References

[AAF]

[AAG]

[AAQ]

[ABV]

[ACM]

[AGD]

[ALF]

[ALG]

[AME]

[ARE]

[ARG]

[BALl [BCM] [BLI] [BNT] [CBT]

Books

G. Shimura, Arithmetic2ty in the Theory of Automorphic Forms, Math­ematical Surveys and Monographs 82, American Mathematical Society, Providence, RI, 2000. S. S. Gelbart, Automorphic Forms on Adele Groups, Annals of Math. Studies 83, Princeton University Press, Princeton, NJ, 1975. M.-F. Vigneras, Anthmetique des algebres de quaternions, Lecture Notes in Mathematics 800, Springer, New York, 1980. D. Mumford, Abelian Varieties, TIFR Studies in Mathematics, Oxford University Press, New York, 1994. G. Shimura, Abelian Varieties w2th Complex Multiplication and Modular Functions, Princeton University Press, Princeton, NJ, 1998. A. Borel and G. D. Mostow, editors, Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math. IX, American Mathematical Society, Providence, RI, 1966. K. Iwasawa, Algebraic FUnctions, Translation from the 1973 Japanese edition by Goro Kato. Translations of Mathematical Monographs, 118, American Mathematical Society, Providence, RI, 1993. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977. N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Math. Studies 108, Princeton University Press, Princeton, NJ, 1985. J.-P. Serre, Abelian £-Ad2c Representations and Elliphc Curves, Ben­jamin, New York, 1968. G. Cornell and J. H. Silverman, editors, Arithmetic Geometry, Springer, New York, 1986. N. Bourbaki, Algebre, Hermann, Paris, 1958. N. Bourbaki, Algbre Commutative, Hermann, Paris, 1961-1965. N. Bourbaki, Groupes et Algebres de L2e, Hermann, Paris, 1972-1985. A. Weil, Basic Number Theory, Springer, New York, 1974. W. Messing, The Crystals Associated to Barsoth-Tate Groups; With Ap­plications to Abelwn Schemes, Lecture Notes in Mathematics 264, New York, Springer, 1972.

376

[CDR]

[CFT] [CGP]

[CPS] [CRT]

[CSM]

[DAV]

[DGH]

[ECH]

[EGA]

[EPE]

[FAN]

[FGA]

[GAN] [GCH]

[GCS]

[GIT]

[GME]

[HAL]

[HMS]

[IAT]

[ICF]

References

A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Annals of Math. Studies 94, Princeton University Press, Princeton, NJ, 1980. E. Artin and J. Tate, Class Field Theory, Benjamin, New York, 1968. K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics 87, Springer, New York, 1982. G. Shimura, Collected Papers, I, II, III, IV, Springer, New York, 2002. H. Matsumura, Commutative Ring Theory, Cambridge Studies in Ad­vanced Mathematics 8, Cambridge Univ. Press, New York, 1986. C.-L. Chai, Compactijication of Siegel Moduli Schemes, LMS Lecture Note Series 107, Cambridge University Press, New York, 1985. G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties, Springer, New York, 1990. J. Tilouine, Deformations of Galois Representations and Hecke Algebras (published for the Mehta Research Institute of Mathematics and Mathe­matical Physics, Allahabad), Narosa Publishing House, New Delhi, 1996. J. S. Milne, Etale Cohomology, Princeton University Press, Princeton, NJ, 1980. A. Grothendieck and J. Dieudonne, Elements de Geometrie Algebrique, Publications IRES 4 (1960), 8 (1961), 11 (1961), 17 (1963), 20 (1964), 24 (1965), 28 (1966), 32 (1967). G. Shimura, Euler Products and Eisenstein Series, CBMS Regional Con­ference Series 93, American Mathematical Society, Providence, RI, 1997. D. Ramakrishnan and R. J. Valenza, Fourier Analysis on Number Fields, Graduate Texts in Mathematics 186, Springer, New York, 1999. A. Grothendieck, Fondements de la Geometrie Algebrique, Seminaire Bourbaki expo no.149 (1956/57), 182 (1958/59), 190 (1959/60), 195 (1959/60),212 (1960/61), 221 (1960/61), 232 (1961/62), Benjamin, New York, 1966. M. Lazard, Groupes Analytiques p-Adiques, Publications IHES 26, 1965. J.-P. Serre, Galo~s Cohomology, in Monographs in Mathematics, Springer, New York, 2002. M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Annals of Math. Studies 151, Princeton University Press, Princeton, NJ, 200l. D. Mumford, Geometric Invariant Theory, Ergebnisse 34, Springer, New York, 1965. H. Hida, Geometric Modular Forms and Elliptic Curves, World Scientific, Singapore, 2000. P. J. Hilton and U. Stammback, A Course in Homological Algebra, Grad­uate Texts in Mathematics 4, Springer, New York, 1970. P. Deligne, J. S. Milne, A. Ogus and K.-y. Shih, Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics 900, Springer, New York, 1982. G. Shimura, Introduction to the Arithmet~c Theory of Automorphic Func­tions, Princeton University Press, Princeton, NJ, and Iwanami Shoten, Tokyo, 1971. L. C. Washington, Introduction to Cyclotomic Fields, Graduate Text in Mathematics, 83, Springer, New York, 1980.

