On component groups of Jacobians of Drinfeld modular curves

Mihran Papikian[1]

  • [1] Stanford University, Department of Mathematics, Stanford, CA 94305 (USA)

Annales de l'Institut Fourier (2004)

  • Volume: 54, Issue: 7, page 2163-2199
  • ISSN: 0373-0956

Abstract

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Let J 0 ( 𝔫 ) be the Jacobian variety of the Drinfeld modular curve X 0 ( 𝔫 ) over 𝔽 q ( t ) , where 𝔫 is an ideal in 𝔽 q [ t ] . Let 0 B J 0 ( 𝔫 ) A 0 be an exact sequence of abelian varieties. Assume B , as a subvariety of J 0 ( 𝔫 ) , is stable under the action of the Hecke algebra 𝕋 End ( J 0 ( 𝔫 ) ) . We give a criterion which is sufficient for the exactness of the induced sequence of component groups 0 Φ B , Φ J , Φ A , 0 of the Néron models of these abelian varieties over 𝔽 q [ [ 1 t ] ] . This criterion is always satisfied when either A or B is one-dimensional. Moreover, we prove that the sequence of component groups is always exact on -power torsion for any prime not dividing ( q - 1 ) . In particular, the sequence is always exact when q = 2 .

How to cite

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Papikian, Mihran. "On component groups of Jacobians of Drinfeld modular curves." Annales de l'Institut Fourier 54.7 (2004): 2163-2199. <http://eudml.org/doc/116171>.

@article{Papikian2004,
abstract = {Let $J_0(\{\mathfrak \{n\}\})$ be the Jacobian variety of the Drinfeld modular curve $X_0(\{\mathfrak \{n\}\})$ over $\{\mathbb \{F\}\}_q(t)$, where $\{\mathfrak \{n\}\}$ is an ideal in $\{\mathbb \{F\}\}_q[t]$. Let $0\rightarrow B \rightarrow J_0(\{\mathfrak \{n\}\}) \rightarrow A \rightarrow 0$ be an exact sequence of abelian varieties. Assume $B$, as a subvariety of $J_0(\{\mathfrak \{n\}\})$ , is stable under the action of the Hecke algebra $\{\mathbb \{T\}\} \subset $ End $ (J_0(\{\mathfrak \{n\}\}))$. We give a criterion which is sufficient for the exactness of the induced sequence of component groups $0 \rightarrow \Phi _\{B, \infty \} \rightarrow \Phi _\{J, \infty \} \rightarrow \Phi _\{A, \infty \} \rightarrow 0$ of the Néron models of these abelian varieties over $\{\mathbb \{F\}\}_q [\![\{1 \over t\}]\!]$. This criterion is always satisfied when either $A$ or $B$ is one-dimensional. Moreover, we prove that the sequence of component groups is always exact on $\ell $-power torsion for any prime $\ell $ not dividing $(q-1)$. In particular, the sequence is always exact when $q=2$.},
affiliation = {Stanford University, Department of Mathematics, Stanford, CA 94305 (USA)},
author = {Papikian, Mihran},
journal = {Annales de l'Institut Fourier},
keywords = {Component groups; Drinfeld modular curves; monodromy pairing; component groups},
language = {eng},
number = {7},
pages = {2163-2199},
publisher = {Association des Annales de l'Institut Fourier},
title = {On component groups of Jacobians of Drinfeld modular curves},
url = {http://eudml.org/doc/116171},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Papikian, Mihran
TI - On component groups of Jacobians of Drinfeld modular curves
JO - Annales de l'Institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 7
SP - 2163
EP - 2199
AB - Let $J_0({\mathfrak {n}})$ be the Jacobian variety of the Drinfeld modular curve $X_0({\mathfrak {n}})$ over ${\mathbb {F}}_q(t)$, where ${\mathfrak {n}}$ is an ideal in ${\mathbb {F}}_q[t]$. Let $0\rightarrow B \rightarrow J_0({\mathfrak {n}}) \rightarrow A \rightarrow 0$ be an exact sequence of abelian varieties. Assume $B$, as a subvariety of $J_0({\mathfrak {n}})$ , is stable under the action of the Hecke algebra ${\mathbb {T}} \subset $ End $ (J_0({\mathfrak {n}}))$. We give a criterion which is sufficient for the exactness of the induced sequence of component groups $0 \rightarrow \Phi _{B, \infty } \rightarrow \Phi _{J, \infty } \rightarrow \Phi _{A, \infty } \rightarrow 0$ of the Néron models of these abelian varieties over ${\mathbb {F}}_q [\![{1 \over t}]\!]$. This criterion is always satisfied when either $A$ or $B$ is one-dimensional. Moreover, we prove that the sequence of component groups is always exact on $\ell $-power torsion for any prime $\ell $ not dividing $(q-1)$. In particular, the sequence is always exact when $q=2$.
LA - eng
KW - Component groups; Drinfeld modular curves; monodromy pairing; component groups
UR - http://eudml.org/doc/116171
ER -

