The Iwasawa $\mu$-invariant of $p$-adic Hecke $L$-functions

Pages 41-137 from Volume 172 (2010), Issue 1 by Haruzo Hida

Abstract

For an odd prime $p$, we compute the $\mu$-invariant of the anticyclotomic Katz $p$-adic $L$-function of a $p$-ordinary CM field if the conductor of the branch character is a product of primes split over the maximal real subfield. Except for rare cases where the root number of the $p$-adic functional equation is congruent to $-1$ modulo $p$, the invariant vanishes.

  • [Art69] Go to document M. Artin, "Algebraic approximation of structures over complete local rings," Inst. Hautes Études Sci. Publ. Math., iss. 36, pp. 23-58, 1969.
    @article {Art69, MRKEY = {0268188},
      AUTHOR = {Artin, M.},
      TITLE = {Algebraic approximation of structures over complete local rings},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {36},
      YEAR = {1969},
      PAGES = {23--58},
      ISSN = {0073-8301},
      MRCLASS = {14.05},
      MRNUMBER = {42 \#3087},
      MRREVIEWER = {H. Kurke},
      URL = {http://www.numdam.org/item?id=PMIHES_1969__36__23_0},
      ZBLNUMBER = {0181.48802},
      }
  • [Sh00] G. Shimura, Arithmeticity in the Theory of Automorphic Forms, Providence, RI: Amer. Math. Soc., 2000.
    @book {Sh00, MRKEY = {1780262},
      AUTHOR = {Shimura, Goro},
      TITLE = {Arithmeticity in the Theory of Automorphic Forms},
      SERIES = {Math. Surveys Monogr.},
      NUMBER = {82},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2000},
      PAGES = {x+302},
      ISBN = {0-8218-2671-9},
      MRCLASS = {11F67 (11-02 11F30 11F41 11F55)},
      MRNUMBER = {2001k:11086},
      MRREVIEWER = {Hideshi Takayanagi},
      ZBLNUMBER = {0967.11001},
      }
  • [Mumab] D. Mumford, Abelian Varieties, Published for the Tata Institute of Fundamental Research, Bombay, 1970, vol. 5.
    @book {Mumab, MRKEY = {0282985},
      AUTHOR = {Mumford, David},
      TITLE = {Abelian Varieties},
      SERIES = {Tata Institute of Fundamental Research Studies in Mathematics},
      VOLUME={ 5},
      PUBLISHER = {Published for the Tata Institute of Fundamental Research, Bombay},
      YEAR = {1970},
      PAGES = {viii+242},
      MRCLASS = {14.51},
      MRNUMBER = {44 \#219},
      MRREVIEWER = {J. S. Milne},
      ZBLNUMBER = {0223.14022},
      }
  • [Sh98] G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, Princeton, NJ: Princeton Univ. Press, 1998.
    @book {Sh98, MRKEY = {1492449},
      AUTHOR = {Shimura, Goro},
      TITLE = {Abelian Varieties with Complex Multiplication and Modular Functions},
      SERIES = {Princeton Math. Series},
      NUMBER = {46},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1998},
      PAGES = {xvi+218},
      ISBN = {0-691-01656-9},
      MRCLASS = {11G15 (11F46 14K22)},
      MRNUMBER = {99e:11076},
      MRREVIEWER = {Rolf Berndt},
      ZBLNUMBER = {0908.11023},
      }
  • [Har] R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977, vol. 52.
    @book {Har, MRKEY = {0463157},
      AUTHOR = {Hartshorne, Robin},
      TITLE = {Algebraic Geometry},
      SERIES = {Grad. Texts Math.},
      VOLUME={52},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {xvi+496},
      ISBN = {0-387-90244-9},
      MRCLASS = {14-01},
      MRNUMBER = {57 \#3116},
      MRREVIEWER = {Robert Speiser},
      ZBLNUMBER = {0367.14001},
      }
  • [Ci90] Go to document C. Chai, "Arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces," Ann. of Math., vol. 131, iss. 3, pp. 541-554, 1990.
    @article {Ci90, MRKEY = {1053489},
      AUTHOR = {Chai, Ching-Li},
      TITLE = {Arithmetic minimal compactification of the {H}ilbert-{B}lumenthal moduli spaces},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {131},
      YEAR = {1990},
      NUMBER = {3},
      PAGES = {541--554},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11G15 (11F67 11G18)},
      MRNUMBER = {91i:11063},
      MRREVIEWER = {Alexey A. Panchishkin},
      DOI = {10.2307/1971469},
      ZBLNUMBER = {0754.14030},
      }
  • [Ci95] Go to document C. Chai, "Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli," Invent. Math., vol. 121, iss. 3, pp. 439-479, 1995.