[LAP]

[LFE]

[MFG]

[MFM] [MTV]

[NAZ]

[NMD]

[RAG]

[RTP]

[SFT] [SGA]

[SGL]

[TEB]

[TFG]

References 377

H. Yoshida, Lectures on Absolute CM Period, Mathematical Surveys and Monographs 106, American Mathematical Society, Providence, RI, 2003. H. Hida, Elementary Theory of L-Functions and Eisenstein Series, LMSST 26, Cambridge University Press, Cambridge, England, 1993. H. Hida, Modular Forms and Galois Cohomology, Cambridge Studies in Advanced Mathematics 69, Cambridge University Press, Cambridge, England, 2000. T. Miyake, Modular Forms, Springer, New York-Tokyo, 1989. U. Jannsen, S. Kleiman and J.-P. Serre, Motives, Proc. Symp. Pure Math. 55 Part 1 and 2, American Mathematical Society, Providence, RI, 1994. N. Koblitz, p-Adic Numbers, p-Adic Analysis, and Zeta Functions, Grad­uate Text in Mathematics, 58, Springer, New York, 1977. S. Bosch, W. Liitkebohmert, and M. Raynaud, Neron Models, Springer, New York, 1990. J. C. Jantzen, Representations of Algebraic Groups, Academic Press, Or­lando, FL, 1987. G. Faltings and G. Wiistholtz et aI, Rational Points, Aspects of Mathe­matics E6, Friedr. Vieweg & Sohn, Braunschweig, 1992. G. E. Breadon, Sheaf Theory, McGraw-Hill, New York, 1967. A. Grothendieck, Revetements Etale et Groupe Fondamental, Seminaire de geometrie algebrique, Lecture Notes in Mathematics 224, Springer, New York, 1971. H. Hida, On the Search of Genuine p-Adic Modular L-Functions for GL(n), Mem. SMF 67, 1996. G. Kempf, F. Knudson, D. Mumford and B. Saint-Donat, Toroidal Em­beddings I, Lecture Notes in Mathematics 339, Springer, New York, 1973. A Grothendieck and J. P. Murre, The Tame Fundamental Group of a Formal Neighborhood of a Divisor with Normal Crossings on a Scheme, Lecture Notes in Mathematics 208, Springer, New York, 1971.

Articles

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[BI] D. Blasius, Elliptic curves, Hilbert modular forms, and the Hodge con­jecture, in Contributions to Automorphic Forms, Geometry, and Number Theory, a supplement of the American Journal of Mathematics, Johns Hopkins University Press, Baltimore, MD, 2004.

[BlR] D. Blasius and J. D. Rogawski, Motives for Hilbert modular forms, In­ventiones Math. 114 (1993), 55-87.

[BlRl] D. Blasius and J. D. Rogawski, Zeta functions of Shimura varieties, In "Motive": [MTV], Proc. Symp. Pure Math. 55 (1994) Part 2, 525-571.

[Bo] A. Borel, Reduction theory of arithmetic groups, Proc. Symp. Pure Math. 9 (1966), 20-25 (CEuvres II 679-684, No.74).

[BoS] A. Borel and J. P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1974), 436-491 (Borel's CEuvres III 244-299 No.98).

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[BrCDT] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q or Wild 3-adic exercises, Journal AMS 14 (2001), 843-939.

