Cubic structures, equivariant Euler characteristics and lattices of modular forms

Abstract

We use the theory of cubic structures to give a fixed point Riemann-Roch formula for the equivariant Euler characteristics of coherent sheaves on projective flat schemes over $\mathbb{Z}$ with a tame action of a finite abelian group. This formula supports a conjecture concerning the extent to which such equivariant Euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus. Our results are applied to study the module structure of modular forms having Fourier coefficients in a ring of algebraic integers, as well as the action of diamond Hecke operators on the Mordell-Weil groups and Tate-Shafarevich groups of Jacobians of modular curves.

  • [A] A. Abbes, "The Grothendieck-Ogg-Shafarevich formula for arithmetic surfaces," J. Algebraic Geom., vol. 9, iss. 3, pp. 529-576, 2000.
    @article{A, MRKEY = {1752013},
      AUTHOR = {Abbes, Ahmed},
      TITLE = {The {G}rothendieck-{O}gg-{S}hafarevich formula for arithmetic surfaces},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {9},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {529--576},
      ISSN = {1056-3911},
      MRCLASS = {14G40 (14F20 14G15)},
      MRNUMBER = {2001b:14039},
      ZBLNUMBER = {0986.14015},
      MRREVIEWER = {Carlo Gasbarri},
      }
  • [BFQ] Go to document P. Baum, W. Fulton, and G. Quart, "Lefschetz-Riemann-Roch for singular varieties," Acta Math., vol. 143, iss. 3-4, pp. 193-211, 1979.
    @article{BFQ, MRKEY = {549774},
      AUTHOR = {Baum, Paul and Fulton, William and Quart, George},
      TITLE = {Lefschetz-{R}iemann-{R}och for singular varieties},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {143},
      YEAR = {1979},
      NUMBER = {3-4},
      PAGES = {193--211},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {14F12 (14C40)},
      MRNUMBER = {82c:14022a},
      DOI = {10.1007/BF02392092},
      ZBLNUMBER = {0454.14009},
      MRREVIEWER = {Allen B. Altman},
      }
  • [BLR] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, New York: Springer-Verlag, 1990.
    @book{BLR, 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},
      ZBLNUMBER = {0705.14001},
      MRREVIEWER = {James Milne},
      }
  • [Br] L. Breen, Fonctions thêta et théorème du cube, New York: Springer-Verlag, 1983.
    @book{Br, MRKEY = {823233},
      AUTHOR = {Breen, Lawrence},
      TITLE = {Fonctions thêta et théorème du cube},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {980},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1983},
      PAGES = {xiii+115},
      ISBN = {3-540-12002-5},
      MRCLASS = {14K25},
      MRNUMBER = {87d:14036},
      ZBLNUMBER = {0558.14029},
      MRREVIEWER = {N. Yui},
      }
  • [BG] Go to document J. P. Buhler and B. H. Gross, "Arithmetic on elliptic curves with complex multiplication. II," Invent. Math., vol. 79, iss. 1, pp. 11-29, 1985.
    @article{BG, MRKEY = {774527},
      AUTHOR = {Buhler, Joe P. and Gross, Benedict H.},
      TITLE = {Arithmetic on elliptic curves with complex multiplication. {II}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {79},
      YEAR = {1985},
      NUMBER = {1},
      PAGES = {11--29},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11G40 (11G05 14K22)},
      MRNUMBER = {86j:11066},
      DOI = {10.1007/BF01388654},
      ZBLNUMBER = {0584.14027},
      MRREVIEWER = {Kenneth Kramer},
      }
  • [BF] D. Burns and M. Flach, "Tamagawa numbers for motives with $($non-commutative$)$ coefficients," Doc. Math., vol. 6, pp. 501-570, 2001.
    @article{BF, MRKEY = {1884523},
      AUTHOR = {Burns, D. and Flach, M.},
      TITLE = {Tamagawa numbers for motives with $($non-commutative$)$ coefficients},
      JOURNAL = {Doc. Math.},
      FJOURNAL = {Documenta Mathematica},
      VOLUME = {6},
      YEAR = {2001},
      PAGES = {501--570},
      ISSN = {1431-0635},
      MRCLASS = {11G40 (11R65)},
      MRNUMBER = {2002m:11055},
      ZBLNUMBER = {1052.11077},
      MRREVIEWER = {Cornelius Greither},
      }
  • [CNT] Go to document P. Cassou-Noguès and M. J. Taylor, "Un élément de Stickelberger quadratique," J. Number Theory, vol. 37, iss. 3, pp. 307-342, 1991.