    @article {Ci95, MRKEY = {1353306},
      AUTHOR = {Chai, Ching-Li},
      TITLE = {Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {121},
      YEAR = {1995},
      NUMBER = {3},
      PAGES = {439--479},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11G10 (11G25 14G20 14K02 14K10 14K15)},
      MRNUMBER = {96f:11082},
      MRREVIEWER = {F. Oort},
      DOI = {10.1007/BF01884309},
      ZBLNUMBER = {0990.11039},
      }
  • [Ci01] C. Chai, "A rigidity result for $p$-divisible formal groups," Asian J. Math., vol. 12, iss. 2, pp. 193-202, 2008.
    @article {Ci01, MRKEY = {2439259},
      AUTHOR = {Chai, Ching-Li},
      TITLE = {A rigidity result for {$p$}-divisible formal groups},
      JOURNAL = {Asian J. Math.},
      FJOURNAL = {The Asian Journal of Mathematics},
      VOLUME = {12},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {193--202},
      ISSN = {1093-6106},
      MRCLASS = {14L05 (14G35 14K10)},
      MRNUMBER = {2009f:14089},
      MRREVIEWER = {Eva Viehmann},
      ZBLNUMBER = {1153.14030},
      }
  • [Ci02] Go to document C. Chai, Families of ordinary abelian varieties: canonical coordinates, $p$-adic monodromy, Tate-linear subvarieties and Hecke orbits, preprint, 2003.
    @misc{Ci02,
      author = {Chai, Ching-Li},
      TITLE={Families of ordinary abelian varieties: canonical coordinates, $p$-adic monodromy, Tate-linear subvarieties and Hecke orbits, preprint},
      YEAR={2003},
      URL={http://www.math.upenn.edu/~chai/papers.html},
     }
  • [Me72] W. Messing, The Crystals Associated to Barsotti-Tate Groups: With Applications to Abelian Schemes, New York: Springer-Verlag, 1972, vol. 264.
    @book {Me72, MRKEY = {0347836},
      AUTHOR = {Messing, William},
      TITLE = {The Crystals Associated to {B}arsotti-{T}ate Groups: With Applications to Abelian Schemes},
      SERIES = {Lecture Notes in Math.},
      VOLUME={264},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1972},
      PAGES = {iii+190},
      MRCLASS = {14L05 (14B20 14D15 14F30 14K10)},
      MRNUMBER = {50 \#337},
      MRREVIEWER = {F. Oort},
      ZBLNUMBER = {0243.14013},
      }
  • [D69] Go to document P. Deligne, "Variétés abéliennes ordinaires sur un corps fini," Invent. Math., vol. 8, pp. 238-243, 1969.
    @article {D69, MRKEY = {0254059},
      AUTHOR = {Deligne, Pierre},
      TITLE = {Variétés abéliennes ordinaires sur un corps fini},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {8},
      YEAR = {1969},
      PAGES = {238--243},
      ISSN = {0020-9910},
      MRCLASS = {14.51},
      MRNUMBER = {40 \#7270},
      MRREVIEWER = {J. S. Milne},
      DOI = {10.1007/BF01406076},
      ZBLNUMBER = {0179.26201},
      }
  • [D71] P. Deligne, "Travaux de Shimura," in Sém. Bourbaki, 23ème année (1970/71), Exp. No. 389, New York: Springer-Verlag, 1971, vol. 244, pp. 123-165.
    @incollection {D71, MRKEY = {0498581},
      AUTHOR = {Deligne, Pierre},
      TITLE = {Travaux de {S}himura},
      BOOKTITLE = {Sém. {B}ourbaki, 23ème année (1970/71), {E}xp. {N}o. 389},
      PAGES = {123--165},
      SERIES={ Lecture Notes in Math.},
      VOLUME={244},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1971},
      MRCLASS = {14G25 (10D25 14K15)},
      MRNUMBER = {58 \#16675},
      ZBLNUMBER = {0225.14007},
      }
  • [D79] P. Deligne, "Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques," in Automorphic Forms, Representations and $L$-Functions, Providence, R.I.: Amer. Math. Soc., 1979, vol. XXXIII, pp. 247-289.
    @incollection {D79, MRKEY = {546620},
      AUTHOR = {Deligne, Pierre},
      TITLE = {Variétés de {S}himura: interprétation modulaire, et techniques de construction de modèles canoniques},
      BOOKTITLE = {Automorphic Forms, Representations and {$L$}-Functions},
      VENUE={{P}roc. {S}ympos. {P}ure {M}ath., {O}regon {S}tate {U}niv., {C}orvallis, {O}re., 1977},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME={XXXIII},
      PAGES = {247--289},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1979},
      MRCLASS = {10D20 (14D20 14G25 14K15)},
      MRNUMBER = {81i:10032},
      MRREVIEWER = {J. S. Milne},
      ZBLNUMBER = {0437.14012},
      }
  • [FC90] G. Faltings and C. Chai, Degeneration of Abelian Varieties, New York: Springer-Verlag, 1990, vol. 22.
