On the Birch-Swinnerton-Dyer quotients modulo squares

Abstract

Let $A$ be an abelian variety over a number field $K$. An identity between the $L$-functions $L(A/K_i,s)$ for extensions $K_i$ of $K$ induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of Ш, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of $A$ over extensions of $K$. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of $Ш(E/K(E[2]))[6^\infty]$ and some restrictions on the reduction at primes above 2 and 3: the parity of the Mordell-Weil rank of $E/K$ agrees with the parity of the analytic rank, as determined by the root number. We also prove the $p$-parity conjecture for all elliptic curves over $\mathbb{Q}$ and all primes $p$: the parities of the $p^\infty$-Selmer rank and the analytic rank agree.

  • [BD] Go to document M. Bertolini and H. Darmon, "Iwasawa’s main conjecture for elliptic curves over anticyclotomic $\Bbb Z_p$-extensions," Ann. of Math., vol. 162, iss. 1, pp. 1-64, 2005.
    @article {BD, MRKEY = {2178960},
      AUTHOR = {Bertolini, M. and Darmon, H.},
      TITLE = {Iwasawa's main conjecture for elliptic curves over anticyclotomic {$\Bbb Z_p$}-extensions},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {162},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {1--64},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11R23 (11G05)},
      MRNUMBER = {2006g:11218},
      MRREVIEWER = {Benjamin V. Howard},
      DOI = {10.4007/annals.2005.162.1},
      ZBLNUMBER={1093.11037},
      }
  • [Bir] B. J. Birch, "Conjectures concerning elliptic curves," in Proc. Sympos. Pure Math., Vol. VIII, Providence, R.I.: Amer. Math. Soc., 1965, pp. 106-112.
    @incollection {Bir, MRKEY = {0174558},
      AUTHOR = {Birch, B. J.},
      TITLE = {Conjectures concerning elliptic curves},
      BOOKTITLE = {Proc. {S}ympos. {P}ure {M}ath., {V}ol. {\rm VIII}},
      PAGES = {106--112},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1965},
      MRCLASS = {14.49 (14.40)},
      MRNUMBER = {30 \#4759},
      MRREVIEWER = {J. W. S. Cassels},
      ZBLNUMBER = {0238.14011},
      }
  • [BS] Go to document B. J. Birch and N. M. Stephens, "The parity of the rank of the Mordell-Weil group," Topology, vol. 5, pp. 295-299, 1966.
    @article {BS, MRKEY = {0201379},
      AUTHOR = {Birch, B. J. and Stephens, N. M.},
      TITLE = {The parity of the rank of the {M}ordell-{W}eil group},
      JOURNAL = {Topology},
      FJOURNAL = {Topology. An International Journal of Mathematics},
      VOLUME = {5},
      YEAR = {1966},
      PAGES = {295--299},
      ISSN = {0040-9383},
      MRCLASS = {14.40 (10.12)},
      MRNUMBER = {34 \#1263},
      MRREVIEWER = {J. W. S. Cassels},
      DOI = {10.1016/0040-9383(66)90021-8},
      ZBLNUMBER = {0146.42401},
      }
  • [BFH] Go to document D. Bump, S. Friedberg, and J. Hoffstein, "Nonvanishing theorems for $L$-functions of modular forms and their derivatives," Invent. Math., vol. 102, iss. 3, pp. 543-618, 1990.
    @article {BFH, MRKEY = {1074487},
      AUTHOR = {Bump, Daniel and Friedberg, Solomon and Hoffstein, Jeffrey},
      TITLE = {Nonvanishing theorems for {$L$}-functions of modular forms and their derivatives},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {102},
      YEAR = {1990},
      NUMBER = {3},
      PAGES = {543--618},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F66 (11F67 11G05 11G40)},
      MRNUMBER = {92a:11058},
      MRREVIEWER = {Alexey A. Panchishkin},
      DOI = {10.1007/BF01233440},
      ZBLNUMBER = {0721.11023},
      }
  • [CasIV] Go to document J. W. S. Cassels, "Arithmetic on curves of genus $1$. IV. Proof of the hauptvermutung," J. Reine Angew. Math., vol. 211, pp. 95-112, 1962.
