The rationality of Stark-Heegner points over genus fields of real quadratic fields

Abstract

We study the algebraicity of Stark-Heegner points on a modular elliptic curve $E$. These objects are $p$-adic points on $E$ given by the values of certain $p$-adic integrals, but they are conjecturally defined over ring class fields of a real quadratic field $K$. The present article gives some evidence for this algebraicity conjecture by showing that linear combinations of Stark-Heegner points weighted by certain genus characters of $K$ are defined over the predicted quadratic extensions of $K$. The non-vanishing of these combinations is also related to the appropriate twisted Hasse-Weil $L$-series of $E$ over $K$, in the spirit of the Gross-Zagier formula for classical Heegner points.

  • [bertolini_darmon_inv] Go to document M. Bertolini and H. Darmon, "Heegner points, $p$-adic $L$-functions, and the Cerednik-Drinfeld uniformization," Invent. Math., vol. 131, iss. 3, pp. 453-491, 1998.
    @article {bertolini_darmon_inv, MRKEY = {1614543},
      AUTHOR = {Bertolini, Massimo and Darmon, Henri},
      TITLE = {Heegner points, {$p$}-adic {$L$}-functions, and the {C}erednik-{D}rinfeld uniformization},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {131},
      YEAR = {1998},
      NUMBER = {3},
      PAGES = {453--491},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11G40 (11G05 14G10 14G20)},
      MRNUMBER = {99f:11080},
      MRREVIEWER = {Bas Edixhoven},
      DOI = {10.1007/s002220050211},
      }
  • [bd_hida] Go to document M. Bertolini and H. Darmon, "Hida families and rational points on elliptic curves," Invent. Math., vol. 168, iss. 2, pp. 371-431, 2007.
    @article {bd_hida, MRKEY = {2289868},
      AUTHOR = {Bertolini, Massimo and Darmon, Henri},
      TITLE = {Hida families and rational points on elliptic curves},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {168},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {371--431},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F67 (11F85 11G05 11G40)},
      MRNUMBER = {2008c:11076},
      MRREVIEWER = {Benjamin V. Howard},
      DOI = {10.1007/s00222-007-0035-4},
      ZBLNUMBER = {1129.11025},
      }
  • [bertolini_darmon_green] M. Bertolini, H. Darmon, and P. Green, "Periods and points attached to quadratic algebras," in Heegner points and Rankin $L$-series, Cambridge: Cambridge Univ. Press, 2004, pp. 323-367.
    @incollection {bertolini_darmon_green, MRKEY = {2083218},
      AUTHOR = {Bertolini, Massimo and Darmon, Henri and Green, Peter},
      TITLE = {Periods and points attached to quadratic algebras},
      BOOKTITLE = {Heegner points and {R}ankin {$L$}-series},
      SERIES = {Math. Sci. Res. Inst. Publ.},
      NUMBER = {49},
      PAGES = {323--367},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2004},
      MRCLASS = {11F67 (11F41 11F85 11G05)},
      MRNUMBER = {2005e:11062},
      MRREVIEWER = {Alexey A. Panchishkin},
      }
  • [bdi] Go to document M. Bertolini, H. Darmon, and A. Iovita, "Families of modular forms on definite quaternion algebras and Teitelbaum’s conjecture,".
    @techreport{bdi,
      author = {Bertolini, Massimo and Darmon, Henri and Iovita, A.},
      TITLE = {Families of modular forms on definite quaternion algebras and Teitelbaum's conjecture},
      url = {http://newrobin.mat.unimi.it/users/mbertoli/bdi.pdf},
      }
  • [darmon_hpxh] Go to document H. Darmon, "Integration on $\mathcal H_p\times\mathcal H$ and arithmetic applications," Ann. of Math., vol. 154, iss. 3, pp. 589-639, 2001.
