants and optimal curves: conductors $60000$ - $400000$ ,https://raw.githubusercontent.com/JohnCremona/ecdata/master/doc/manin.txt (accessed on December 28, 2016).Google Scholar
Darmon, H., Diamond, F. and Taylor, R., Fermat’s last theorem , in Elliptic curves, modular forms & Fermat’s last theorem, Hong Kong, 1993 (International Press, Cambridge, MA, 1997), 2–140; MR 1605752.Google Scholar
Deligne, P. and Rapoport, M., Les schémas de modules de courbes elliptiques , in Modular functions of one variable, II, Proc. int. summer school, Antwerp, 1972, Lecture Notes in Mathematics, vol. 349 (Springer, Berlin, 1973), 143–316; MR 0337993 (49 #2762).Google Scholar
Edixhoven, B., On the Manin constants of modular elliptic curves , in Arithmetic algebraic geometry, Texel, 1989, Progress in Mathematics, vol. 89 (Birkhäuser Boston, Boston, MA, 1991), 25–39; MR 1085254.Google Scholar
Edixhoven, B., The weight in Serre’s conjectures on modular forms , Invent. Math. 109 (1992), 563–594, doi:10.1007/BF01232041; MR 1176206.Google Scholar
Edixhoven, B., Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one , J. Inst. Math. Jussieu 5 (2006), 1–34; MR 2195943.Google Scholar
Flynn, E. V., Leprévost, F., Schaefer, E. F., Stein, W. A., Stoll, M. and Wetherell, J. L., Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves , Math. Comp. 70 (2001), 1675–1697, doi:10.1090/S0025-5718-01-01320-5;MR 1836926.Google Scholar
Greenberg, R. and Vatsal, V., On the Iwasawa invariants of elliptic curves , Invent. Math. 142 (2000), 17–63, doi:10.1007/s002220000080; MR 1784796.Google Scholar
Gross, B. H., A tameness criterion for Galois representations associated to modular forms (mod p) , Duke Math. J. 61 (1990), 445–517, doi:10.1215/S0012-7094-90-06119-8;MR 1074305.Google Scholar
Illusie, L., Grothendieck’s existence theorem in formal geometry , in Fundamental algebraic geometry, Mathematical Surveys and Monographs, vol. 123 (American Mathematical Society, Providence, RI, 2005), 179–233; MR 2223409.Google Scholar
Joyce, A., The Manin constant of an optimal quotient of J 0(431) , J. Number Theory 110 (2005), 325–330, doi:10.1016/j.jnt.2004.07.008; MR 2122612.Google Scholar
Kilford, L. J. P., Some non-Gorenstein Hecke algebras attached to spaces of modular forms , J. Number Theory 97 (2002), 157–164, doi:10.1006/jnth.2002.2803; MR 1939142.Google Scholar
Kilford, L. J. P. and Wiese, G., On the failure of the Gorenstein property for Hecke algebras of prime weight , Exp. Math. 17 (2008), 37–52; MR 2410114.Google Scholar
Knudsen, F. F. and Mumford, D., The projectivity of the moduli space of stable curves. I. Preliminaries on ‘det’ and ‘Div’ , Math. Scand. 39 (1976), 19–55; MR 0437541.Google Scholar
Ling, S. and Oesterlé, J., The Shimura subgroup of J 0(N) , in Courbes modulaires et courbes de Shimura, Orsay, 1987/1988, Astérisque, vol. 196–197 (Société Mathématique de France, 1991), 171–203; MR 1141458.Google Scholar
Liu, Q., Lorenzini, D. and Raynaud, M., Néron models, Lie algebras, and reduction of curves of genus one , Invent. Math. 157 (2004), 455–518, doi:10.1007/s00222-004-0342-y;MR 2092767.Google Scholar
The LMFDB Collaboration, The L-functions and Modular Forms Database,http://www.lmfdb.org(accessed on November 27, 2016).Google Scholar
Manin, J. I., Cyclotomic fields and modular curves , Uspekhi Mat. Nauk 26 (1971), 7–71; MR 0401653.Google Scholar
Mazur, B., Modular curves and the Eisenstein ideal , Publ. Math. Inst. Hautes Études Sci. 47 (1977), 33–186; MR 488287.Google Scholar