    @article {CasIV, MRKEY = {0163915},
      AUTHOR = {Cassels, J. W. S.},
      TITLE = {Arithmetic on curves of genus {$1$}. {IV}. {P}roof of the hauptvermutung},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {211},
      YEAR = {1962},
      PAGES = {95--112},
      ISSN = {0075-4102},
      MRCLASS = {14.49 (12.40)},
      MRNUMBER = {29 \#1214},
      MRREVIEWER = {P. Abellanas},
      DOI = {10.1515/crll.1962.211.95},
      ZBLNUMBER={0106.03706},
      }
  • [CasVIII] Go to document J. W. S. Cassels, "Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer," J. Reine Angew. Math., vol. 217, pp. 180-199, 1965.
    @article {CasVIII, MRKEY = {0179169},
      AUTHOR = {Cassels, J. W. S.},
      TITLE = {Arithmetic on curves of genus 1. {VIII}. {O}n conjectures of {B}irch and {S}winnerton-{D}yer},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {217},
      YEAR = {1965},
      PAGES = {180--199},
      ISSN = {0075-4102},
      MRCLASS = {14.48 (14.49)},
      MRNUMBER = {31 \#3420},
      MRREVIEWER = {T. Ono},
      DOI = {10.1515/crll.1965.217.180},
      ZBLNUMBER={0241.14017},
      }
  • [CFKS] J. Coates, T. Fukaya, K. Kato, and R. Sujatha, "Root numbers, Selmer groups, and non-commutative Iwasawa theory," J. Algebraic Geom., vol. 19, iss. 1, pp. 19-97, 2010.
    @article {CFKS, MRKEY = {2551757},
      AUTHOR = {Coates, John and Fukaya, Takako and Kato, Kazuya and Sujatha, Ramdorai},
      TITLE = {Root numbers, {S}elmer groups, and non-commutative {I}wasawa theory},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {19},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {19--97},
      ISSN = {1056-3911},
      MRCLASS = {22Exx (11R23 11S23)},
      MRNUMBER = {2551757},
      }
  • [CS] Go to document J. Coates and R. Sujatha, "Appendix to Computations in non-commutative Iwasawa theory, by T. Dokchitser and V. Dokchitser," Proc. Lond. Math. Soc., vol. 94, iss. 1, pp. 211-272, 2007.
    @article {CS, MRKEY = {2294995},
      AUTHOR={Coates, John and Sujatha, R.},
      TITLE = {appendix to Computations in non-commutative {I}wasawa theory, by T. Dokchitser and V. Dokchitser},
      JOURNAL = {Proc. Lond. Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {94},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {211--272},
      ISSN = {0024-6115},
      MRCLASS = {11G40 (11G05 11R23)},
      MRNUMBER = {2008g:11106},
      MRREVIEWER = {Manfred Kolster},
      DOI = {10.1112/plms/pdl014},
      ZBLNUMBER={pre05129595},
      }
  • [CV] C. Cornut and V. Vatsal, "Nontriviality of Rankin-Selberg $L$-functions and CM points," in $L$-Functions and Galois Representations, Cambridge: Cambridge Univ. Press, 2007, pp. 121-186.
    @incollection {CV, MRKEY = {2392354},
      AUTHOR = {Cornut, Christophe and Vatsal, Vinayak},
      TITLE = {Nontriviality of {R}ankin-{S}elberg {$L$}-functions and {CM} points},
      BOOKTITLE = {{$L$}-Functions and {G}alois Representations},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      NUMBER = {320},
      PAGES = {121--186},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2007},
      MRCLASS = {11G18 (11F66 11F70 11G15)},
      MRNUMBER = {2009m:11088},
      MRREVIEWER = {Benjamin V. Howard},
      ZBLNUMBER = {1153.11025},
      }
  • [DN] C. Diem and N. Naumann, "On the structure of Weil restrictions of abelian varieties," J. Ramanujan Math. Soc., vol. 18, iss. 2, pp. 153-174, 2003.