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[Du] M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktio­nenkorper, Abhandlungen Math. Sem. Hansischen Universitat 14 (1941), 197-272.

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[H85] H. Hida, A p-adic measure attached to the zeta functions associated with two elliptic modular forms. I, Inventiones Math. 79 (1985), 159-195.

[H86a] H. Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. Ec. Norm. Sup. 4th series 19 (1986), 231-273.

[H86b]

[H88]

[H89a]

[H89b]

[H91]

[H94]

[H95]

[H98]

[H02]

[H03a]

[H03b]

[HM]

[HT]

[HI]

[Hz]

[1]

[K]

[K1]

[K2]

[K3]

[Kn]

[Kn1]

References 379

H. Hida, Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms, Inventiones Math. 85 (1986), 545-613. H. Hida, On p-adic Hecke algebras for G L2 over totally real fields, Ann. of Math. 128 (1988), 295- 384. H. Hida, On nearly ordinary Hecke algebras for GL(2) over totally real fields, Adv. Studies in Pure Math. 17 (1989), 139-169. H. Hida, Nearly ordinary Hecke algebras and Galois representations of several variables, Proc. JAMl Inaugural Conference, Supplement to Amer. J. Math. (1989), 115-134. H. Hida, On p-adic L-functions of GL(2) x GL(2) over totally real fields, Ann. Inst. Fourier 41 (1991),311-391. H. Hida, On the critical values of L-functions of GL(2) and GL(2) x GL(2), Duke Math. J. 74 (1994), 431-529. H. Hida, Control theorems of p-ordinary cohomology groups for SL(n), Bull. SMF 123 (1995), 425-475. H. Hida, Automorphic induction and Leopoldt type conjectures for GL(n), Asian J. Math. 2 (1998),667-710. H. Hida, Control theorems of coherent sheaves on Shimura varieties of PEL-type, Journal of the Inst. of Math. Jussieu, 2002 1, 1-76 (preprint downloadable at www.math.ucla.edu{bida). H. Hida, p-Adic automorphic forms on reductive groups, to ap­pear in The Proceedings of the Automorphic Semester of the Year 2000 at l'Institut Henri Poincare (preprint downloadable at www.math.ucla.edu{bida). H. Hida, Automorphism groups of Shimura varieties of PEL type, preprint, 2003 (preprint downloadable at www.math.ucla.edu{bida). H. Hida and Y. Maeda, Non-abelian base-change for totally real fields, Special Issue of Pacific J. Math. in memory of Olga Taussky Todd, 189-217, 1997. H. Hida and J. Tilouine, Anticyclotomic Katz p-adic L-functions and congruence modules, Ann. Sci. Ec. Norm. Sup. 4th series 26 (1993), 189-259. D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 37 (2000), 407-436. A. Hurwitz, Uber die Entwicklungkoeffizienten der lemniskatischen Funk­tionen, Math. Ann. 51 (1899), 196- 226 (Werke II, 342-373). J. Igusa, Kroneckerian model of fields of elliptic modular functions, Amer. J. Math. 81 (1959),561-577. N. M. Katz, p-adic properties of modular schemes and modular forms, Lecture notes in Math. 350 (1973), 70189. N. M. Katz, Higher congruences between modular forms, Annals of Math. 101 (1975),332-367. N. M. Katz, Serre-Tate local moduli, In "Surfaces Algebriques," Lecture Notes in Math. 868 (1978), 138-202 N. M. Katz, p-adic L-functions for CM fields, Inventiones Math. 49 (1978), 199-297. M. Kneser, Strong approximation, in [AGD]: Proc. Symp. Pure Math. IX (1966), 187-196. M. Kneser, Hasse principle for HI of simply connected groups, in [AGD]: Proc. Symp. Pure Math. IX (1966), 159-163.

380 References

[Ko] R. Kottwitz, Points on Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373-444.

[L] M. J. Larsen, Arithmetic compactification of some Shimura surfaces, CRM Publications 13 (1992), 31-45.

[LaR] R. P. Langlands and M. Rapoport, Shimuravarietiiten und Gerben, J. Reine. Angew. Math. 378 (1987), 113-220.

[MaM] Y. Matsushima and S. Murakami, On vector bundle valued harmonic forms on symmetric spaces, Ann. of Math. 78 (1963), 365-416.