    @book {FC90, MRKEY = {1083353},
      AUTHOR = {Faltings, Gerd and Chai, Ching-Li},
      TITLE = {Degeneration of Abelian Varieties},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME= {22},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1990},
      PAGES = {xii+316},
      ISBN = {3-540-52015-5},
      MRCLASS = {14K10 (11G10 14D20 14K25)},
      MRNUMBER = {92d:14036},
      MRREVIEWER = {Min Ho Lee},
      ZBLNUMBER = {0744.14031},
      }
  • [DR80] Go to document P. Deligne and K. A. Ribet, "Values of abelian $L$-functions at negative integers over totally real fields," Invent. Math., vol. 59, iss. 3, pp. 227-286, 1980.
    @article {DR80, MRKEY = {579702},
      AUTHOR = {Deligne, Pierre and Ribet, Kenneth A.},
      TITLE = {Values of abelian {$L$}-functions at negative integers over totally real fields},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {59},
      YEAR = {1980},
      NUMBER = {3},
      PAGES = {227--286},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {12A70 (10D21)},
      MRNUMBER = {81m:12019},
      MRREVIEWER = {Neal Koblitz},
      DOI = {10.1007/BF01453237},
      ZBLNUMBER = {0434.12009},
      }
  • [ECH] J. S. Milne, Étale Cohomology, Princeton, N.J.: Princeton Univ. Press, 1980.
    @book {ECH, MRKEY = {559531},
      AUTHOR = {Milne, James S.},
      TITLE = {\'{E}tale Cohomology},
      SERIES = {Princeton Math. Series},
      NUMBER = {33},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, N.J.},
      YEAR = {1980},
      PAGES = {xiii+323},
      ISBN = {0-691-08238-3},
      MRCLASS = {14-02 (14F20 18F99)},
      MRNUMBER = {81j:14002},
      MRREVIEWER = {G. Horrocks},
      ZBLNUMBER = {0433.14012},
      }
  • [EGA] A. Grothendieck and J. Dieudonné, Eléments de Géométrie Algébrique
    @book{EGA,
      author={Grothendieck, A. and Dieudonné, J.},
      TITLE={Eléments de Géométrie Algébrique},
      SERIES={Publ. {\rm IHES}},
      NOTE={{\bf 4} (1960), {\bf 8} (1961), {\bf 11} (1961), {\bf 17} (1963), {\bf 20} (1964), {\bf 24} (1965), {\bf 28} (1966), {\bf 32} (1967)},
      }
  • [FW79] Go to document B. Ferrero and L. C. Washington, "The Iwasawa invariant $\mu _{p}$ vanishes for abelian number fields," Ann. of Math., vol. 109, iss. 2, pp. 377-395, 1979.
    @article {FW79, MRKEY = {528968},
      AUTHOR = {Ferrero, Bruce and Washington, Lawrence C.},
      TITLE = {The {I}wasawa invariant {$\mu _{p}$} vanishes for abelian number fields},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {109},
      YEAR = {1979},
      NUMBER = {2},
      PAGES = {377--395},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {12A35 (12A50)},
      MRNUMBER = {81a:12005},
      MRREVIEWER = {Wells Johnson},
      DOI = {10.2307/1971116},
      ZBLNUMBER = {0443.12001},
      }
  • [F99] Go to document T. Finis, "Divisibility of anticyclotomic $L$-functions and theta functions with complex multiplication," Ann. of Math., vol. 163, iss. 3, pp. 767-807, 2006.
    @article {F99, MRKEY = {2215134},
      AUTHOR = {Finis, Tobias},
      TITLE = {Divisibility of anticyclotomic {$L$}-functions and theta functions with complex multiplication},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {163},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {767--807},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11R23 (11S40)},
      MRNUMBER = {2007m:11151},
      MRREVIEWER = {David Y. Jao},
      DOI = {10.4007/annals.2006.163.767},
      ZBLNUMBER = {1111.11047},
      }
  • [F06] Go to document T. Finis, "The $\mu$-invariant of anticyclotomic $L$-functions of imaginary quadratic fields," J. Reine Angew. Math., vol. 596, pp. 131-152, 2006.