    @article{CNT, MRKEY = {1096447},
      AUTHOR = {Cassou-Nogu{è}s, Ph. and Taylor, M. J.},
      TITLE = {Un élément de {S}tickelberger quadratique},
      JOURNAL = {J. Number Theory},
      FJOURNAL = {Journal of Number Theory},
      VOLUME = {37},
      YEAR = {1991},
      NUMBER = {3},
      PAGES = {307--342},
      ISSN = {0022-314X},
      CODEN = {JNUTA9},
      MRCLASS = {11R29 (11G05 11R04)},
      MRNUMBER = {92e:11125},
      DOI = {10.1016/S0022-314X(05)80046-0},
      ZBLNUMBER = {0719.11075},
      MRREVIEWER = {Ernst Kleinert},
      }
  • [CW] C. Chevalley and A. Weil, "Über das Verhalten der Integrale. I: Gattung bei Automorphismen des Funktionenkörpers," Abh. Math. Semin. Hamb. Univ., vol. 10, pp. 358-361, 1934.
    @article{CW,
      author={Chevalley, Claude and Weil, A.},
      title={{Ü}ber das {V}erhalten der {I}ntegrale. {I}: {G}attung bei {A}utomorphismen des {F}unktionenkörpers},
      journal="Abh. Math. Semin. Hamb. Univ. ", volume="10", pages="358--361", year="1934", ZBLNUMBER = {0009.16001},
      JFMNUMBER = {60.0098.01},
      }
  • [C] Go to document T. Chinburg, "Galois structure of de Rham cohomology of tame covers of schemes," Ann. of Math., vol. 139, iss. 2, pp. 443-490, 1994.
    @article{C, MRKEY = {1274097},
      AUTHOR = {Chinburg, Ted},
      TITLE = {Galois structure of de {R}ham cohomology of tame covers of schemes},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {139},
      YEAR = {1994},
      NUMBER = {2},
      PAGES = {443--490},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11R33 (14F40 14G10)},
      MRNUMBER = {95h:11125a},
      DOI = {10.2307/2946586},
      ZBLNUMBER = {0828.14007},
      MRREVIEWER = {Boas Erez},
      }
  • [CEPT1] Go to document T. Chinburg, B. Erez, G. Pappas, and M. J. Taylor, "Tame actions of group schemes: integrals and slices," Duke Math. J., vol. 82, iss. 2, pp. 269-308, 1996.
    @article{CEPT1, MRKEY = {1387229},
      AUTHOR = {Chinburg, Ted and Erez, Boas and Pappas, Georgios and Taylor, Martin J.},
      TITLE = {Tame actions of group schemes: {i}ntegrals and slices},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {82},
      YEAR = {1996},
      NUMBER = {2},
      PAGES = {269--308},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {14L30 (11R33)},
      MRNUMBER = {97d:14066},
      DOI = {10.1215/S0012-7094-96-08212-5},
      ZBLNUMBER = {0907.14021},
      MRREVIEWER = {Nigel Byott},
      }
  • [CEPT2] Go to document T. Chinburg, B. Erez, G. Pappas, and M. J. Taylor, "$\epsilon$-constants and the Galois structure of de Rham cohomology," Ann. of Math., vol. 146, iss. 2, pp. 411-473, 1997.
    @article{CEPT2, MRKEY = {1477762},
      AUTHOR = {Chinburg, Ted and Erez, Boas and Pappas, Georgios and Taylor, Martin J.},
      TITLE = {{$\epsilon$}-constants and the {G}alois structure of de {R}ham cohomology},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {146},
      YEAR = {1997},
      NUMBER = {2},
      PAGES = {411--473},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {14F40 (11G09 11R33)},
      MRNUMBER = {98j:14025},
      DOI = {10.2307/2952466},
      MRREVIEWER = {Nigel Byott},
      ZBLNUMBER = {0939.14009},
      }
  • [CEPT3] T. Chinburg, B. Erez, G. Pappas, and M. J. Taylor, "Riemann-Roch type theorems for arithmetic schemes with a finite group action," J. Reine Angew. Math., vol. 489, pp. 151-187, 1997.
    @article{CEPT3, MRKEY = {1461208},
      AUTHOR = {Chinburg, Ted and Erez, Boas and Pappas, Georgios and Taylor, Martin J.},
      TITLE = {Riemann-{R}och type theorems for arithmetic schemes with a finite group action},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {489},
      YEAR = {1997},
      PAGES = {151--187},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {14C40 (11R33 14C35 14G40 14L30 19L10)},
      MRNUMBER = {98e:14006},
      ZBLNUMBER = {0903.19001},
      MRREVIEWER = {Nigel Byott},
      }
  • [CES] B. Conrad, B. Edixhoven, and W. Stein, "$J_1(p)$ has connected fibers," Doc. Math., vol. 8, pp. 331-408, 2003.