[Mi] J. Milne, Canonical models of (mixed) Shimura varieties and automor­phic vector bundles, Perspective Math. 10 (1990), 283-414.

[Mil] J. Milne, The points on a Shimura variety modulo a prime of good re­duction, CRM Publications 13 (1992) ([ZFP]), 151-253

[Mi2] J. Milne, Shimura varieties and motives, Proc. Symp. Pure Math. 55 (1994) ([MTV]) Part 2, 447-523.

[MiS] J. Milne and K-y. Shih, Automorphism groups of Shimura varieties and reciprocity laws, Amer. J. Math. 103 (1981), 911-935.

[Mt] T. Miyake, On automorphism groups of the fields of automorphic func­tions, Ann. of Math. 95 (1972), 243-252.

[Mu] D. Mumford, Biextensions of formal groups, in Algebraic Geometry, Ox­ford Univ. Press, Oxford (1969), 307-322.

[Mu1] D. Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24 (1972), 239-272.

[P] A. A. Panchishkin, On the Siegel-Eisenstein measure and its applica­tions, in "Proceedings of the Conference on p-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998)". Israel J. Math. 120 (2000), part B, 467-509.

[Ra] M. Rapoport, Compactifications de l'espace de modules de Hilbert­Blumenthal, Compo Math. 36 (1978), 255-335.

[Ri] K. A. Ribet, P-adic interpolation via Hilbert modular forms, Proc. Symp. Pure Math. 29 (1975), 581-592

[SBT] 1. Shepherd-Barron and R. Taylor, Mod 2 and mod 5 icosahedral repre­sentations, J. AMS 10 (1997), 283-298.

[Sc] A. J. Scholl, Motives for modular forms, Inventiones Math. 100 (1990), 419-430.

[Se] J.-P, Serre, Geometrie algebrique et geometrie analytique, Annales Inst. Fourier 6 (1955), 1-42 (CEuvres I 402-443, No. 32).

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[Se2] J.-P, Serre, Proprietes galoisiennes des points d'ordre fini des courbes elliptiques, Inventiones Math. 15 (1972), 259-331 (CEuvres III 1-73, No. 94).

[Se3] J.-P. Serre, Formes modulaires et fonctions zeta p-adiques, Lecture notes in Math. 350 (1973), 191-268 (CEuvres III 95-172, No. 97).

[Se4] J.-P. Serre, Quelques applications du theoreme de densite de Chebotarev, Publ. IRES. 54 (1981), 323-401 (CEuvres III 563-641, No. 125).

[Sh] G. Shimura, On analytic families of polarized abelian varieties and auto­morphic functions, Ann. of Math. 78 (1963), 149-192 ([63b] in [CPS] I).

[Sh1] G. Shimura, On modular correspondences for Sp(N, £::) and their con­gruence relations, Proc. NAS 49 (1963), 824-828 ([63d] in [CPS] I).

[Sh2]

[Sh3]

[Sh4]

[Sh5]

[Sh6]

[Sh7]

[Sh8]

lSi]

[SkW]

[Sp]

[T]

[Tl]

[T2]

[Ta]

[Tal]

[Ta2]

[TaW]

[TiU]

[Tt]

[Ttl]

[Wa]

[We]

References 381

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[Zl] J. G. Zarhin, Endomorphisms of abelian varieties over fields of finite characteristics, Math. USSR Izvestija 9 (2) (1975), 255-260.

[Zi] T. Zink, Uber die Schlechte Reduktion einiger Shimuramanifaltigkeiten, Compositio Math. 45 (1981), 15-107.

Symbol Index

(-, ')N, 103 (z),180 (3), 125, 173

00*, 167 .:1(E,w),52 .:10('Jt) , 165 .:10 (pOO'Jt) , 168 .:1H , 235 .:1N , 235

E~, 170 ES, 170 i\('Jt), Tl('Jt), r('Jt), 108 /-LN,71

/-L'J1, 103 Tio(T), 69 Til(S,S), Til(X,X), 133,216 p~, 7 "-'.k,119

"-'.;, "-'.'" E' 171 w(f), 246, 248

B-ALG, ALG/B , 68 AE ,354 C(1l,b,qw),110 c('\), 100 D(m,n),239 Erl ('J1), 103

i E x /s , 264 e(z), e(x), 24, 103 ET/ x ,216 F~, 100

J, J(p), 365 Gk(rdN); R), 78, 86 Gk(r1 (N); 1fp), Grd(rdN); R), 87 G"JK, E; R), hd'Jt, x; R), 177 S:i, 6 h~ord(K, E; R), 189 H,q,233 H~ms(P, Pi), Isoms(P, Pi), 297