    @article {darmon_hpxh, MRKEY = {1884617},
      AUTHOR = {Darmon, Henri},
      TITLE = {Integration on {$\mathcal H\sb p\times\mathcal H$} and arithmetic applications},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {154},
      YEAR = {2001},
      NUMBER = {3},
      PAGES = {589--639},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11G40 (11F67 11F85 11G18 11R37)},
      MRNUMBER = {2003j:11067},
      MRREVIEWER = {Jeremy T. Teitelbaum},
      DOI = {10.2307/3062142},
      ZBLNUMBER = {1035.11027},
      }
  • [dasgupta_thesis] S. Dasgupta, "Gross-Stark units, Stark-Heegner points, and class fields of real quadratic fields," PhD Thesis , Univ. of California, Berkeley, 2004.
    @phdthesis {dasgupta_thesis,
      author = {Dasgupta, Samit},
      TITLE = {Gross-Stark units, Stark-Heegner points, and class fields of real quadratic fields},
      type = {Ph.D. thesis},
      school = {Univ. of California, Berkeley},
      year = {2004},
      }
  • [dasgupta_paper] Go to document S. Dasgupta, "Stark-Heegner points on modular Jacobians," Ann. Sci. École Norm. Sup., vol. 38, iss. 3, pp. 427-469, 2005.
    @article {dasgupta_paper, MRKEY = {2166341},
      AUTHOR = {Dasgupta, Samit},
      TITLE = {Stark-{H}eegner points on modular {J}acobians},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {38},
      YEAR = {2005},
      NUMBER = {3},
      PAGES = {427--469},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {11G18},
      MRNUMBER = {2006e:11080},
      MRREVIEWER = {Benjamin V. Howard},
      DOI = {10.1016/j.ansens.2005.03.002},
      }
  • [darmon_dasgupta] Go to document H. Darmon and S. Dasgupta, "Elliptic units for real quadratic fields," Ann. of Math., vol. 163, iss. 1, pp. 301-346, 2006.
    @article {darmon_dasgupta, MRKEY = {2195136},
      AUTHOR = {Darmon, Henri and Dasgupta, Samit},
      TITLE = {Elliptic units for real quadratic fields},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {163},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {301--346},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11G16 (11F85 11R37)},
      MRNUMBER = {2007a:11079},
      MRREVIEWER = {Benjamin V. Howard},
      DOI = {10.4007/annals.2006.163.301},
      ZBLNUMBER = {1130.11030},
      }
  • [darmon_green] Go to document H. Darmon and P. Green, "Elliptic curves and class fields of real quadratic fields: algorithms and evidence," Experiment. Math., vol. 11, iss. 1, pp. 37-55, 2002.
    @article {darmon_green, MRKEY = {1960299},
      AUTHOR = {Darmon, Henri and Green, Peter},
      TITLE = {Elliptic curves and class fields of real quadratic fields: algorithms and evidence},
      JOURNAL = {Experiment. Math.},
      FJOURNAL = {Experimental Mathematics},
      VOLUME = {11},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {37--55},
      ISSN = {1058-6458},
      MRCLASS = {11G40 (11F67 11G05)},
      MRNUMBER = {2004c:11112},
      MRREVIEWER = {Chandrashekhar Khare},
      URL = {http://projecteuclid.org/getRecord?id=euclid.em/1057860313},
      ZBLNUMBER = {1040.11048},
      }
  • [darmon_pollack] Go to document H. Darmon and R. Pollack, "Efficient calculation of Stark-Heegner points via overconvergent modular symbols," Israel J. Math., vol. 153, pp. 319-354, 2006.
    @article {darmon_pollack, MRKEY = {2254648},
      AUTHOR = {Darmon, Henri and Pollack, Robert},
      TITLE = {Efficient calculation of {S}tark-{H}eegner points via overconvergent modular symbols},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {153},
      YEAR = {2006},