References

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  1. S. Bosch, W. Lütkebohmert, Degenerating abelian varieties, Topology 30 (1991), 653-698 Zbl0761.14015MR1133878
  2. S. Bosch, W. Lütkebohmert, Formal and rigid geometry I, Math. Ann. 295 (1993), 291-317 Zbl0808.14017MR1202394
  3. S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, (1990), Springer Zbl0705.14001MR1045822
  4. B. Conrad, Irreducible components of rigid spaces, Ann. Inst. Fourier 49 (1999), 473-541 Zbl0928.32011MR1697371
  5. B. Conrad, W. Stein, Component groups of purely toric quotients, Math. Research Letters 8 (2001), 745-766 Zbl1081.11040MR1879817
  6. P. Deligne, Formes modulaires et représentations de G L ( 2 ) , 349 (1973), 55-105, Springer Zbl0271.10032
  7. V. Drinfeld, Elliptic modules, Math. Sbornik 94 (1974), 594-627 Zbl0321.14014MR384707
  8. M. Emerton, Optimal quotients of modular Jacobians, Math. Ann. 327 (2003), 429-458 Zbl1061.11018MR2021024
  9. J. Fresnel, M. van der Put, Géométrie analytique rigide et applications, (1981), Birkhäuser Zbl0479.14015MR644799
  10. E.-U. Gekeler, Automorphe Formen über 𝔽 q ( T ) mit kleinem Führer, Abh. Math. Sem. Univ. Hamburg 55 (1985), 111-146 Zbl0564.10026MR831522
  11. E.-U. Gekeler, Über Drinfeld'sche Modulkurven vom Hecke-Typ, Comp. Math. 57 (1986), 219-236 Zbl0599.14032MR827352
  12. E.-U. Gekeler, Analytic construction of Weil curves over function fields, J. Th. nombres Bordeaux 7 (1995), 27-49 Zbl0846.11037MR1413565
  13. E.-U. Gekeler, Improper Eisenstein series on Bruhat-Tits trees, Manuscripta Math. 86 (1995), 367-391 Zbl0884.11025MR1323798
  14. E.-U. Gekeler, On the cuspidal divisor group of a Drinfeld modular curve, Doc. Math. J. DMV 2 (1997), 351-374 Zbl0895.11024MR1487469
  15. E.-U. Gekeler, U. Nonnengardt, Fundamental domains of some arithmetic groups over function fields, Internat. J. Math. 6 (1995), 689-708 Zbl0858.11025MR1351161
  16. E.-U. Gekeler, M. Reversat, Jacobians of Drinfeld modular curves, J. reine angew. Math. 476 (1996), 27-93 Zbl0848.11029MR1401696
  17. S. Gelbart, Automorphic forms on adele groups, (1975), Princeton Univ. Press Zbl0329.10018MR379375
  18. L. Gerritzen, M. van der Put, Schottky groups and Mumford curves, 817 (1980), Springer Zbl0442.14009MR590243
  19. A. Grothendieck, Groupes de type mulitplicatif: homomorphismes dans un schéma en groupes, SGA 3 exposé IX (1970) 
  20. A. Grothendieck, Modèles de Néron et monodromie, SGA 7 exposé IX (1972) Zbl0248.14006
  21. A. Grothendieck, J. Dieudonné, Étude cohomologique des faisceaux cohérents : EGA III, Publ. Math. IHÉS 11 (1962) Zbl0118.36206
  22. B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHÉS 47 (1977), 33-186 Zbl0394.14008MR488287
  23. D. Mumford, Abelian varieties, (1970), Oxford Univ. Press Zbl0326.14012MR282985
  24. D. Mumford, An analytic construction of degenerating curves over complete local rings, Comp. Math. 24 (1972), 129-174 Zbl0228.14011MR352105
  25. M. Reversat, Sur les revêtements de Schottky des courbes modulaires de Drinfeld, Arch. Math. 66 (1996), 378-387 Zbl0853.14014MR1383902
  26. K. Ribet, Letter to J.-F. Mestre, (1987) 
  27. K. Ribet, On the modular representations of G a l ( ¯ / ) arising from modular forms, Invent. Math. 100 (1990), 431-476 Zbl0773.11039MR1047143
  28. J-P. Serre, Trees, (1980), Springer Zbl0548.20018MR607504
  29. W. Stein, The refined Eisenstein conjecture, (1999) 
  30. A. Tamagawa, The Eisenstein quotient of the Jacobian variety of a Drinfeld modular curve, Publ. RIMS, Kyoto Univ. 31 (1995), 204-246 Zbl1045.11510MR1329480
  31. M. van der Put, A note on p -adic uniformization, Proc. Nederl. Akad. Wetensch. 90 (1987), 313-318 Zbl0624.32018MR914089
  32. M. van der Put, Discrete groups, Mumford curves and theta functions, Ann. Fac. Sci. Toulouse 1 (1992), 399-438 Zbl0789.14020MR1225666
  33. D. Zagier, Modular parametrizations of elliptic curves, Canad. Math. Bull. 28 (1985), 372-384 Zbl0579.14027MR790959
  34. L. Illusie, Réalisation -adique de l’accouplement de monodromie d’après A. Grothendieck, Astérisque 196-197 (1991), 27-44 Zbl0781.14011MR1141455
  35. A. Grothendieck, J. Dieudonné, Étude cohomologique des faisceaux cohérents : EGA III, Publ. Math., Inst. Hautes Étud. Sci. 17 (1963) Zbl0122.16102MR163911

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