4346:
4306:
4326:
4316:
4336:
2139:
of the union of images of algebraic groups under non-trivial rational maps; alternatively one may take images of abelian varieties; another definition is the union of all subvarieties that are not of general type. For abelian varieties the definition would be the union of all translates of proper
2162:
shows that points of small height in projective space lie in a finite number of hyperplanes. A quantitative form of the theorem, in which the number of subspaces containing all solutions, was also obtained by
Schmidt, and the theorem was generalised by Schlickewei (1977) to allow more general
1203:. One of the main objectives of Diophantine geometry is to classify cases where the Hasse principle holds. Generally that is for a large number of variables, when the degree of an equation is held fixed. The Hasse principle is often associated with the success of the
1207:. When the circle method works, it can provide extra, quantitative information such as asymptotic number of solutions. Reducing the number of variables makes the circle method harder; therefore failures of the Hasse principle, for example for
1374:'s classical method for Diophantine equations. It became one half of the standard proof of the Mordell–Weil theorem, with the other being an argument with height functions (q.v.). Descent is something like division by two in a group of
1922:
is a real number which describes the rate of growth of the number of rational points on the variety with respect to the embedding defined by the divisor. It has similar formal properties to the abscissa of convergence of the
792:. The arithmetic genus is larger than the geometric genus, and the height of a point may be bounded in terms of the arithmetic genus. Obtaining similar bounds involving the geometric genus would have significant consequences.
955:
of an algebraic curve or compact
Riemann surface, with a point or more removed corresponding to the 'infinite places' of a number field. This idea is more precisely encoded in the theory that
1891:
as provided by the general theory of heights. It can be defined from a general height by a limiting process; there are also formulae, in the sense that it is a sum of local contributions.
2332:
2135:
in an algebraic variety is the subset in which one might expect to find many rational points. The precise definition varies according to context. One definition is the
2779:
1163:
3693:
511:
and the order of pole of its Hasse–Weil L-function. It has been an important landmark in
Diophantine geometry since the mid-1960s, with results such as the
1127:
605:
1828:
or classical height of a vector of rational numbers is the maximum absolute value of the vector of coprime integers obtained by multiplying through by a
915:
is, for the school of
Grothendieck, one terminal point of development. It has the disadvantage of being quite hard to compute with. The reason that the
4329:
2379:
is an algebraic subgroup intersecting a subvariety of a torus or abelian variety in a set of unusually large dimension, such as is involved in the
2244:, an analogue of the Birch–Swinnerton-Dyer conjecture (q.v.), leading quickly to a clarification of the latter and a recognition of its importance.
4029:
2505:, e.g. better estimates for curves of the number of points than come from Weil's basic theorem of 1940. The latter turn out to be of interest for
2213:
2208:
was eventually proved. For abelian varieties, and in particular the Birch–Swinnerton-Dyer conjecture (q.v.), the
Tamagawa number approach to a
2176:
4002:
3969:
3824:
3751:
3660:
3431:
3030:
2824:
1860:
as a sum of local contributions. The global Néron symbol, which is the sum of the local symbols, is just the negative of the height pairing.
621:
500:
1046:
allow clearing of denominators by multiplying by a common scalar. For a given, single point one can do this and not leave a common factor
2497:, made public around 1949, on local zeta-functions. The proof was completed in 1973. Those being proved, there remain extensions of the
2205:
1295:
1204:
4380:
1070:
can become larger when linear terms reduce to 0 (the geometric formulation shows it is not the fault of a single set of coordinates).
3935:
3893:
3859:
3324:
3126:
1732:
1723:
states that there should be a uniform bound on the number of such points, depending only on the genus and the field of definition.
2561:
is an effective procedure for assigning a height function to any divisor on smooth projective variety over a number field (or to
1836:, or of a polynomial, regarded as a vector of coefficients, or of an algebraic number, from the height of its minimal polynomial.