    @article {F06, MRKEY = {2254809},
      AUTHOR = {Finis, Tobias},
      TITLE = {The {$\mu$}-invariant of anticyclotomic {$L$}-functions of imaginary quadratic fields},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {596},
      YEAR = {2006},
      PAGES = {131--152},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {11R23},
      MRNUMBER = {2007g:11139},
      MRREVIEWER = {Francesc Bars},
      DOI = {10.1515/CRELLE.2006.056},
      ZBLNUMBER = {1111.11048},
      }
  • [Gil85] Go to document R. Gillard, "Fonctions $L$ $p$-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes," J. Reine Angew. Math., vol. 358, pp. 76-91, 1985.
    @article {Gil85, MRKEY = {797675},
      AUTHOR = {Gillard, Roland},
      TITLE = {Fonctions {$L$} {$p$}-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {358},
      YEAR = {1985},
      PAGES = {76--91},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {11R23},
      MRNUMBER = {87a:11106},
      MRREVIEWER = {Y{û}ji Kida},
      DOI = {10.1515/crll.1985.358.76},
      ZBLNUMBER = {0551.12011},
      }
  • [Gil91] Go to document R. Gillard, "Remarques sur l’invariant mu d’Iwasawa dans le cas CM," Sém. Théor. Nombres Bordeaux, vol. 3, iss. 1, pp. 13-26, 1991.
    @article {Gil91, MRKEY = {1116098},
      AUTHOR = {Gillard, Roland},
      TITLE = {Remarques sur l'invariant mu d'{I}wasawa dans le cas {CM}},
      JOURNAL = {Sém. Théor. Nombres Bordeaux},
      FJOURNAL = {Séminaire de Théorie des Nombres de Bordeaux. Série 2},
      VOLUME = {3},
      YEAR = {1991},
      NUMBER = {1},
      PAGES = {13--26},
      ISSN = {0989-5558},
      MRCLASS = {11R23 (11R42)},
      MRNUMBER = {92m:11123},
      MRREVIEWER = {Robert Gold},
      URL = {http://jtnb.cedram.org/item?id=JTNB_1991__3_1_13_0},
      ZBLNUMBER = {0732.11060},
      }
  • [Mum] D. Mumford, Geometric Invariant Theory, New York: Springer-Verlag, 1965, vol. 34.
    @book {Mum, MRKEY = {0214602},
      AUTHOR = {Mumford, David},
      TITLE = {Geometric Invariant Theory},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME={34},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1965},
      PAGES = {vi+145},
      MRCLASS = {14.55},
      MRNUMBER = {35 \#5451},
      MRREVIEWER = {F. Oort},
      ZBLNUMBER = {0147.39304},
      }
  • [H00] H. Hida, Geometric Modular Forms and Elliptic Curves, River Edge, NJ: World Sci. Publ. Co., 2000.
    @book {H00, MRKEY = {1794402},
      AUTHOR = {Hida, Haruzo},
      TITLE = {Geometric Modular Forms and Elliptic Curves},
      PUBLISHER = {World Sci. Publ. Co.},
      ADDRESS = {River Edge, NJ},
      NOTE={a list of errata available at www.math.ucla.edu/$\sim$hida},
      YEAR = {2000},
      PAGES = {xii+361},
      ISBN = {981-02-4337-5},
      MRCLASS = {11F33 (11F80 11G05)},
      MRNUMBER = {2001j:11022},
      MRREVIEWER = {Jacques Tilouine},
      ZBLNUMBER = {0960.11032},
      }
  • [H00a] Go to document H. Hida, "Control theorems of coherent sheaves on Shimura varieties of PEL type," J. Inst. Math. Jussieu, vol. 1, iss. 1, pp. 1-76, 2002.
    @article {H00a, MRKEY = {1954939},
      AUTHOR = {Hida, Haruzo},
      TITLE = {Control theorems of coherent sheaves on {S}himura varieties of {PEL} type},
      JOURNAL = {J. Inst. Math. Jussieu},
      FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu},
      VOLUME = {1},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {1--76},
      ISSN = {1474-7480},
      MRCLASS = {11G18 (11F33 11F41 11F60 14F05 14G35)},
      MRNUMBER = {2003m:11086},
      MRREVIEWER = {Jacques Tilouine},
      DOI = {10.1017/S1474748002000014},
      ZBLNUMBER = {1039.11041},
      }
  • [H06] H. Hida, "Anticyclotomic main conjectures," Doc. Math., iss. Extra Vol., pp. 465-532, 2006.
    @article {H06, MRKEY = {2290595},
      AUTHOR = {Hida, Haruzo},
      TITLE = {Anticyclotomic main conjectures},
      JOURNAL = {Doc. Math.},
      FJOURNAL = {Documenta Mathematica},
      YEAR = {2006},
      NUMBER = {Extra Vol.},
      PAGES = {465--532},
      ISSN = {1431-0635},
      MRCLASS = {11R23 (11F30 11F33 11F60 11G15 11G18)},
      MRNUMBER = {2007m:11152},
      MRREVIEWER = {David Y. Jao},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {05165905},
      }
  • [H05] H. Hida, "Non-vanishing modulo $p$ of Hecke $L$-values and application," in $L$-Functions and Galois Representations, Cambridge: Cambridge Univ. Press, 2007, pp. 207-269.
