Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Arithmetic of abelian varieties with constrained torsion


Authors: Christopher Rasmussen and Akio Tamagawa
Journal: Trans. Amer. Math. Soc. 369 (2017), 2395-2424
MSC (2010): Primary 11G10; Secondary 11F80, 14K15
DOI: https://doi.org/10.1090/tran/6790
Published electronically: July 15, 2016
MathSciNet review: 3592515
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over $ K$ whose $ \ell $-power torsion fields are arithmetically constrained for some rational prime $ \ell $. Such arithmetic constraints are related to an unresolved question of Ihara regarding the kernel of the canonical outer Galois representation on the pro-$ \ell $ fundamental group of $ \mathbb{P}^1-\{0,1,\infty \}$.

Under GRH, we demonstrate the set of classes is finite for any fixed $ K$ and any fixed dimension. Without GRH, we prove a semistable version of the result. In addition, several unconditional results are obtained when the degree of $ K/\mathbb{Q}$ and the dimension of abelian varieties are not too large through a careful analysis of the special fiber of such abelian varieties. In some cases, the results (viewed as a bound on the possible values of $ \ell $) are uniform in the degree of the extension $ K/\mathbb{Q}$.


References [Enhancements On Off] (What's this?)

  • [AI88] Greg Anderson and Yasutaka Ihara, Pro-$ l$ branched coverings of $ {\bf P}^1$ and higher circular $ l$-units, Ann. of Math. (2) 128 (1988), no. 2, 271-293. MR 960948 (89f:14023), https://doi.org/10.2307/1971443
  • [Bro12] Francis Brown, Mixed Tate motives over $ \mathbb{Z}$, Ann. of Math. (2) 175 (2012), no. 2, 949-976. MR 2993755, https://doi.org/10.4007/annals.2012.175.2.10
  • [CGH$^+$96] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert $ W$ function, Adv. Comput. Math. 5 (1996), no. 4, 329-359. MR 1414285 (98j:33015), https://doi.org/10.1007/BF02124750
  • [Coh77] Paul Moritz Cohn, Algebra. Vol. 2, John Wiley & Sons, London-New York-Sydney, 1977. With errata to Vol. I. MR 0530404 (58 #26625)
  • [Ell71] P. D. T. A. Elliott, The least prime $ k-{\rm th}$-power residue, J. London Math. Soc. (2) 3 (1971), 205-210. MR 0281686 (43 #7401)
  • [Fal83] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349-366 (German). MR 718935 (85g:11026a), https://doi.org/10.1007/BF01388432
  • [GL06] Robert M. Guralnick and Martin Lorenz, Orders of finite groups of matrices, Groups, rings and algebras, Contemp. Math., vol. 420, Amer. Math. Soc., Providence, RI, 2006, pp. 141-161. MR 2279238 (2008e:20007), https://doi.org/10.1090/conm/420/07974
  • [HB92] D. R. Heath-Brown, Zero-free regions for Dirichlet $ L$-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3) 64 (1992), no. 2, 265-338. MR 1143227 (93a:11075), https://doi.org/10.1112/plms/s3-64.2.265
  • [Iha86] Yasutaka Ihara, Profinite braid groups, Galois representations and complex multiplications, Ann. of Math. (2) 123 (1986), no. 1, 43-106. MR 825839 (87c:11055), https://doi.org/10.2307/1971352
  • [Lan94] Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723 (95f:11085)
  • [LO77] J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic number fields: $ L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London, 1977, pp. 409-464. MR 0447191 (56:5506)
  • [Maz78] B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129-162. MR 482230 (80h:14022), https://doi.org/10.1007/BF01390348
  • [Mil86] J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 103-150. MR 861974
  • [Min87] Hermann Minkowski, Zur Theorie der positiven quadratischen Formen, J. Reine Angew. Math. 101 (1887), 196-202 (German). MR 1580123, https://doi.org/10.1515/crll.1887.101.196
  • [Mom95] Fumiyuki Momose, Isogenies of prime degree over number fields, Compositio Math. 97 (1995), no. 3, 329-348. MR 1353278 (97f:11039)
  • [Mum70] David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970. MR 0282985 (44 #219)