H(X, Y)/s, W· s ' (X, Y)/s, 263 i oo , ip, 12 Ig, Igs0(p r 'J1) , IgK , 164 Jac(V), 45 .R, .R(p), 365 [KgK'], 165 M(c, r(N)), 112 9J1 d ,N, 277

9J1cu, 9J1~ir, 366, 367 Pic(V), Pico (V), 45 Picx/s, 270 R, R(S, .:1s ), 227, 228

ResR' / RH/ R', 97 SCH/ B , B-SCH, 68 Sd'Jt) , si('Jt), S('Jt), 108 S(E), 252 §/1H:,305 Sh(G, X), Sh(p), 303, 309, 313, 325 S",(K,E;R),177 s;:ord(K, E; R), 189 Tate u ,b(q),116 Tv(Y), Uv(Y), 174, 188

Vcusp(i'o('Jt)), Vcusp(ro('Jt)), 125 Xc, Xcu, 305, 306 3,105

Statement Index

(E), 143

(Abel), 69 (Al-4), (A4'), 277, 278 (Bl-2),319 (e),345 (el-3),211 (eT),322 (Dl-4),321 (dl-2), 342 (det), 99, 268, 308 (E), 119, 338 (El-3),68 (exO-3), 170 (F), 345 (Fl-2),212 (GO-2), 76 (G3), 79 (Hpl-2),335 (Ism), 106 (Ll-2),304

(ord), 353, 369 (Pl-3), 59, 350 (pel-4), no (Pc1-4),283 (pol), 134, 312, 315 (pos),305 (QS), 219 (q-exp) , 119 (rc1-2),325 (rml-4),99 (S), 218 (Sl-3),218 (sl-3), 164 (SAl-3), 193 (SB),203 (S81-2),201 (Se),322 (se), 305 (ul-4),355 (unr),304 (Vl-4),37

Subject Index

Abelian scheme, 98, 264 of CM type, 4 formal,284 ordinary, 101 ordinary semi-abelian, 118 polarization of, 99, 306 Raynaud extension of, 284 rigidity of endomorphisms, 264 semi-abelian scheme, 116, 118, 265 Weil pairing of, 103, 350 with complex multiplication, 4

Abel's theorem, 56 Algebra

double coset, 165, 227 Hecke, 177, 189, 191 Iwasawa,8 opposite, 351

Automorphic, factor, 6 function field, 364

Automorphic form, central character, 171 classical, 330, 335 cusp form, 119 false, 329, 335 A-adic,91 p-adic analytic family of, 91 true, 330, 335

Cartier duality, 71 Cases A, B, C, D, 316 Category,

categorical quotient, 211

direct product, 210 direct sum, 210 fibered category, 98, 99 fibered product, 210 fibered sum, 210 final object, 210 Galois, 212 initial object, 210 test object, 86

Character, arithmetic, 176 Hecke, 176 totally even, 177

CM field, 23, 154 CM point, 4, 154, 303 Cohomology,

interior, 233 Compactification,

minimal, 284 toroidal, 284

Cone, decomposition, 110 simplicial, 110

Congruence subgroup, 6 cocompact, 6

Correspondence algebraic, 164 Jacquet-Langlands and Shimizu, 202

Curve, Igusa,82

Differential, holomorphic, 46 of the first kind, 46

388 Subject Index

Divisor, 45 algebraic equivalence of, 45 degree of, 45 linear equivalence of, 45

Duality, Cartier, 71

Elliptic curve, 21, 68 j-invariant of, 22 Tate, 78 Weierstrass equation of, 52

Elliptic modular form, p-adic, 10 q-expansion of, 63

Frobenius, absolute, 80 element, 20 relative, 80

Function field, arithmetic automorphic, 364 smooth model of, 42

Functor, exact functor, 212 fiber functor, 99 fundamental, 212 left exact, 212 Picard, 69, 270 representability, 72, 75, 209 representable, 209 right exact, 212