    @article {DN, MRKEY = {1995864},
      AUTHOR = {Diem, Claus and Naumann, Niko},
      TITLE = {On the structure of {W}eil restrictions of abelian varieties},
      JOURNAL = {J. Ramanujan Math. Soc.},
      FJOURNAL = {Journal of the Ramanujan Mathematical Society},
      VOLUME = {18},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {153--174},
      ISSN = {0970-1249},
      MRCLASS = {14K15 (11G10)},
      MRNUMBER = {2004m:14096},
      MRREVIEWER = {Antoine Chambert-Loir},
      ZBLNUMBER = {1080.14536},
      }
  • [TV-P] Go to document T. Dokchitser and V. Dokchitser, "Parity of ranks for elliptic curves with a cyclic isogeny," J. Number Theory, vol. 128, iss. 3, pp. 662-679, 2008.
    @article {TV-P, MRKEY = {2389862},
      AUTHOR={Dokchitser, Tim and Dokchitser, Vladimir},
      TITLE = {Parity of ranks for elliptic curves with a cyclic isogeny},
      JOURNAL = {J. Number Theory},
      FJOURNAL = {Journal of Number Theory},
      VOLUME = {128},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {662--679},
      ISSN = {0022-314X},
      CODEN = {JNUTA9},
      MRCLASS = {11G05 (11G07 11G40)},
      MRNUMBER = {2009c:11079},
      MRREVIEWER = {Anupam Saikia},
      DOI = {10.1016/j.jnt.2007.02.008},
      ZBLNUMBER = {05242979},
      }
  • [Selfduality] T. Dokchitser and V. Dokchitser, "Self-duality of Selmer groups," Math. Proc. Cam. Phil. Soc., vol. 146, pp. 257-267, 2009.
    @article{Selfduality,
      author={Dokchitser, Tim and Dokchitser, Vladimir},
      TITLE = {Self-duality of Selmer groups},
      YEAR={2009},
      JOURNAL={Math. Proc. Cam. Phil. Soc.},
      VOLUME={146},
      PAGES={257--267},
      MRNUMBER={2010a:11219},
      ZBLNUMBER={pre05532372},
      }
  • [Tamroot] T. Dokchitser and V. Dokchitser, "Regulator constants and the parity conjecture," Invent. Math, vol. 178, pp. 23-71, 2009.
    @article{Tamroot,
      author={Dokchitser, Tim and Dokchitser, Vladimir},
      TITLE={Regulator constants and the parity conjecture},
      YEAR={2009},
      JOURNAL={Invent. Math},
      VOLUME={178},
      PAGES={23--71},
      MRNUMBER={2534092},
      ZBLNUMBER={pre05602515},
      }
  • [Fis] Go to document V. Dokchitser, "Root numbers of non-abelian twists of elliptic curves," Proc. London Math. Soc., vol. 91, iss. 2, pp. 300-324, 2005.
    @article {Fis, MRKEY = {2167089},
      AUTHOR = {Dokchitser, Vladimir},
      TITLE = {Root numbers of non-abelian twists of elliptic curves},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {91},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {300--324},
      ISSN = {0024-6115},
      CODEN = {PLMTAL},
      MRCLASS = {11G05 (11G40)},
      MRNUMBER = {2006f:11060},
      MRREVIEWER = {Chandan Singh Dalawat},
      DOI = {10.1112/S0024611505015261},
      ZBLNUMBER = {1076.11042},
      }
  • [Gre] Go to document R. Greenberg, "On the Birch and Swinnerton-Dyer conjecture," Invent. Math., vol. 72, iss. 2, pp. 241-265, 1983.
    @article {Gre, MRKEY = {700770},
      AUTHOR = {Greenberg, Ralph},
      TITLE = {On the {B}irch and {S}winnerton-{D}yer conjecture},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {72},
      YEAR = {1983},
      NUMBER = {2},
      PAGES = {241--265},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11G40 (11G15 14G25 14K07)},
      MRNUMBER = {85c:11052},
      MRREVIEWER = {Kenneth A. Ribet},
      DOI = {10.1007/BF01389322},
      ZBLNUMBER = {0546.14015},
      }
  • [Guo] Go to document L. Guo, "General Selmer groups and critical values of Hecke $L$-functions," Math. Ann., vol. 297, iss. 2, pp. 221-233, 1993.