    @article{CES, MRKEY = {2029169},
      AUTHOR = {Conrad, Brian and Edixhoven, Bas and Stein, William},
      TITLE = {{$J\sb 1(p)$} has connected fibers},
      JOURNAL = {Doc. Math.},
      FJOURNAL = {Documenta Mathematica},
      VOLUME = {8},
      YEAR = {2003},
      PAGES = {331--408},
      ISSN = {1431-0635},
      MRCLASS = {11G18 (11F11 14H40)},
      MRNUMBER = {2004k:11094},
      MRREVIEWER = {Alessandra Bertapelle},
      ZBLNUMBER = {1101.14311},
      }
  • [De] P. Deligne, "Le déterminant de la cohomologie," in Current Trends in Arithmetical Algebraic Geometry, Providence, RI: Amer. Math. Soc., 1987, pp. 93-177.
    @incollection{De, MRKEY = {902592},
      AUTHOR = {Deligne, P.},
      TITLE = {Le déterminant de la cohomologie},
      BOOKTITLE = {Current Trends in Arithmetical Algebraic Geometry},
      VENUE = {{A}rcata, {CA},
      1985},
      SERIES = {Contemp. Math.},
      NUMBER = {67},
      PAGES = {93--177},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1987},
      MRCLASS = {32L10 (11G25 14F12 14H99 32G13 58G11)},
      MRNUMBER = {89b:32038},
      ZBLNUMBER = {0629.14008},
      MRREVIEWER = {Daniel S. Freed},
      }
  • [DR] P. Deligne and M. Rapoport, "Les schémas de modules de courbes elliptiques," in Modular Functions of One Variable. II, Deligne, P. and Kuyk, W., Eds., New York: Springer-Verlag, 1973, pp. 143-316.
    @incollection{DR, MRKEY = {0337993},
      AUTHOR = {Deligne, P. and Rapoport, M.},
      EDITOR = {Deligne, Pierre and Kuyk, Willem},
      TITLE = {Les schémas de modules de courbes elliptiques},
      BOOKTITLE = {Modular Functions of One Variable. {II}},
      VENUE = {Antwerp, 1972},
      PAGES = {143--316},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {349},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1973},
      MRCLASS = {14K10 (10D05)},
      MRNUMBER = {49 \#2762},
      ZBLNUMBER = {0281.14010},
      MRREVIEWER = {T. Oda},
      }
  • [Du] Go to document F. Ducrot, "Cube structures and intersection bundles," J. Pure Appl. Algebra, vol. 195, iss. 1, pp. 33-73, 2005.
    @article{Du, MRKEY = {2100310},
      AUTHOR = {Ducrot, Fran{ç}ois},
      TITLE = {Cube structures and intersection bundles},
      JOURNAL = {J. Pure Appl. Algebra},
      FJOURNAL = {Journal of Pure and Applied Algebra},
      VOLUME = {195},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {33--73},
      ISSN = {0022-4049},
      CODEN = {JPAAA2},
      MRCLASS = {14C17 (14C20 18D10)},
      MRNUMBER = {2005g:14016},
      DOI = {10.1016/j.jpaa.2004.06.002},
      ZBLNUMBER = {1067.14007},
      MRREVIEWER = {Adrian Langer},
      }
  • [F] M. Flach, "The equivariant Tamagawa number conjecture: A survey," in Stark’s Conjectures: Recent Work and New Directions, Providence, RI: Amer. Math. Soc., 2004, pp. 79-125.
    @incollection{F, MRKEY = {2088713},
      AUTHOR = {Flach, Matthias},
      TITLE = {The equivariant {T}amagawa number conjecture: {A} survey},
      BOOKTITLE = {Stark's Conjectures: {R}ecent Work and New Directions},
      SERIES = {Contemp. Math.},
      NUMBER = {358},
      PAGES = {79--125},
      INVNOTE = {With an appendix by C. Greither},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2004},
      MRCLASS = {11G40 (11G18 11R23 11R33)},
      MRNUMBER = {2005k:11132},
      ZBLNUMBER = {1070.11025},
      MRREVIEWER = {Nigel Byott},
      }
  • [Fr] A. Fröhlich, Galois Module Structure of Algebraic Integers, New York: Springer-Verlag, 1983.
