On the Birch-Swinnerton-Dyer quotients modulo squares

Tim Dokchitser; Vladimir Dokchitser

doi:10.4007/annals.2010.172.567

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.

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[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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@techreport{YZZ, author={Yuan, X. and Zhang, S. and Zhang, W.}, TITLE={Heights of CM points I: Gross-Zagier formula}, YEAR={2008}, TYPE={preprint}, }

On the Birch-Swinnerton-Dyer quotients modulo squares

Abstract

Authors