      PAGES = {319--354},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {11F67 (11G05 11G40)},
      MRNUMBER = {2007k:11077},
      MRREVIEWER = {Samit Dasgupta},
      DOI = {10.1007/BF02771789},
      ZBLNUMBER = {1157.11028},
      }
  • [greenberg_stevens] Go to document R. Greenberg and G. Stevens, "$p$-adic $L$-functions and $p$-adic periods of modular forms," Invent. Math., vol. 111, iss. 2, pp. 407-447, 1993.
    @article {greenberg_stevens, MRKEY = {1198816},
      AUTHOR = {Greenberg, Ralph and Stevens, Glenn},
      TITLE = {{$p$}-adic {$L$}-functions and {$p$}-adic periods of modular forms},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {111},
      YEAR = {1993},
      NUMBER = {2},
      PAGES = {407--447},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11G40 (11F67)},
      MRNUMBER = {93m:11054},
      MRREVIEWER = {Nigel Boston},
      DOI = {10.1007/BF01231294},
      ZBLNUMBER = {0778.11034},
      }
  • [zagier] W. Kohnen and D. Zagier, "Modular forms with rational periods," in Modular Forms (Durham, 1983), Chichester: Horwood, 1984, pp. 197-249.
    @incollection {zagier, MRKEY = {803368},
      AUTHOR = {Kohnen, W. and Zagier, D.},
      TITLE = {Modular forms with rational periods},
      BOOKTITLE = {Modular Forms ({D}urham, 1983)},
      PAGES = {197--249},
      PUBLISHER = {Horwood},
      ADDRESS = {Chichester},
      YEAR = {1984},
      MRCLASS = {11F67 (11F11)},
      MRNUMBER = {87h:11043},
      MRREVIEWER = {Jing Yu},
      ZBLNUMBER = {0618.10019},
      }
  • [popa] Go to document A. A. Popa, "Central values of Rankin $L$-series over real quadratic fields," Compos. Math., vol. 142, iss. 4, pp. 811-866, 2006.
    @article {popa, MRKEY = {2249532},
      AUTHOR = {Popa, Alexandru A.},
      TITLE = {Central values of {R}ankin {$L$}-series over real quadratic fields},
      JOURNAL = {Compos. Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {142},
      YEAR = {2006},
      NUMBER = {4},
      PAGES = {811--866},
      ISSN = {0010-437X},
      MRCLASS = {11F67 (11F27 11F70)},
      MRNUMBER = {2007m:11070},
      MRREVIEWER = {Gergely Harcos},
      DOI = {10.1112/S0010437X06002259},
      ZBLNUMBER = {1144.11041},
      }
  • [siegel] C. L. Siegel, Advanced Analytic Number Theory, Second ed., Bombay: Tata Institute of Fundamental Research, 1980.
    @book {siegel, MRKEY = {659851},
      AUTHOR = {Siegel, Carl Ludwig},
      TITLE = {Advanced Analytic Number Theory},
      SERIES = {TIFR Studies in Math.},
      NUMBER = {9},
      EDITION = {Second},
      PUBLISHER = {Tata Institute of Fundamental Research},
      ADDRESS = {Bombay},
      YEAR = {1980},
      PAGES = {v+268},
      MRCLASS = {10-01 (10Dxx 10Hxx)},
      MRNUMBER = {83m:10001},
      ZBLNUMBER = {0478.10001},
      }
  • [weil] A. Weil, Elliptic Functions According to Eisenstein and Kronecker, New York: Springer-Verlag, 1976.
    @book {weil, MRKEY = {0562289},
      AUTHOR = {Weil, Andr{é}},
      TITLE = {Elliptic Functions According to {E}isenstein and {K}ronecker},
      NOTE = {{\it Ergeb. Math. Grenzgeb.\/} {\bf 88}},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1976},
      PAGES = {ii+93},
      ISBN = {3-540-07422-8},
      MRCLASS = {10DXX (01A55 10-03)},
      MRNUMBER = {58 \#27769a},
      MRREVIEWER = {S. Chowla},
      ZBLNUMBER = {0318.33004},
      }

Authors

Massimo Bertolini

Dipartimento di Matematica
Università degli Studi di Milano
Via Saldini, 50
20133 Milano
Italy

Henri Darmon

Department of Mathematics
McGill University
805 Shebrooke Street West
Montreal, QC  H3A 2K6
Canada