    @book{Fr, MRKEY = {717033},
      AUTHOR = {Fr{ö}hlich, Albrecht},
      TITLE = {Galois Module Structure of Algebraic Integers},
      SERIES = {Ergeb. Math. Grenzgeb.},
      NUMBER = {1},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1983},
      PAGES = {x+262},
      ISBN = {3-540-11920-5},
      MRCLASS = {11R33 (11R32 11R65)},
      MRNUMBER = {85h:11067},
      MRREVIEWER = {J. Browkin},
      ZBLNUMBER = {0501.12012},
      }
  • [Fu] W. Fulton, Intersection Theory, 2nd ed., New York: Springer-Verlag, 1998.
    @book{Fu, MRKEY = {1644323},
      AUTHOR = {Fulton, William},
      TITLE = {Intersection Theory},
      SERIES = {Ergeb. Math. Grenzgeb.},
      NUMBER = {2},
      EDITION = {2nd},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1998},
      PAGES = {xiv+470},
      ISBN = {3-540-62046-X; 0-387-98549-2},
      MRCLASS = {14C17 (14-02)},
      MRNUMBER = {99d:14003},
      ZBLNUMBER = {0885.14002},
      }
  • [G] B. H. Gross, "On the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication," in Number Theory Related to Fermat’s Last Theorem, Koblitz, N., Ed., Mass.: Birkhäuser, 1982, pp. 219-236.
    @incollection{G, MRKEY = {685298},
      AUTHOR = {Gross, Benedict H.},
      EDITOR = {Koblitz, Neal},
      TITLE = {On the conjecture of {B}irch and {S}winnerton-{D}yer for elliptic curves with complex multiplication},
      BOOKTITLE = {Number Theory Related to {F}ermat's Last Theorem},
      VENUE = {Cambridge, MA, 1981},
      SERIES = {Progr. Math.},
      NUMBER = {26},
      PAGES = {219--236},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Mass.},
      YEAR = {1982},
      MRCLASS = {14G10 (12A70 14K22)},
      MRNUMBER = {84e:14020},
      ZBLNUMBER = {0506.14040},
      MRREVIEWER = {M. A. Kenku},
      }
  • [GM] A. Grothendieck and J. P. Murre, The Tame Fundamental Group of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme, New York: Springer-Verlag, 1971.
    @book{GM, MRKEY = {0316453},
      AUTHOR = {Grothendieck, Alexander and Murre, Jacob P.},
      TITLE = {The Tame Fundamental Group of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {208},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1971},
      PAGES = {viii+133},
      MRCLASS = {14E20 (14B20)},
      MRNUMBER = {47 \#5000},
      ZBLNUMBER = {0216.33001},
      MRREVIEWER = {T. Oda},
      }
  • [GS] Go to document H. Gillet and C. Soulé, "An arithmetic Riemann-Roch theorem," Invent. Math., vol. 110, iss. 3, pp. 473-543, 1992.
    @article{GS, MRKEY = {1189489},
      AUTHOR = {Gillet, Henri and Soul{é},
      Christophe},
      TITLE = {An arithmetic {R}iemann-{R}och theorem},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {110},
      YEAR = {1992},
      NUMBER = {3},
      PAGES = {473--543},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14G40 (14C15 14C40 19E99 32L10 58G26)},
      MRNUMBER = {94f:14019},
      DOI = {10.1007/BF01231343},
      ZBLNUMBER = {0777.14008},
      MRREVIEWER = {Douglas L. Ulmer},
      }
  • [I] L. Illusie, "Théorie de Brauer et caractéristique d’Euler-Poincaré," Astérisque, vol. 82–83, pp. 173-219, 1978–79.
    @article{I,
      author = {L. Illusie},
      TITLE = {Théorie de {B}rauer et caractéristique d'{E}uler-{P}oincaré},
      JOURNAL = {Astérisque},
      VOLUME = {82--83},
      YEAR = {1978--79},
      PAGES = {173--219},
      }
  • [KL] D. S. Kubert and S. Lang, Modular Units, New York: Springer-Verlag, 1981.
    @book{KL, MRKEY = {648603},
      AUTHOR = {Kubert, Daniel S. and Lang, Serge},
      TITLE = {Modular Units},
      SERIES = {Grundle. Math. Wissens.},
      NUMBER = {244},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1981},
      PAGES = {xiii+358},
      ISBN = {0-387-90517-0},
      MRCLASS = {12A45 (10D05 14G25)},
      MRNUMBER = {84h:12009},
      ZBLNUMBER = {0492.12002},
      MRREVIEWER = {Roland Gillard},
      }
  • [KM] N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Princeton, NJ: Princeton Univ. Press, 1985.