    @incollection {H05, MRKEY = {2392356},
      AUTHOR = {Hida, Haruzo},
      TITLE = {Non-vanishing modulo {$p$} of {H}ecke {$L$}-values and application},
      BOOKTITLE = {{$L$}-Functions and {G}alois Representations},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      NUMBER = {320},
      PAGES = {207--269},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2007},
      MRCLASS = {11F41 (11F60 11R23)},
      MRNUMBER = {2392356},
      ZBLNUMBER = {1156.11023},
      }
  • [H07] Go to document H. Hida, "Irreducibility of the Igusa tower," Acta Math. Sin. $($Engl. Ser.$)$, vol. 25, iss. 1, pp. 1-20, 2009.
    @article {H07, MRKEY = {2465518},
      AUTHOR = {Hida, Haruzo},
      TITLE = {Irreducibility of the {I}gusa tower},
      JOURNAL = {Acta Math. Sin. $($Engl. Ser.$)$},
      FJOURNAL = {Acta Mathematica Sinica (English Series)},
      VOLUME = {25},
      YEAR = {2009},
      NUMBER = {1},
      PAGES = {1--20},
      ISSN = {1439-8516},
      MRCLASS = {11G18 (11G15 11G25)},
      MRNUMBER = {2465518},
      DOI = {10.1007/s10114-008-6490-z},
      ZBLNUMBER = {05629478},
      }
  • [H10] Go to document H. Hida, "Vanishing of the $\mu$-invariant of $p$-Adic Hecke $L$-functions," , preprint , 2009.
    @techreport{H10,
      author = {Hida, Haruzo},
      TITLE = {Vanishing of the $\mu$-invariant of $p$-Adic Hecke $L$-functions},
      TYPE={preprint},
      YEAR={2009},
      URL={http://www.math.ucla.edu/~hida} , }
  • [Lum06] Go to document H. Hida, "Irreducibility of the Igusa tower over unitary Shimura varieties," in Shahidi Anniversary Volume, Cambridge, MA: Clay Math. Inst.
    @incollection {Lum06, MRKEY = {2465518},
      AUTHOR = {Hida, Haruzo},
      TITLE = {Irreducibility of the Igusa tower over unitary Shimura varieties},
      BOOKTITLE={Shahidi Anniversary Volume},
      SERIES={Clay Math. Proc.\/},
      PUBLISHER={Clay Math. Inst.},
      ADDRESS={Cambridge, MA},
      URL={http://www.math.ucla.edu/~hida},
      NOTE={to appear},
      }
  • [HT93] Go to document H. Hida and J. Tilouine, "Anti-cyclotomic Katz $p$-adic $L$-functions and congruence modules," Ann. Sci. École Norm. Sup., vol. 26, iss. 2, pp. 189-259, 1993.
    @article {HT93, MRKEY = {1209708},
      AUTHOR = {Hida, H. and Tilouine, J.},
      TITLE = {Anti-cyclotomic {K}atz {$p$}-adic {$L$}-functions and congruence modules},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {26},
      YEAR = {1993},
      NUMBER = {2},
      PAGES = {189--259},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {11F80 (11F33 11F85)},
      MRNUMBER = {93m:11044},
      MRREVIEWER = {Nigel Boston},
      URL = {http://www.numdam.org/item?id=ASENS_1993_4_26_2_189_0},
      ZBLNUMBER = {0778.11061},
      NOTE={see Appendix {\em Ann. of Math.} \textbf{172} (2010), p.~134},
      }
  • [Jg98] Go to document A. J. de Jong, "Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic," Invent. Math., vol. 134, iss. 2, pp. 301-333, 1998.
    @article {Jg98, MRKEY = {1650324},
      AUTHOR = {de Jong, A. J.},
      TITLE = {Homomorphisms of {B}arsotti-{T}ate groups and crystals in positive characteristic},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {134},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {301--333},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14L05 (13A35 14F30)},
      MRNUMBER = {2000f:14070a},
      MRREVIEWER = {R. T. Hoobler},
      DOI = {10.1007/s002220050266},
      ZBLNUMBER = {0929.14029},
      }
  • [K73] N. M. Katz, "$p$-adic properties of modular schemes and modular forms," in Modular Functions of One Variable, III, New York: Springer-Verlag, 1973, vol. 350, pp. 69-190.