Mazur, B., Rational isogenies of prime degree , Invent. Math. 44 (1978), 129–162; doi:10.1007/BF01390348; MR 482230.Google Scholar
Mazur, B. and Messing, W., Universal extensions and one dimensional crystalline cohomology, Lecture Notes in Mathematics, vol. 370 (Springer, New York, NY, 1974); MR 0374150.Google Scholar
Mazur, B. and Ribet, K. A., Two-dimensional representations in the arithmetic of modular curves , in Courbes modulaires et courbes de Shimura, Orsay, 1987/1988, Astérisque, vol. 196-197 (Société Mathématique de France, 1991), 215–255; MR 1141460.Google Scholar
Mazur, B. and Wiles, A., Class fields of abelian extensions of Q , Invent. Math. 76 (1984), 179–330, doi:10.1007/BF01388599; MR 742853.Google Scholar
Oda, T., The first de Rham cohomology group and Dieudonné modules , Ann. Sci. Éc. Norm. Supér. (4) 2 (1969), 63–135; MR 0241435 (39 #2775).Google Scholar
Parent, P., Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres , J. Reine Angew. Math. 506 (1999), 85–116, doi:10.1515/crll.1999.009; MR 1665681.Google Scholar
Polishchuk, A., Abelian varieties, theta functions and the Fourier transform, Cambridge Tracts in Mathematics, vol. 153 (Cambridge University Press, Cambridge, 2003); MR 1987784.Google Scholar
Raynaud, M., Hauteurs et isogénies , in Seminar on arithmetic bundles: the Mordell conjecture, Paris, 1983/84, Astérisque, vol. 127 (Société Mathématique de France, 1985), 199–234; MR 801923.Google Scholar
Ribet, K. A., Congruence relations between modular forms , in Proc. int. congress of mathematicians, vols 1 and 2, Warsaw, 1983 (PWN, Warsaw, 1984), 503–514; MR 804706.Google Scholar
Ribet, K. A., On modular representations of Gal( Q /Q) arising from modular forms , Invent. Math. 100 (1990), 431–476, doi:10.1007/BF01231195; MR 1047143.Google Scholar
Ribet, K. A. and Stein, W. A., Lectures on Serre’s conjectures , in Arithmetic algebraic geometry, Park City, UT, 1999, IAS/Park City Mathematics Series, vol. 9 (American Mathematical Society, Providence, RI, 2001), 143–232; MR 1860042.Google Scholar
The Sage Developers, SageMath, the Sage Mathematics Software System (Version 7.4),http://www.sagemath.org(2016).Google Scholar
Gille, P. and Polo, P. (eds), Schémas en groupes (SGA 3), Volume I, Propriétés générales des schémas en groupes, Documents Mathématiques (Paris), vol. 7 (Société Mathématique de France, Paris, 2011); revised and annotated edition of the 1970 original; MR 2867621.Google Scholar
Stein, W. A., The first few nonmaximal orders attached to weight two newforms on $\unicode[STIX]{x1D6E4}_{0}(N)$ , Preprint (1999), http://wstein.org/Tables/nonmaximal.dvi (accessed on December 28, 2016).Google Scholar
Stevens, G., Stickelberger elements and modular parametrizations of elliptic curves , Invent. Math. 98 (1989), 75–106, doi:10.1007/BF01388845; MR 1010156.Google Scholar
Ullmo, E., Hauteur de Faltings de quotients de J 0(N), discriminants d’algèbres de Hecke et congruences entre formes modulaires , Amer. J. Math. 122 (2000), 83–115; MR 1737258.Google Scholar
Wiles, A., Modular curves and the class group of Q(𝜁 p ) , Invent. Math. 58 (1980), 1–35, doi:10.1007/BF01402272; MR 570872.Google Scholar
Wiles, A., Modular elliptic curves and Fermat’s last theorem , Ann. of Math. (2) 141 (1995), 443–551, doi:10.2307/2118559; MR 1333035 (96d:11071).Google Scholar
Yoo, H., The index of an Eisenstein ideal and multiplicity one , Math. Z. 282 (2016), 1097–1116, doi:10.1007/s00209-015-1579-4; MR 3473658.Google Scholar