    @book{KM, MRKEY = {772569},
      AUTHOR = {Katz, Nicholas M. and Mazur, Barry},
      TITLE = {Arithmetic Moduli of Elliptic Curves},
      SERIES = {Ann. Math. Studies},
      NUMBER = {108},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1985},
      PAGES = {xiv+514},
      ISBN = {0-691-08349-5; 0-691-08352-5},
      MRCLASS = {11G05 (11F11 14G25 14K15)},
      MRNUMBER = {86i:11024},
      ZBLNUMBER = {0576.14026},
      MRREVIEWER = {Kenneth A. Ribet},
      }
  • [KS] Go to document N. M. Katz and P. Sarnak, "Zeroes of zeta functions and symmetry," Bull. Amer. Math. Soc., vol. 36, iss. 1, pp. 1-26, 1999.
    @article{KS, MRKEY = {1640151},
      AUTHOR = {Katz, Nicholas M. and Sarnak, Peter},
      TITLE = {Zeroes of zeta functions and symmetry},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {American Mathematical Society. Bulletin. New Series},
      VOLUME = {36},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {1--26},
      ISSN = {0273-0979},
      CODEN = {BAMOAD},
      MRCLASS = {11M41 (11F66 11G40)},
      MRNUMBER = {2000f:11114},
      DOI = {10.1090/S0273-0979-99-00766-1},
      ZBLNUMBER = {0921.11047},
      MRREVIEWER = {Daniel A. Goldston},
      }
  • [KMu] F. F. Knudsen and D. Mumford, "The projectivity of the moduli space of stable curves. I: Preliminaries on “det” and “Div”," Math. Scand., vol. 39, iss. 1, pp. 19-55, 1976.
    @article{KMu, MRKEY = {0437541},
      AUTHOR = {Knudsen, Finn Faye and Mumford, David},
      TITLE = {The projectivity of the moduli space of stable curves. {I}: {P}reliminaries on ``det'' and ``{D}iv''},
      JOURNAL = {Math. Scand.},
      FJOURNAL = {Mathematica Scandinavica},
      VOLUME = {39},
      YEAR = {1976},
      NUMBER = {1},
      PAGES = {19--55},
      ISSN = {0025-5521},
      MRCLASS = {14H10 (14F05 14C05)},
      MRNUMBER = {55 \#10465},
      ZBLNUMBER = {0343.14008},
      MRREVIEWER = {P. E. Newstead},
      }
  • [La1] S. Lang, Algebraic Number Theory, 2nd ed., New York: Springer-Verlag, 1994.
    @book{La1, MRKEY = {1282723},
      AUTHOR = {Lang, Serge},
      TITLE = {Algebraic Number Theory},
      SERIES = {Grad. Texts Math.},
      NUMBER = {110},
      EDITION = {2nd},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1994},
      PAGES = {xiv+357},
      ISBN = {0-387-94225-4},
      MRCLASS = {11Rxx (11-01 11-02)},
      MRNUMBER = {95f:11085},
      ZBLNUMBER = {0811.11001},
      MRREVIEWER = {M. Ram Murty},
      }
  • [La2] S. Lang, Introduction to Arakelov Theory, New York: Springer-Verlag, 1988.
    @book{La2, MRKEY = {969124},
      AUTHOR = {Lang, Serge},
      TITLE = {Introduction to {A}rakelov Theory},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1988},
      PAGES = {x+187},
      ISBN = {0-387-96793-1},
      MRCLASS = {11G30 (14C17 14H25)},
      MRNUMBER = {89m:11059},
      ZBLNUMBER = {0667.14001},
      MRREVIEWER = {Joseph H. Silverman},
      }
  • [LLR] Go to document Q. Liu, D. Lorenzini, and M. Raynaud, "Néron models, Lie algebras, and reduction of curves of genus one," Invent. Math., vol. 157, iss. 3, pp. 455-518, 2004.
    @article{LLR, MRKEY = {2092767},
      AUTHOR = {Liu, Qing and Lorenzini, Dino and Raynaud, Michel},
      TITLE = {Néron models, {L}ie algebras, and reduction of curves of genus one},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {157},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {455--518},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14G20 (11G07 14H25 14H40)},
      MRNUMBER = {2005m:14039},
      DOI = {10.1007/s00222-004-0342-y},
      ZBLNUMBER = {1060.14037},
      MRREVIEWER = {Frank Herrlich},
      }
  • [Ma] Go to document B. Mazur, "Modular curves and the Eisenstein ideal," Inst. Hautes Études Sci. Publ. Math., iss. 47, pp. 33-186, 1977.