    @incollection {K73, MRKEY = {0447119},
      AUTHOR = {Katz, Nicholas M.},
      TITLE = {{$p$}-adic properties of modular schemes and modular forms},
      BOOKTITLE = {Modular Functions of One Variable, {\rm III}},
      VENUE={{P}roc. {I}nternat. {S}ummer {S}chool, {U}niv. {A}ntwerp, {A}ntwerp, 1972},
      PAGES = {69--190},
      SERIES={ Lecture Notes in Math.},
      VOLUME={350},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1973},
      MRCLASS = {10D15 (14D20)},
      MRNUMBER = {56 \#5434},
      MRREVIEWER = {V. V. Sokurov},
      ZBLNUMBER = {0271.10033},
      }
  • [Kseeib] N. M. Katz, "Serre-Tate local moduli," in Algebraic Surfaces, New York: Springer-Verlag, 1981, vol. 868, pp. 138-202.
    @incollection {Kseeib, MRKEY = {638600},
      AUTHOR = {Katz, Nicholas M.},
      TITLE = {Serre-{T}ate local moduli},
      BOOKTITLE = {Algebraic Surfaces},
      VENUE={{O}rsay, 1976--78},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {868},
      PAGES = {138--202},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1981},
      MRCLASS = {14K10},
      MRNUMBER = {83k:14039b},
      MRREVIEWER = {William E. Lang},
      ZBLNUMBER = {0477.14007},
      }
  • [K78a] Go to document N. M. Katz, "$p$-adic $L$-functions for CM fields," Invent. Math., vol. 49, iss. 3, pp. 199-297, 1978.
    @article {K78a, MRKEY = {513095},
      AUTHOR = {Katz, Nicholas M.},
      TITLE = {{$p$}-adic {$L$}-functions for {CM} fields},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {49},
      YEAR = {1978},
      NUMBER = {3},
      PAGES = {199--297},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {10D25 (12A65 12A67 14K22)},
      MRNUMBER = {80h:10039},
      MRREVIEWER = {V. V. Shokurov},
      DOI = {10.1007/BF01390187},
      ZBLNUMBER = {0417.12003},
      }
  • [Ko92] Go to document R. E. Kottwitz, "Points on some Shimura varieties over finite fields," J. Amer. Math. Soc., vol. 5, iss. 2, pp. 373-444, 1992.
    @article {Ko92, MRKEY = {1124982},
      AUTHOR = {Kottwitz, Robert E.},
      TITLE = {Points on some {S}himura varieties over finite fields},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {5},
      YEAR = {1992},
      NUMBER = {2},
      PAGES = {373--444},
      ISSN = {0894-0347},
      MRCLASS = {11G18 (11G25 14G10 14G35)},
      MRNUMBER = {93a:11053},
      MRREVIEWER = {Min Ho Lee},
      DOI = {10.2307/2152772},
      ZBLNUMBER = {0796.14014},
      }
  • [Ml90] J. S. Milne, "Canonical models of (mixed) Shimura varieties and automorphic vector bundles," in Automorphic Forms, Shimura Varieties, and $L$-Functions, Boston, MA: Academic Press, 1990, vol. 10, pp. 283-414.
    @incollection {Ml90, MRKEY = {1044823},
      AUTHOR = {Milne, J. S.},
      TITLE = {Canonical models of (mixed) {S}himura varieties and automorphic vector bundles},
      BOOKTITLE = {Automorphic Forms, {S}himura Varieties, and {$L$}-Functions},
      SERIES = {Perspect. Math.},
      VOLUME = {10},
      PAGES = {283--414},
      PUBLISHER = {Academic Press},
      ADDRESS = {Boston, MA},
      YEAR = {1990},
      MRCLASS = {11G18 (11F03 11F12 14K22)},
      MRNUMBER = {91a:11027},
      MRREVIEWER = {Min Ho Lee},
      ZBLNUMBER = {0704.14016},
      }
  • [MS81] Go to document J. S. Milne and K. Shih, "Automorphism groups of Shimura varieties and reciprocity laws," Amer. J. Math., vol. 103, iss. 5, pp. 911-935, 1981.
    @article {MS81, MRKEY = {630772},
      AUTHOR = {Milne, J. S. and Shih, Kuang-yen},
      TITLE = {Automorphism groups of {S}himura varieties and reciprocity laws},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {103},
      YEAR = {1981},
      NUMBER = {5},
      PAGES = {911--935},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {12A67 (10D20 14G25)},
      MRNUMBER = {83b:12014},
      MRREVIEWER = {Paul B. Garrett},
      DOI = {10.2307/2374252},
      ZBLNUMBER={0475.14022},
      }
  • [Mt72] Go to document T. Miyake, "On automorphism groups of the fields of automorphic functions," Ann. of Math., vol. 95, pp. 243-252, 1972.
