Abstract
We show how the finiteness of integral points on an elliptic curve over $\mathbb{Q}$ with complex multiplication can be accounted for by the nonvanishing of $L$-functions that leads to bounds for dimensions of Selmer varieties.
-
[BK] S. Bloch and K. Kato, "$L$-functions and Tamagawa numbers of motives," in The Grothendieck Festschrift, Vol. I, Boston, MA: Birkhäuser, 1990, pp. 333-400.
@incollection {BK, MRKEY = {1086888},
AUTHOR = {Bloch, Spencer and Kato, Kazuya},
TITLE = {{$L$}-functions and {T}amagawa numbers of motives},
BOOKTITLE = {The {G}rothendieck {F}estschrift, {V}ol. {\rm I}},
SERIES = {Progr. Math.},
NUMBER = {86},
PAGES = {333--400},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1990},
MRCLASS = {11G40 (11G09 14C35 14F30 14G10)},
MRNUMBER = {92g:11063},
MRREVIEWER = {Ehud de Shalit},
ZBLNUMBER = {0768.14001},
} -
[deligne] P. Deligne, "Le groupe fondamental de la droite projective moins trois points," in Galois groups over ${\bf Q}$ (Berkeley, CA, 1987), New York: Springer-Verlag, 1989, pp. 79-297.
@incollection {deligne, MRKEY = {1012168},
AUTHOR = {Deligne, P.},
TITLE = {Le groupe fondamental de la droite projective moins trois points},
BOOKTITLE = {Galois groups over {${\bf Q}$} ({B}erkeley, {CA},
1987)},
SERIES = {Math. Sci. Res. Inst. Publ.},
NUMBER = {16},
PAGES = {79--297},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1989},
MRCLASS = {14G25 (11G35 11M06 11R70 14F35 19E99 19F27)},
MRNUMBER = {90m:14016},
MRREVIEWER = {James Milne},
ZBLNUMBER = {0742.14022},
} -
[kim1]
M. Kim, "The motivic fundamental group of $\Bbb P^1\setminus\{0,1,\infty\}$ and the theorem of Siegel," Invent. Math., vol. 161, iss. 3, pp. 629-656, 2005.@article {kim1, MRKEY = {2181717},
AUTHOR = {Kim, Minhyong},
TITLE = {The motivic fundamental group of {$\bold P^1\setminus\{0,1,\infty\}$} and the theorem of {S}iegel},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {161},
YEAR = {2005},
NUMBER = {3},
PAGES = {629--656},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {11G30 (11G55 14F30 14F42)},
MRNUMBER = {2006k:11119},
MRREVIEWER = {Tam{á}s Szamuely},
DOI = {10.1007/s00222-004-0433-9},
ZBLNUMBER = {1090.14006},
} -
[kim2] M. Kim, "The unipotent Albanese map and Selmer varieties for curves," Publ. Res. Inst. Math. Sci., vol. 45, iss. 1, pp. 89-133, 2009.
@article {kim2, MRKEY = {2512779},
AUTHOR = {Kim, Minhyong},
TITLE = {The unipotent {A}lbanese map and {S}elmer varieties for curves},
JOURNAL = {Publ. Res. Inst. Math. Sci.},
FJOURNAL = {Kyoto University. Research Institute for Mathematical Sciences. Publications},
VOLUME = {45},
YEAR = {2009},
NUMBER = {1},
PAGES = {89--133},
ISSN = {0034-5318},
CODEN = {KRMPBV},
MRCLASS = {14G05 (11G30)},
MRNUMBER = {2512779},
DOI = {10.2977/prims/1234361156},
ZBLNUMBER = {1165.14020},
} -
[KT] M. Kim and A. Tamagawa, "The $l$-component of the unipotent Albanese map," Math. Ann., vol. 340, iss. 1, pp. 223-235, 2008.
@article {KT, MRKEY = {2349775},
AUTHOR = {Kim, Minhyong and Tamagawa, Akio},
TITLE = {The {$l$}-component of the unipotent {A}lbanese map},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {340},
YEAR = {2008},
NUMBER = {1},
PAGES = {223--235},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {11G30 (11G25 11G55 14F30 14G20)},
MRNUMBER = {2009a:11132},
MRREVIEWER = {Dipendra Prasad},
DOI = {10.1007/s00208-007-0151-x},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {1126.14035},
} -
[rubin] K. Rubin, "On the main conjecture of Iwasawa theory for imaginary quadratic fields," Invent. Math., vol. 93, iss. 3, pp. 701-713, 1988.
@article {rubin, MRKEY = {952288},
AUTHOR = {Rubin, Karl},
TITLE = {On the main conjecture of {I}wasawa theory for imaginary quadratic fields},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {93},
YEAR = {1988},
NUMBER = {3},
PAGES = {701--713},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {11R23 (11G40)},
MRNUMBER = {89j:11105},
MRREVIEWER = {Leslie Jane Federer},
DOI = {10.1007/BF01410205},
ZBLNUMBER = {0673.12004},
} -
[serre] J. Serre, Lie algebras and Lie groups, Second ed., New York: Springer-Verlag, 1992.
@book {serre, MRKEY = {1176100},
AUTHOR = {Serre, Jean-Pierre},
TITLE = {Lie algebras and {L}ie groups},
SERIES = {Lecture Notes in Math.},
NUMBER = {1500},
EDITION = {Second},
NOTE = {1964 lectures given at Harvard University},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1992},
PAGES = {viii+168},
ISBN = {3-540-55008-9},
MRCLASS = {17-01 (17Bxx 22-01)},
MRNUMBER = {93h:17001},
ZBLNUMBER = {0742.17008},
}
Full Article