Fundamental group, of (C, F), 212 path,215 pro-object, 213

Group, Barsotti-Tate, 346 cocenter, 326 expanding semi-, 165, 228 formal,346 maximal at p, 307 p-divisible, 346 Picard,45 quasi-split, 218 special unitary, 305 split, 218 symplectic, 224 types A, B, C, D, 316

unitary, 224 Group scheme,

Barsotti-Tate, 346 cyclic subgroup, 126, 127 etale cyclic subgroup, 126, 127 formal,346 locally free, 71 p-divisible, 346

Hasse, invariant, 118 principle, 319

Hecke, algebra for GL(2), 177 character, 176 module, 7 nearly ordinary Hecke algebra, 189 operator, 128, 165 polynomial, 235 universal p-nearly ordinary Hecke

algebra, 191 Hermitian form,

anisotropic, 237 isotropic, 238 null space, 236 radical, 236 signature, 237 totally isotropic, 238

Hermitian symmetric domain bounded realization, 240 unbounded realization, 240

Igusa, curve, 82 tower, 119, 295, 330

Induction, algebrogeometric, 231 continuous, 333 regularly induced, 227 smooth,226

Involution, of the first and the second kind, 304 positive, 304

Isogeny, 265 degree of, 265 etale, 168 prime-to-E, 138 symmetric, 99

Iwasawa function, 11

Jacobian, 45 Jacquet module, 226

Level structure, r-structure, 124 r(N), r 1 (1J1)-structure, 102, 103 Fo(IJ1)-structure, 127

Modular form, central character, 171 cusp form, 119 A-adic form, 91 p-adic analytic family, 91

Module, of finite corank, 339 Hecke,7 Jacquet, 226 Tate, 307

Modulus, Moduli, 75 coarse, 72, 75 fine, 72 representability, 72, 75, 209

Morphism, epimorphism, 209 monomorphism, 209 separable, 216

Nearly ordinary, Hecke algebra, 189 representation, 234 universal Hecke algebra, 191

Norm, reduced, 199

Null space, 236

Object, connected, 212 final, 210 fundamental pro-object, 213 initial, 210 test, 86

Operator, diamond, 125, 171, 173 Hecke, 128, 165

p-adic, analytic family of, 91 ring, 127

Parameter,

Subject Index 389

adapted to a differential, 51 Path, 215 Picard,

functor, 69, 270 group, 45 scheme, 69

Pliicker coordinate, 256 Polarization, 99, 307

c-polarization, 100 D-linear, 307 symmetric, 99

Polygon, 235 Hodge, 235 Newton, 235

Positive, involution, 304 totally, 100

Radical, of Hermitian space, 236

Raynaud extension, 284 Reflex,

field, 132, 306, 325 Representation,

admissible, 226 continuously induced, 333 nearly p-ordinary, 234 regularly induced, 227 schematic, 76, 332 smooth, 226 universal, 331

Residue, field, 41 theorem, 48

Riemann-Roch theorem, 49 Rigidity,

of endomorphisms, 264

Scheme, abelian, 98, 264 ample invertible sheaf on, 276 Hilbert, 261 moduli, 75 ordinary abelian, 101 Picard, 69 quasi-projective, 261 semi-abelian, 116, 118, 265 stable point of, 273 strongly projective, 265

390 Subject Index

strongly quasi-projective, 265 Wei! restriction of, 97

Sheaf, ample invertible, 276 jppj, 346 of cusp forms, 119

Shimura variety, abelian type, 326 CM point of, 4, 154, 303 geometrically irreducible component

of, 141 neutral component of, 142 reflex field, 132, 306, 325 special point of, 325

Tate, Barsotti~, 346 curve, 78 module, 307 semi-abelian scheme, 116, 118, 265

Test object, 86 Theorem,

Abel's, 56 of Chai~Faltings, 283, 297 of Faltings~Zarhin, 141

residue, 48 Riemann~Roch, 49 vertical control, 9

Trace, reduced, 199

U niformizer, 42

Valuation ring, 37 discrete, 37 DVR,37 uniformizer of, 42

Weierstrass equation, 52 Weight, 5

dominant, 5, 231 double digit, 171 positive, 331 regular, 231

Wei!, pairing, 103, 350 restriction, 97

Zariski topology, 44 Zariski~Riemann space, 39