    @article{Ma, MRKEY = {488287},
      AUTHOR = {Mazur, B.},
      TITLE = {Modular curves and the {E}isenstein ideal},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {47},
      YEAR = {1977},
      PAGES = {33--186},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {14G25 (10D05)},
      MRNUMBER = {80c:14015},
      URL = {http://www.numdam.org/item?id=PMIHES_1977__47__33_0},
      ZBLNUMBER = {0394.14008},
      MRREVIEWER = {M. Ohta},
      }
  • [McC] L. R. McCulloh, "Galois module structure of abelian extensions," J. Reine Angew. Math., vol. 375/376, pp. 259-306, 1987.
    @article{McC, MRKEY = {882300},
      AUTHOR = {McCulloh, Leon R.},
      TITLE = {Galois module structure of abelian extensions},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {375/376},
      YEAR = {1987},
      PAGES = {259--306},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {11R33},
      MRNUMBER = {88k:11080},
      ZBLNUMBER = {0619.12008},
      MRREVIEWER = {Lindsay N. Childs},
      }
  • [P1] Go to document G. Pappas, "Galois modules and the theorem of the cube," Invent. Math., vol. 133, iss. 1, pp. 193-225, 1998.
    @article{P1, MRKEY = {1626489},
      AUTHOR = {Pappas, Georgios},
      TITLE = {Galois modules and the theorem of the cube},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {133},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {193--225},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11R33 (14C40 14L30)},
      MRNUMBER = {99d:11122},
      DOI = {10.1007/s002220050244},
      ZBLNUMBER = {0923.14030},
      MRREVIEWER = {Nigel Byott},
      }
  • [P2] Go to document G. Pappas, "Cubic structures and ideal class groups," Ann. Sci. École Norm. Sup., vol. 38, iss. 3, pp. 471-503, 2005.
    @article{P2, MRKEY = {2166342},
      AUTHOR = {Pappas, Georgios},
      TITLE = {Cubic structures and ideal class groups},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {38},
      YEAR = {2005},
      NUMBER = {3},
      PAGES = {471--503},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {11G45 (14L15 19D50)},
      MRNUMBER = {2006j:11090},
      DOI = {10.1016/j.ansens.2005.03.001},
      ZBLNUMBER = {1135.11033},
      MRREVIEWER = {Adrian Langer},
      }
  • [P3] Go to document G. Pappas, "Galois module structure of unramified covers," Math. Ann., vol. 341, iss. 1, pp. 71-97, 2008.
    @article{P3, MRKEY = {2377470},
      AUTHOR = {Pappas, Georgios},
      TITLE = {Galois module structure of unramified covers},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {341},
      YEAR = {2008},
      NUMBER = {1},
      PAGES = {71--97},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11R32 (11R34 14F43 19A49)},
      MRNUMBER = {2377470},
      DOI = {10.1007/s00208-007-0183-2},
      ZBLNUMBER = {1138.11050},
      }
  • [P4] Go to document G. Pappas, "Galois module structure and the $\gamma$-filtration," Compositio Math., vol. 121, iss. 1, pp. 79-104, 2000.
    @article{P4, MRKEY = {1753111},
      AUTHOR = {Pappas, Georgios},
      TITLE = {Galois module structure and the {$\gamma$}-filtration},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {121},
      YEAR = {2000},
      NUMBER = {1},
      PAGES = {79--104},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14F05 (11R33 14C35 19A31 19E08)},
      MRNUMBER = {2001f:14032},
      DOI = {10.1023/A:1001722414377},
      ZBLNUMBER = {0968.14026},
      MRREVIEWER = {Nigel Byott},
      }
  • [P5] G. Pappas, "Chow groups and cubic structures," , preprint , 2003.
    @techreport{P5,
      author = {Pappas, Georgios},
      TITLE = {Chow groups and cubic structures},
      TYPE = {preprint},
      year = {2003},
      }
  • [Ra] M. Raynaud, Anneaux Locaux Henséliens, New York: Springer-Verlag, 1970.
    @book{Ra, MRKEY = {0277519},
      AUTHOR = {Raynaud, Michel},
      TITLE = {Anneaux Locaux Henséliens},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {169},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1970},
      PAGES = {v+129},
      MRCLASS = {13.95},
      MRNUMBER = {43 \#3252},
      ZBLNUMBER = {0203.05102},
      MRREVIEWER = {J.-P. Lafon},
      }
  • [Ri] Go to document D. S. Rim, "Modules over finite groups," Ann. of Math., vol. 69, pp. 700-712, 1959.