    @article {Mt72, MRKEY = {0300977},
      AUTHOR = {Miyake, Toshitsune},
      TITLE = {On automorphism groups of the fields of automorphic functions},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {95},
      YEAR = {1972},
      PAGES = {243--252},
      ISSN = {0003-486X},
      MRCLASS = {10D20 (14G99 32N05)},
      MRNUMBER = {46 \#137},
      MRREVIEWER = {William F. Hammond},
      DOI = {10.2307/1970798},
      ZBLNUMBER = {0249.10021},
      }
  • [BLR90] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, New York: Springer-Verlag, 1990.
    @book {BLR90, MRKEY = {1045822},
      AUTHOR = {Bosch, Siegfried and L{ü}tkebohmert, Werner and Raynaud, Michel},
      TITLE = {Néron Models},
      SERIES = {Ergeb. Math. Grenzgeb.},
      NUMBER = {21},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1990},
      PAGES = {x+325},
      ISBN = {3-540-50587-3},
      MRCLASS = {14K15 (11G10 14L15)},
      MRNUMBER = {91i:14034},
      MRREVIEWER = {James Milne},
      ZBLNUMBER={0705.14001},
      }
  • [PAF] H. Hida, $p$-Adic Automorphic Forms on Shimura Varieties, New York: Springer-Verlag, 2004.
    @book {PAF, MRKEY = {2055355},
      AUTHOR = {Hida, Haruzo},
      TITLE = {{$p$}-Adic Automorphic Forms on {S}himura Varieties},
      SERIES = {Springer Monogr. in Math.},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2004},
      PAGES = {xii+390},
      ISBN = {0-387-20711-2},
      MRCLASS = {11F33 (11F46 11F85 11G18)},
      MRNUMBER = {2005e:11054},
      MRREVIEWER = {Jacques Tilouine},
      ZBLNUMBER = {1055.11032},
      }
  • [Rap78] Go to document M. Rapoport, "Compactifications de l’espace de modules de Hilbert-Blumenthal," Compositio Math., vol. 36, iss. 3, pp. 255-335, 1978.
    @article {Rap78, MRKEY = {515050},
      AUTHOR = {Rapoport, M.},
      TITLE = {Compactifications de l'espace de modules de {H}ilbert-{B}lumenthal},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {36},
      YEAR = {1978},
      NUMBER = {3},
      PAGES = {255--335},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14D20 (10D20)},
      MRNUMBER = {80j:14009},
      MRREVIEWER = {Jean Giraud},
      URL = {http://www.numdam.org/item?id=CM_1978__36_3_255_0},
      ZBLNUMBER = {0386.14006},
      }
  • [Sh66] Go to document G. Shimura, "Moduli and fibre systems of abelian varieties," Ann. of Math., vol. 83, pp. 294-338, 1966.
    @article {Sh66, MRKEY = {0199190},
      AUTHOR = {Shimura, Goro},
      TITLE = {Moduli and fibre systems of abelian varieties},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {83},
      YEAR = {1966},
      PAGES = {294--338},
      ISSN = {0003-486X},
      MRCLASS = {14.51 (10.68)},
      MRNUMBER = {33 \#7339},
      MRREVIEWER = {S. Koizumi},
      DOI = {10.2307/1970434},
      ZBLNUMBER = {0141.37503},
      }
  • [Sh70] Go to document G. Shimura, "On canonical models of arithmetic quotients of bounded symmetric domains, I," Ann. of Math., vol. 91, pp. 144-222, 1970.
    @article {Sh70, MRKEY = {0257031},
      AUTHOR = {Shimura, Goro},
      TITLE = {On canonical models of arithmetic quotients of bounded symmetric domains, I},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {91},
      YEAR = {1970},
      PAGES = {144--222},
      ISSN = {0003-486X},
      MRCLASS = {10.68},
      MRNUMBER = {41 \#1686},
      MRREVIEWER = {K. Iyanaga},
      DOI = {10.2307/1970604},
      ZBLNUMBER = {0237.14009},
      }
  • [Sh70b] G. Shimura, "On canonical models of arithmetic quotients of bounded symmetric domains, II," Ann. of Math., vol. 92, pp. 528-549, 1970.
    @article {Sh70b, MRKEY = {0257031},
      AUTHOR = {Shimura, Goro},
      TITLE = {On canonical models of arithmetic quotients of bounded symmetric domains, II},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {92},
      YEAR = {1970},
      PAGES = {528--549},
      MRCLASS = {10D05},
      MRNUMBER = {45 \#1840},
      MRREVIEWER = {I. I. Pjateckii-Sapiro},
      ZBLNUMBER = {0237.14010},
      }
  • [Sh75] Go to document G. Shimura, "On some arithmetic properties of modular forms of one and several variables," Ann. of Math., vol. 102, iss. 3, pp. 491-515, 1975.