  • [PR07] Matthew Papanikolas and Christopher Rasmussen, On the torsion of Jacobians of principal modular curves of level $ 3^n$, Arch. Math. (Basel) 88 (2007), no. 1, 19-28. MR 2289596 (2007k:11094), https://doi.org/10.1007/s00013-006-1740-8
  • [Ras04] C. Rasmussen, On the fields of 2-power torsion of certain elliptic curves, Math. Res. Lett. 11 (2004), no. 4, 529-538. MR 2092905 (2005f:11114), https://doi.org/10.4310/MRL.2004.v11.n4.a10
  • [Ras09] C. Rasmussen, On elliptic curves of conductor $ 11^2$ and an open question of Ihara, Algebraic number theory and related topics 2007, RIMS Kôkyûroku Bessatsu, B12, Res. Inst. Math. Sci. (RIMS), Kyoto, 2009, pp. 101-113. MR 2605776 (2012a:14056)
  • [Ray74] Michel Raynaud, Schémas en groupes de type $ (p,\dots , p)$, Bull. Soc. Math. France 102 (1974), 241-280 (French). MR 0419467 (54 #7488)
  • [RT08] Christopher Rasmussen and Akio Tamagawa, A finiteness conjecture on abelian varieties with constrained prime power torsion, Math. Res. Lett. 15 (2008), no. 6, 1223-1231. MR 2470396 (2009k:11101), https://doi.org/10.4310/MRL.2008.v15.n6.a12
  • [Ser72] Jean-Pierre Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259-331 (French). MR 0387283 (52 #8126)
  • [Ser79] J.-P. Serre, Arithmetic groups, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 105-136. MR 564421 (82b:22021)
  • [SGA72] Groupes de monodromie en géométrie algébrique. I, Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (SGA 7 I); Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim, Lecture Notes in Mathematics, Vol. 288, Springer-Verlag, Berlin-New York, 1972 (French). MR 0354656 (50 #7134)
  • [Sha02] Romyar T. Sharifi, Relationships between conjectures on the structure of pro-$ p$ Galois groups unramified outside $ p$, Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999), Proc. Sympos. Pure Math., vol. 70, Amer. Math. Soc., Providence, RI, 2002, pp. 275-284. MR 1935409 (2004c:11204), https://doi.org/10.1090/pspum/070/1935409
  • [ST68] Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492-517. MR 0236190 (38 #4488)
  • [SZ98] A. Silverberg and Yu. G. Zarhin, Subgroups of inertia groups arising from abelian varieties, J. Algebra 209 (1998), no. 1, 94-107. MR 1652189 (99j:11069), https://doi.org/10.1006/jabr.1998.7523
  • [Tam95] Akio Tamagawa, The Eisenstein quotient of the Jacobian variety of a Drinfel'd modular curve, Publ. Res. Inst. Math. Sci. 31 (1995), no. 2, 203-246. MR 1329480 (96b:11077), https://doi.org/10.2977/prims/1195164439
  • [TO70] John Tate and Frans Oort, Group schemes of prime order, Ann. Sci. École Norm. Sup. (4) 3 (1970), 1-21. MR 0265368 (42 #278)
  • [Wei48] André Weil, Variétés abéliennes et courbes algébriques, Actualités Sci. Ind., no. 1064 = Publ. Inst. Math. Univ. Strasbourg 8 (1946), Hermann & Cie., Paris, 1948 (French). MR 0029522 (10,621d)
  • [Zar85] Yu. G. Zarhin, A finiteness theorem for unpolarized abelian varieties over number fields with prescribed places of bad reduction, Invent. Math. 79 (1985), no. 2, 309-321. MR 778130 (86d:14041), https://doi.org/10.1007/BF01388976

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11G10, 11F80, 14K15

Retrieve articles in all journals with MSC (2010): 11G10, 11F80, 14K15


Additional Information

Christopher Rasmussen
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
Email: crasmussen@wesleyan.edu

Akio Tamagawa
Affiliation: Research Institute for Mathematical Sciences, Kyoto 606-8502, Japan
Email: tamagawa@kurims.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/tran/6790
Received by editor(s): October 30, 2013
Received by editor(s) in revised form: April 3, 2015
Published electronically: July 15, 2016
Additional Notes: The first author was partially supported by JSPS kakenhi Grant Number $19 ⋅07028$.
The second author was supported by JSPS kakenhi Grant Numbers 22340006, 15H03609.
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society