    @article {Guo, MRKEY = {1241803},
      AUTHOR = {Guo, Li},
      TITLE = {General {S}elmer groups and critical values of {H}ecke {$L$}-functions},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {297},
      YEAR = {1993},
      NUMBER = {2},
      PAGES = {221--233},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11G40 (11G05 11R23)},
      MRNUMBER = {95b:11064},
      MRREVIEWER = {Alexis Michel},
      DOI = {10.1007/BF01459498},
      ZBLNUMBER = {0789.14018},
      }
  • [HV] Y. Hachimori and O. Venjakob, "Completely faithful Selmer groups over Kummer extensions," Doc. Math., iss. Extra Vol., pp. 443-478, 2003.
    @article {HV, MRKEY = {2046605},
      AUTHOR = {Hachimori, Yoshitaka and Venjakob, Otmar},
      TITLE = {Completely faithful {S}elmer groups over {K}ummer extensions},
      NOTE = {Extra volume: Kazuya Kato's fiftieth birthday},
      JOURNAL = {Doc. Math.},
      FJOURNAL = {Documenta Mathematica},
      YEAR = {2003},
      NUMBER = {Extra Vol.},
      PAGES = {443--478},
      ISSN = {1431-0635},
      MRCLASS = {11G05 (14K15)},
      MRNUMBER = {2005b:11072},
      MRREVIEWER = {Ramdorai Sujatha},
      ZBLNUMBER = {1117.14046},
      }
  • [Kim] Go to document B. D. Kim, "The parity conjecture for elliptic curves at supersingular reduction primes," Compos. Math., vol. 143, iss. 1, pp. 47-72, 2007.
    @article {Kim, MRKEY = {2295194},
      AUTHOR = {Kim, Byoung Du},
      TITLE = {The parity conjecture for elliptic curves at supersingular reduction primes},
      JOURNAL = {Compos. Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {143},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {47--72},
      ISSN = {0010-437X},
      MRCLASS = {11G05 (11G40 11R23)},
      MRNUMBER = {2007k:11091},
      MRREVIEWER = {Benjamin V. Howard},
      DOI = {10.1112/S0010437X06002569},
      ZBLNUMBER = {1169.11022},
      }
  • [Kob] Go to document S. Kobayashi, "The local root number of elliptic curves with wild ramification," Math. Ann., vol. 323, iss. 3, pp. 609-623, 2002.
    @article {Kob, MRKEY = {1923699},
      AUTHOR = {Kobayashi, Shin-ichi},
      TITLE = {The local root number of elliptic curves with wild ramification},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {323},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {609--623},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11G07 (11G40 11R32)},
      MRNUMBER = {2004b:11083},
      MRREVIEWER = {Anupam Saikia},
      DOI = {10.1007/s002080200318},
      ZBLNUMBER = {1117.11034},
      }
  • [Kol] V. A. Kolyvagin, "Euler systems," in The Grothendieck Festschrift, Vol. II, Boston, MA: Birkhäuser, 1990, pp. 435-483.
    @incollection {Kol, MRKEY = {1106906},
      AUTHOR = {Kolyvagin, V. A.},
      TITLE = {Euler systems},
      BOOKTITLE = {The {G}rothendieck {F}estschrift, {V}ol. {II}},
      SERIES = {Progr. Math.},
      NUMBER = {87},
      PAGES = {435--483},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1990},
      MRCLASS = {11R34 (11G05 11G40 11R29)},
      MRNUMBER = {92g:11109},
      MRREVIEWER = {Karl Rubin},
      ZBLNUMBER = {0742.14017},
      }
  • [Kra] Go to document K. Kramer, "Arithmetic of elliptic curves upon quadratic extension," Trans. Amer. Math. Soc., vol. 264, iss. 1, pp. 121-135, 1981.