    @article{Ri, MRKEY = {0104721},
      AUTHOR = {Rim, Dock Sang},
      TITLE = {Modules over finite groups},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {69},
      YEAR = {1959},
      PAGES = {700--712},
      ISSN = {0003-486X},
      MRCLASS = {18.00},
      MRNUMBER = {21 \#3474},
      DOI = {10.2307/1970033},
      ZBLNUMBER = {0092.26104},
      MRREVIEWER = {T. Nakayama},
      }
  • [Ro1] Go to document P. Roberts, "Abelian extensions of regular local rings," Proc. Amer. Math. Soc., vol. 78, iss. 3, pp. 307-310, 1980.
    @article{Ro1, MRKEY = {553363},
      AUTHOR = {Roberts, Paul},
      TITLE = {Abelian extensions of regular local rings},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {78},
      YEAR = {1980},
      NUMBER = {3},
      PAGES = {307--310},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {13H05 (13H10)},
      MRNUMBER = {81a:13017},
      DOI = {10.2307/2042312},
      ZBLNUMBER = {0436.13011},
      MRREVIEWER = {R. A. Wiegand},
      }
  • [Ro2] P. C. Roberts, Multiplicities and Chern Classes in Local Algebra, Cambridge Univ. Press, 1998.
    @book{Ro2, MRKEY = {1686450},
      AUTHOR = {Roberts, Paul C.},
      TITLE = {Multiplicities and {C}hern Classes in Local Algebra},
      SERIES = {Cambridge Tracts in Mathematics},
      NUMBER = {133},
      PUBLISHER = {Cambridge Univ. Press},
      INVADDRESS = {Cambridge},
      YEAR = {1998},
      PAGES = {xii+303},
      ISBN = {0-521-47316-0},
      MRCLASS = {13D22 (13H15 14C17)},
      MRNUMBER = {2001a:13029},
      ZBLNUMBER = {0917.13007},
      MRREVIEWER = {Hans-Bjørn Foxby},
      }
  • [SGA4a] Go to document Théorie des Topos et Cohomologie Étale des Schémas. Tome 1: Théorie des topos. Exp. I-IV., Artin, M., Grothendieck, A., and Verdier, J. L., Eds., New York: Springer-Verlag, 1972.
    @book{SGA4a, MRKEY = {0354652},
      key = {SGA\,4\,I},
      TITLE = {Théorie des Topos et Cohomologie Étale des Schémas. {T}ome 1: {T}héorie des topos. Exp. I-IV.},
      VENUE = {Séminaire de Géométrie Algébrique du Bois-Marie 1963--1964 (SGA 4)},
      EDITOR = {Artin, M. and Grothendieck, A. and Verdier, J. L.},
      INVNOTE = {Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {269},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1972},
      PAGES = {xix+525},
      MRCLASS = {14-06},
      MRNUMBER = {50 \#7130},
      DOI = {10.1007/BFb0081551},
      ISBN = {978-3-540-05896-0},
      ZBLNUMBER = {0234.00007},
      }
  • [SGA4b] Go to document Théorie des topos et cohomologie étale des schémas. Tome 2. Exposes V à VIII., Artin, M., Grothendieck, A., and Verdier, J. L., Eds., New York: Springer-Verlag, 1972.
    @book{SGA4b, key = {SGA\,4\,II},
      MRKEY = {0354653},
      TITLE = {Théorie des topos et cohomologie étale des schémas. {T}ome 2. Exposes V à VIII.},
      VENUE = {Séminaire de Géométrie Algébrique du Bois-Marie 1963--1964 (SGA 4)},
      EDITOR = {Artin, M. and Grothendieck, A. and Verdier, J. L.},
      INVNOTE = {Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {270},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1972},
      PAGES = {iv+418},
      MRCLASS = {14-06},
      MRNUMBER = {50 \#7131},
      DOI = {10.1007/BFb0061319},
      ISBN = {978-3-540-06012-3},
      ZBLNUMBER = {0237.00012},
      }
  • [SGA4c] Go to document Théorie des Topos et Cohomologie Étale des Schémas. Tome 3. Exposés IX à XIX., Artin, M., Grothendieck, A., and Verdier, J. L., Eds., New York: Springer-Verlag, 1973.