    @article {Sh75, MRKEY = {0491519},
      AUTHOR = {Shimura, Goro},
      TITLE = {On some arithmetic properties of modular forms of one and several variables},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {102},
      YEAR = {1975},
      NUMBER = {3},
      PAGES = {491--515},
      ISSN = {0003-486X},
      MRCLASS = {10D25 (10D15 10D20)},
      MRNUMBER = {58 \#10758},
      DOI = {10.2307/1971041},
      ZBLNUMBER = {0327.10028},
      }
  • [Sin84] Go to document W. Sinnott, "On the $\mu $-invariant of the $\Gamma $-transform of a rational function," Invent. Math., vol. 75, iss. 2, pp. 273-282, 1984.
    @article {Sin84, MRKEY = {732547},
      AUTHOR = {Sinnott, W.},
      TITLE = {On the {$\mu $}-invariant of the {$\Gamma $}-transform of a rational function},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {75},
      YEAR = {1984},
      NUMBER = {2},
      PAGES = {273--282},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11S40 (11R23 11S50)},
      MRNUMBER = {85g:11112},
      MRREVIEWER = {Lawrence Washington},
      DOI = {10.1007/BF01388565},
      ZBLNUMBER = {0531.12004},
      }
  • [Vat00a] Go to document V. Vatsal, "Uniform distribution of Heegner points," Invent. Math., vol. 148, iss. 1, pp. 1-46, 2002.
    @article {Vat00a, MRKEY = {1892842},
      AUTHOR = {Vatsal, Vinayak},
      TITLE = {Uniform distribution of {H}eegner points},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {148},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {1--46},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11G40 (11G05 11R23)},
      MRNUMBER = {2003j:11070},
      MRREVIEWER = {Massimo Bertolini},
      DOI = {10.1007/s002220100183},
      ZBLNUMBER = {1119.11035},
      }
  • [Vat00b] Go to document V. Vatsal, "Special values of anticyclotomic $L$-functions," Duke Math. J., vol. 116, iss. 2, pp. 219-261, 2003.
    @article {Vat00b, MRKEY = {1953292},
      AUTHOR = {Vatsal, Vinayak},
      TITLE = {Special values of anticyclotomic {$L$}-functions},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {116},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {219--261},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {11G40 (11F33 11F67 11G18 11R23)},
      MRNUMBER = {2004e:11069b},
      MRREVIEWER = {Jan Nekov{á}{\v{r}}},
      DOI = {10.1215/S0012-7094-03-11622-1},
      ZBLNUMBER = {1065.11048},
      }
  • [VICM] V. Vatsal, "Special values of $L$-functions modulo $p$," in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 501-514.
    @incollection {VICM, MRKEY = {2275607},
      AUTHOR = {Vatsal, Vinayak},
      TITLE = {Special values of {$L$}-functions modulo {$p$}},
      BOOKTITLE = {International {C}ongress of {M}athematicians. {V}ol. {\rm II}},
      PAGES = {501--514},
      PUBLISHER = {Eur. Math. Soc., Zürich},
      YEAR = {2006},
      MRCLASS = {11G40 (11F33 11R23)},
      MRNUMBER = {2008d:11065},
      MRREVIEWER = {Thomas A. Weston},
      ZBLNUMBER = {1101.14028},
      }
  • [Z75] J. G. Zarhin, "Endomorphisms of Abelian varieties over fields of finite characteristic," Izv. Akad. Nauk SSSR Ser. Mat., vol. 39, iss. 2, pp. 272-277, 471, 1975.
    @article {Z75, MRKEY = {0371897},
      AUTHOR = {Zarhin, Ju. G.},
      TITLE = {Endomorphisms of {A}belian varieties over fields of finite characteristic},
      JOURNAL = {Izv. Akad. Nauk {\rm SSSR} Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {39},
      YEAR = {1975},
      NUMBER = {2},
      PAGES = {272--277, 471},
      ISSN = {0373-2436},
      MRCLASS = {14G15 (14K15)},
      MRNUMBER = {51 \#8114},
      MRREVIEWER = {J.-E. Bertin},
      ZBLNUMBER={0345.14014},
      }

Mathematical Subject Classification
Primary: , , , ,
DOI
https://doi.org/10.4007/annals.2010.172.41
MR
2680417
zbMATH
1223.11131
Milestones
Received: 7 December 2004
Revised: 19 October 2008
Accepted: 3 June 2009
Published online: 27 June 2010

Authors

Haruzo Hida

University of California, Department of Mathematics, Box 951555, Los Angeles CA 90095-1555, United States