    @article {Kra, MRKEY = {597871},
      AUTHOR = {Kramer, Kenneth},
      TITLE = {Arithmetic of elliptic curves upon quadratic extension},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {264},
      YEAR = {1981},
      NUMBER = {1},
      PAGES = {121--135},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {14G25 (10B10 14K07)},
      MRNUMBER = {82g:14028},
      MRREVIEWER = {Andrew Bremner},
      DOI = {10.2307/1998414},
      ZBLNUMBER = {0471.14020},
      }
  • [KT] Go to document K. Kramer and J. Tunnell, "Elliptic curves and local $\varepsilon $-factors," Compositio Math., vol. 46, iss. 3, pp. 307-352, 1982.
    @article {KT, MRKEY = {664648},
      AUTHOR = {Kramer, K. and Tunnell, J.},
      TITLE = {Elliptic curves and local {$\varepsilon $}-factors},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {46},
      YEAR = {1982},
      NUMBER = {3},
      PAGES = {307--352},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14K07 (10D40 14G10 14G20)},
      MRNUMBER = {83m:14031},
      MRREVIEWER = {Ernst-Wilhelm Zink},
      URL = {http://www.numdam.org/item?id=CM_1982__46_3_307_0},
      ZBLNUMBER = {0496.14030},
      }
  • [MR] Go to document B. Mazur and K. Rubin, "Finding large Selmer rank via an arithmetic theory of local constants," Ann. of Math., vol. 166, iss. 2, pp. 579-612, 2007.
    @article {MR, MRKEY = {2373150},
      AUTHOR = {Mazur, Barry and Rubin, Karl},
      TITLE = {Finding large {S}elmer rank via an arithmetic theory of local constants},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {166},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {579--612},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11G05 (11G40 11R23)},
      MRNUMBER = {2009a:11127},
      MRREVIEWER = {Benjamin V. Howard},
      DOI = {10.4007/annals.2007.166.579},
      ZBLNUMBER = {05248867},
      }
  • [MilO] Go to document J. S. Milne, "On the arithmetic of abelian varieties," Invent. Math., vol. 17, pp. 177-190, 1972.
    @article {MilO, MRKEY = {0330174},
      AUTHOR = {Milne, J. S.},
      TITLE = {On the arithmetic of abelian varieties},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {17},
      YEAR = {1972},
      PAGES = {177--190},
      ISSN = {0020-9910},
      MRCLASS = {14K15 (14G10 14G25)},
      MRNUMBER = {48 \#8512},
      MRREVIEWER = {T. Ono},
      DOI = {10.1007/BF01425446},
      ZBLNUMBER = {0249.14012},
      }
  • [MilA] J. S. Milne, Arithmetic Duality Theorems, Boston, MA: Academic Press, 1986.
    @book {MilA, MRKEY = {881804},
      AUTHOR = {Milne, J. S.},
      TITLE = {Arithmetic Duality Theorems},
      SERIES = {Perspectives in Mathematics},
      NUMBER = {1},
      PUBLISHER = {Academic Press},
      ADDRESS = {Boston, MA},
      YEAR = {1986},
      PAGES = {x+421},
      ISBN = {0-12-498040-6},
      MRCLASS = {14F20 (11G99 11R34 12G10)},
      MRNUMBER = {88e:14028},
      MRREVIEWER = {Gerd Faltings},
      ZBLNUMBER = {0613.14019},
      }
  • [Mon] Go to document P. Monsky, "Generalizing the Birch-Stephens theorem. I. Modular curves," Math. Z., vol. 221, iss. 3, pp. 415-420, 1996.
    @article {Mon, MRKEY = {1381589},
      AUTHOR = {Monsky, P.},
      TITLE = {Generalizing the {B}irch-{S}tephens theorem. {I}. {M}odular curves},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {221},
      YEAR = {1996},
      NUMBER = {3},
      PAGES = {415--420},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {11G40 (11G05)},
      MRNUMBER = {97a:11103},
      MRREVIEWER = {M. Ram Murty},
      DOI = {10.1007/PL00004518},
      }
  • [MM] Go to document R. M. Murty and K. V. Murty, "Mean values of derivatives of modular $L$-series," Ann. of Math., vol. 133, iss. 3, pp. 447-475, 1991.
