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Shimura G. — Introduction to Arithmetic Theory of Automorphic Functions
Shimura G. — Introduction to Arithmetic Theory of Automorphic Functions

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Название: Introduction to Arithmetic Theory of Automorphic Functions

Автор: Shimura G.


The theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their number-theoretical aspects.

After two chapters geared toward elementary levels, there follows a detailed treatment of the theory of Hecke operators, which associate zeta functions to modular forms. At a more advanced level, complex multiplication of elliptic curves and abelian varieties is discussed. The main question is the construction of abelian extensions of certain algebraic number fields, which is traditionally called "Hilbert's twelfth problem." Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by modular functions, which supplies an indispensable background for the recent proof of Fermat's last theorem by Wiles.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1994

Количество страниц: 288

Добавлена в каталог: 01.06.2008

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Предметный указатель
0-cycle      169 257
1-Cycle      169
Abelian variety      257
Abelian variety with complex multiplication      126ff. 211ff.
Adelization of $GL_{2}$      143—144
Adelization of a simple algebra      241
Affine variety      254
Algebraic correspondence      77 169
Algebraic curve      257
Algebraic number field      xi
Algebraic variety      254
Arithmetic Fuchsian group      247
Automorphic form      28—29
Automorphic function      28 30
Automorphism of an abelian variety      258
Automorphism of an elliptic curve      106ff.
Basic polar divisor      259
Birational map      254
Birationally equivalent      254
Birch — Swinnerton — Dyer conjecture      221
Biregular isomorphism      254
Canonical class      36
Character unramified at a prime      213
Class field      116
CM-field      124
CM-type      125
Cohomology group      223
Commensurable      5
Commensurator      51
Complex multiplication of an abelian variety      126ff.
Complex multiplication of an elliptic curve      102
Conductor of an order      106
Congruence subgroup      20
Conjecture of Birch and Swinnerton — Dyer      221
Conjecture of Hasse and Weil      168
Conjecture of Ramanujan      89
Covering of a Riemann surface      19
Cusp      8 18
Cusps of the modular group      14
Degree of a covering      19
Degree of a divisor      35
Degree of a double coset      51
Degree of a rational map      112 258
Differential form      36 257
Differential form of the first kind      36 257
Discrete subgroup      3
Divisor of a Riemann surface      35
Divisor of an algebraic curve      169
Divisor of an algebraic variety      259
Eigen-function of Hecke operators      77ff.
Eigen-values of Hecke operators      77ff.
Eisenstein series      32—33 78
Elliptic curve      96
Elliptic elements of the modular group      14—15
Elliptic function      98
Elliptic matrix      5
Elliptic point      8 18
Elliptic points of the modular group      14—15
Elliptic transformation      5
Endomorphism of an abelian variety      258;
Endomorphism of an elliptic curve      102ff.
Equivalent under a transformation group      1
Euler characteristic      18
Field of definition      254
Field of moduli of an abelian variety      130—131
Field of moduli of an elliptic curve      98
Field of rationality      254
Formal Dirichlet series      60-61 (see also "zeta-function")
Fourier coefficients      29
Fourier expansion      29
Fourier expansion of $\Delta$      33 50
Fourier expansion of J      33
Frobenius correspondence      177
Frobenius morphism      256
Fuchsian group of the first kind      19
Function of an algebraic variety      254—255
Functional equation of a zeta-function      93
Fundamental domain      15 42
Gauss sum      91
Generic point for meromorphic functions      137
Generic point of a variety      254;
Genus of $\Gamma\backslash\mathfrak{H}^*$,$\Gamma$ a congruence subgroup      23
Genus of a compact Riemann surface      18;
Good reduction modulo a prime      114 213
Hasse — Weil conjecture      168
Hecke operator      76 79
Hecke ring      54
Hecke ring of $SL_{n}(Z)$      55ff.
Hecke ring of a congruence subgroup      65ff.
Homogeneous element of a Hecke ring      60
Homomorphism of an abelian variety      257
Homomorphism of an elliptic curve      96
Hurwitz formula      19
Hyperbolic matrix or transformation      5
Inseparable morphism      112
Inseparable or separable morphism      112
Integral form      30
Invariant measure of the upper half plane      41
Invariant of an elliptic curve      97 99
Involution of the endomorphism ring of an abelian variety      259
Irregular cusp      29
Isogenous, isogeny      96 258
Isotropy subgroup      1
l-adic representation      100 189ff.
L-function      213
Lattice in a complex vector space      98 126 258;
Lattice in a number field      104;
Lattice in a rational vector space      56
Level      20 30
Linear fractional transformation      5
Linearly equivalent      35
Local parameter      17
Locus of a point      254
Loxodromic matrix or transformation      5
Main involution      72 243
Maximal order of a number field      xii 104
Maximal ray class field      116
Measure of a fundamental domain      41 42 44
Mellin transformation      94
Model of $\Gamma\backslash\mathfrak{H}^*$      152
Modular correspondence      77 172ff. 176
Modular equation      110
Modular form      30
Modular function      30
Modular function of level N rational over a cyclotomic field      137
Modular group      14
Morphism      254
Non-singular      255
Normalized embedding      103—104 246
Normalized isomorphism into $End_{Q}(E)$      113
Orbit      1
Order in a number field      104
Order of an elliptic point      9
Origin of an elliptic curve      96
Parabolic matrix or transformation      5
Petersson inner product      75
Polarization      259
Polarized Abelian variety      259
Prime ramified in a quaternion algebra      245
Prime unramified in a quaternion algebra      245
Primitive matrix      108
Principal congruence subgroup      20
Projective variety      254
Proper algebraic correspondence      169
Proper ideal      104
Purely inseparable morphism      112
Quaternion algebra      243
Quotient topology      1
Ramanujan conjecture      89
Ramification index      19
Rational map      254
Reduction modulo a prime      114
Reflex of a CM-type      126
Regular cusp      29
Regular extension      253
Riemann form      258
Riemann surface      17
Riemann — Roch theorem      36
Separable morphism      112
Specialization      253
Stability group      1
Subvariety      254
Theta-series      95
Transformation group      1
Triangle group      251
Universal domain      253
Weierstrass function      98
Weight of an automorphic form      28
Z-lattice      56 104
Zeta-function      89ff.
Zeta-function of a curve      167
Zeta-function of an abelian variety      167—168
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