    @book{SGA4c, invkey = {SGA\,4\,III},
      MRKEY = {0354654},
      TITLE = {Théorie des Topos et Cohomologie Étale des Schémas. {T}ome 3. Exposés IX à XIX.},
      VENUE = {Séminaire de Géométrie Algébrique du Bois-Marie 1963--1964 $=$ SGA 4},
      EDITOR = {Artin, M. and Grothendieck, A. and Verdier, J. L.},
      INVNOTE = {Avec la collaboration de P. Deligne et B. Saint-Donat},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {305},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1973},
      PAGES = {vi+640},
      MRCLASS = {14-06},
      MRNUMBER = {50 \#7132},
      ZBLNUMBER = {0245.00002},
      DOI = {10.1007/BFb0070714},
      ISBN = {978-3-540-06118-2},
      }
  • [SGA6] Théorie des Intersections et Théorème de Riemann-Roch, Berthelot, P., Grothendieck, A., and Illusie, L., Eds., New York: Springer-Verlag, 1971.
    @book{SGA6, invkey = {SGA\,6},
      MRKEY = {0354655},
      EDITOR = {Berthelot, P. and Grothendieck, A. and Illusie, L.},
      TITLE = {Théorie des Intersections et Théorème de {R}iemann-{R}och},
      SERIES = {Lecture Notes in Math.},
      NUMBER = {225},
      VENUE = {S{é}minaire de G{é}om{é}trie Alg{é}brique du Bois-Marie 1966--1967 $=$ SGA 6},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1971},
      PAGES = {xii+700},
      MRCLASS = {14-06},
      MRNUMBER = {50 \#7133},
      ZBLNUMBER = {0218.14001},
      }
  • [T1] Go to document R. W. Thomason, "Lefschetz-Riemann-Roch theorem and coherent trace formula," Invent. Math., vol. 85, iss. 3, pp. 515-543, 1986.
    @article{T1, MRKEY = {848684},
      AUTHOR = {Thomason, R. W.},
      TITLE = {Lefschetz-{R}iemann-{R}och theorem and coherent trace formula},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {85},
      YEAR = {1986},
      NUMBER = {3},
      PAGES = {515--543},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14F20 (11G45 14F15 18F25 19F27)},
      MRNUMBER = {87j:14028},
      DOI = {10.1007/BF01390328},
      ZBLNUMBER = {0653.14005},
      MRREVIEWER = {Gerd Faltings},
      }
  • [T2] Go to document R. W. Thomason, "Une formule de Lefschetz en $K$-théorie équivariante algébrique," Duke Math. J., vol. 68, iss. 3, pp. 447-462, 1992.
    @article{T2, MRKEY = {1194949},
      AUTHOR = {Thomason, R. W.},
      TITLE = {Une formule de {L}efschetz en {$K$}-théorie équivariante algébrique},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {68},
      YEAR = {1992},
      NUMBER = {3},
      PAGES = {447--462},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {19E08 (14C35 14L30)},
      MRNUMBER = {93m:19007},
      DOI = {10.1215/S0012-7094-92-06817-7},
      ZBLNUMBER = {0813.19002},
      MRREVIEWER = {V. P. Snaith},
      }
  • [V] I. Vidal, "Théorie de Brauer et conducteur de Swan," J. Algebraic Geom., vol. 13, iss. 2, pp. 349-391, 2004.
    @article{V, MRKEY = {2047703},
      AUTHOR = {Vidal, Isabelle},
      TITLE = {Théorie de {B}rauer et conducteur de {S}wan},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {13},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {349--391},
      ISSN = {1056-3911},
      MRCLASS = {14F22},
      MRNUMBER = {2005m:14030},
      ZBLNUMBER = {1070.14020},
      MRREVIEWER = {Tihomir A. Petrov},
      }
  • [Wa] L. C. Washington, Introduction to Cyclotomic Fields, New York: Springer-Verlag, 1982.
    @book{Wa, MRKEY = {718674},
      AUTHOR = {Washington, Lawrence C.},
      TITLE = {Introduction to Cyclotomic Fields},
      SERIES = {Grad. Texts in Math.},
      NUMBER = {83},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1982},
      PAGES = {xi+389},
      ISBN = {0-387-90622-3},
      MRCLASS = {11-01 (11R18 11R23)},
      MRNUMBER = {85g:11001},
      ZBLNUMBER = {0484.12001},
      MRREVIEWER = {T. Mets{ä}nkyl{ä}},
      }

Authors

Ted Chinburg

Department of Mathematics
University of Pennsylvania David Rittenhouse Lab
209 South 33rd Street
Philadelphia, PA 19104-6395
United States

Georgios Pappas

Department of Mathematics
Michigan State University
East Lansing, MI 48824
United States

Martin J. Taylor

Department of Mathematics
University of Manchester
M60 1QD
United Kingdom