    @article {MM, MRKEY = {1109350},
      AUTHOR = {Murty, M. Ram and Murty, V. Kumar},
      TITLE = {Mean values of derivatives of modular {$L$}-series},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {133},
      YEAR = {1991},
      NUMBER = {3},
      PAGES = {447--475},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11F67 (11G05 11G40)},
      MRNUMBER = {92e:11050},
      MRREVIEWER = {Daniel Bump},
      DOI = {10.2307/2944316},
      ZBLNUMBER = {0745.11032},
      }
  • [NekS] J. Nekovávr, Selmer Complexes, , 2006, vol. 310.
    @book {NekS, MRKEY = {2333680},
      AUTHOR = {Nekov{á}{\v{r}},
      Jan},
      TITLE = {Selmer Complexes},
      SERIES= {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {310},
      YEAR = {2006},
      PAGES = {viii+559},
      ISSN = {0303-1179},
      ISBN = {978-2-85629-226-6},
      MRCLASS = {11R23 (11F41 11G40 11R34 22E41)},
      MRNUMBER = {2009c:11176},
      MRREVIEWER = {Laurent N. Berger},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {05161833},
      }
  • [NekE] J. Nekovávr, "The Euler system method for CM points on Shimura curves," in $L$-Functions and Galois Representations, Cambridge: Cambridge Univ. Press, 2007, pp. 471-547.
    @incollection {NekE, MRKEY = {2392363},
      AUTHOR = {Nekov{á}{\v{r}},
      Jan},
      TITLE = {The {E}uler system method for {CM} points on {S}himura curves},
      BOOKTITLE = {{$L$}-Functions and {G}alois Representations},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      NUMBER = {320},
      PAGES = {471--547},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2007},
      MRCLASS = {11G15 (11F80 11G18)},
      MRNUMBER = {2010a:11110},
      MRREVIEWER = {Benjamin V. Howard},
      ZBLNUMBER = {1152.11023},
      }
  • [PS] Go to document B. Poonen and M. Stoll, "The Cassels-Tate pairing on polarized abelian varieties," Ann. of Math., vol. 150, iss. 3, pp. 1109-1149, 1999.
    @article {PS, MRKEY = {1740984},
      AUTHOR = {Poonen, Bjorn and Stoll, Michael},
      TITLE = {The {C}assels-{T}ate pairing on polarized abelian varieties},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {150},
      YEAR = {1999},
      NUMBER = {3},
      PAGES = {1109--1149},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11G10 (11G30 14H40 14K15)},
      MRNUMBER = {2000m:11048},
      MRREVIEWER = {Tam{á}s Szamuely},
      DOI = {10.2307/121064},
      ZBLNUMBER = {1024.11040},
      }
  • [RohV] Go to document D. E. Rohrlich, "Variation of the root number in families of elliptic curves," Compositio Math., vol. 87, iss. 2, pp. 119-151, 1993.
    @article {RohV, MRKEY = {1219633},
      AUTHOR = {Rohrlich, David E.},
      TITLE = {Variation of the root number in families of elliptic curves},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {87},
      YEAR = {1993},
      NUMBER = {2},
      PAGES = {119--151},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {11G40 (11G05 11N36)},
      MRNUMBER = {94d:11045},
      MRREVIEWER = {Fernando Q. Gouv{ê}a},
      URL = {http://www.numdam.org/item?id=CM_1993__87_2_119_0},
      ZBLNUMBER = {0791.11026},
      }
  • [SerLi] J. Serre, Linear Representations of Finite Groups, New York: Springer-Verlag, 1977, vol. 42.
    @book {SerLi, MRKEY = {0450380},
      AUTHOR = {Serre, Jean-Pierre},
      TITLE = {Linear Representations of Finite Groups},
      SERIES={ Grad. Texts.Math.},
      VOLUME={42},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {x+170},
      ISBN = {0-387-90190-6},
      MRCLASS = {20CXX},
      MRNUMBER = {56 \#8675},
      MRREVIEWER = {W. Feit},
      ZBLNUMBER = {0355.20006},
      }
  • [Sil2] J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, New York: Springer-Verlag, 1994.
    @book {Sil2, MRKEY = {1312368},
      AUTHOR = {Silverman, Joseph H.},
      TITLE = {Advanced Topics in the Arithmetic of Elliptic Curves},
      SERIES = {Grad. Texts Math.},
      NUMBER = {151},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1994},
      PAGES = {xiv+525},
      ISBN = {0-387-94328-5},
      MRCLASS = {11G05 (11G07 11G15 11G40 14H52)},
      MRNUMBER = {96b:11074},
      MRREVIEWER = {Henri Darmon},
      ZBLNUMBER = {0911.14015},
      }
  • [TatD] J. Tate, "Duality theorems in Galois cohomology over number fields," in Proc. Internat. Congr. Mathematicians, Djursholm: Inst. Mittag-Leffler, 1963, pp. 288-295.
    @incollection {TatD, MRKEY = {0175892},
      AUTHOR = {Tate, John},
      TITLE = {Duality theorems in {G}alois cohomology over number fields},
      BOOKTITLE = {Proc. {I}nternat. {C}ongr. {M}athematicians},
      VENUE={{S}tockholm, 1962},
      PAGES = {288--295},
      PUBLISHER = {Inst. Mittag-Leffler},
      ADDRESS = {Djursholm},
      YEAR = {1963},
      MRCLASS = {10.66},
      MRNUMBER = {31 \#168},
      ZBLNUMBER = {0126.07002},
      }
  • [TatC] J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog.
    @misc{TatC,
      author={Tate, John},
      TITLE = {On the conjectures of Birch and Swinnerton-Dyer and a geometric analog},
      NOTE={ Séminaire Bourbaki, 18e année, 1965/66, no. 306},
      }
  • [TZ] Y. Tian and S. Zhang, Euler system of CM-points on Shimura curves.
    @misc{TZ,
      author={Tian, Y. and Zhang, S.},
      TITLE={Euler system of CM-points on Shimura curves},
      NOTE={in preparation},
      }
  • [Wal] Go to document J. Waldspurger, "Correspondances de Shimura et quaternions," Forum Math., vol. 3, iss. 3, pp. 219-307, 1991.
    @article {Wal, MRKEY = {1103429},
      AUTHOR = {Waldspurger, Jean-Loup},
      TITLE = {Correspondances de {S}himura et quaternions},
      JOURNAL = {Forum Math.},
      FJOURNAL = {Forum Mathematicum},
      VOLUME = {3},
      YEAR = {1991},
      NUMBER = {3},
      PAGES = {219--307},
      ISSN = {0933-7741},
      CODEN = {FOMAEF},
      MRCLASS = {11F70 (11F30 11F32 11F37 22E50)},
      MRNUMBER = {92g:11054},
      MRREVIEWER = {Stephen Gelbart},
      DOI = {10.1515/form.1991.3.219},
      ZBLNUMBER = {0724.11026},
      }
  • [Yu] Go to document H. Yu, "Idempotent relations and the conjecture of Birch and Swinnerton-Dyer," Math. Ann., vol. 327, iss. 1, pp. 67-78, 2003.
    @article {Yu, MRKEY = {2005121},
      AUTHOR = {Yu, Hoseog},
      TITLE = {Idempotent relations and the conjecture of {B}irch and {S}winnerton-{D}yer},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {327},
      YEAR = {2003},
      NUMBER = {1},
      PAGES = {67--78},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11G40},
      MRNUMBER = {2005a:11096},
      MRREVIEWER = {Chandan Singh Dalawat},
      DOI = {10.1007/s00208-003-0427-8},
      ZBLNUMBER = {1089.11036},
      }
  • [YZZ] X. Yuan, S. Zhang, and W. Zhang, "Heights of CM points I: Gross-Zagier formula," , preprint , 2008.
    @techreport{YZZ,
      author={Yuan, X. and Zhang, S. and Zhang, W.},
      TITLE={Heights of CM points I: Gross-Zagier formula},
      YEAR={2008},
      TYPE={preprint},
      }

Authors

Tim Dokchitser

University of Cambridge
Robinson College
Cambridge CB3 9AN
United Kingdom

Vladimir Dokchitser

University of Cambridge
Gonville & Caius College
Cambridge CB2 1TA
United Kingdom