1792:
853:
734:
1244:
1167:
808:
443:
4060:
2521:
decomposition of algebraic numbers in coordinates of points on algebraic varieties. It has remained somewhat under-developed.
1130:. The theory is subtle, in the sense that the freedom to change variables to try to improve matters is rather unobvious: see
901:
2498:
1994:
3290:
Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Vol. I: Arithmetic
4370:
3851:
2466:
1989:, i.e. solubility guaranteed by a number of variables polynomial in the degree of an equation, grew out of studies of the
2364:
1752:
1673:
4319:
4106:
976:
375:
2380:
4375:
4101:
4086:
4022:
2957:
2474:
1375:
126:. Arithmetic geometry has also been defined as the application of the techniques of algebraic geometry to problems in
2209:
3816:
3743:
3680:
3423:
3411:
2816:
1402:
1143:
1119:
730:
4119:
1522:
1228:
466:
4281:
4240:
3285:
3152:
3147:
2877:
van der Geer, G.; Schoof, R. (2000). "Effectivity of
Arakelov divisors and the theta divisor of a number field".
2731:
2502:
2405:
2229:
2104:
1829:
1147:
877:
of an elliptic curve or abelian variety defined over a number field is a measure of its complexity introduced by
601:
508:
134:
2546:
2080:
1869:
1857:
1772:
546:
516:
512:
4345:
4125:
3688:
3074:
2966:
2164:
1135:
1059:
609:
4068:
893:
411:
1612:
is a geometrically irreducible
Zariski-closed subgroup of an affine torus (product of multiplicative groups).
4052:
2506:
1986:
1589:
1521:
flavour, and, on the other hand, implicated in some formulations of arithmetic conjectures. See for example
1175:
1043:
928:
1151:
4309:
4129:
4078:
4015:
2430:
2201:
1526:
1398:
1315:
1299:
845:
816:
640:
632:
597:
520:
3311:
Roessler, Damian (2005). "A note on the Manin–Mumford conjecture". In van der Geer, Gerard; Moonen, Ben;
2002:
4286:
4215:
2315:
2108:
1720:
1652:
1067:
932:
458:
419:
2449:
The initial idea, later somewhat modified, for proving the Weil conjectures (q.v.), was to construct a
4276:
4111:
3357:
3083:
2975:
2426:
2172:
1924:
1901:
1248:
672:
115:
89:
54:
36:
4335:
4245:
4154:
4149:
4143:
4135:
4096:
3843:
3415:
3213:
3211:
Igusa, Jun-Ichi (1974). "Complex powers and asymptotic expansions. I. Functions of certain types".
2741:
2736:
2538:
2470:
1760:
1625:
1621:
1581:
1340:
1332:
857:
841:
688:
680:
644:
462:
383:
351:
111:
58:
2212:
fails on a direct attempt, though it has had heuristic value over many years. Now a sophisticated
4291:
4235:
4139:
4056:
3712:
3524:
3373:
3230:
3169:
3099:
2991:
2904:
2886:
2478:
2006:
1916:
1712:
1656:
1514:
1356:
1171:
1051:
1011:
983:
882:
776:, defined by Vojta. The difference between the two may be compared to the difference between the
565:
119:
44:
3953:
319:
4349:
4205:
3998:
3965:
3931:
3889:
3855:
3820:
3747:
3656:
3427:
3320:
3143:
3122:
3026:
2820:
2462:
2458:
2450:
2438:
2409:
2314:. Algebraically closed fields are of Tsen rank zero. The Tsen rank is greater or equal to the
2084:
1808:
1418:
1414:
1406:
1379:
1352:
1291:
1115:
820:
789:
652:
577:
470:
415:
347:
860:, and was basic in the formulation of the Tate conjecture (q.v.) and numerous other theories.
4210:
4195:
3975:
3941:
3907:
3865:
3830:
3720:
3702:
3633:
3532:
3516:
3437:
3390:
3365:
3330:
3293:
3238:
3222:
3177:
3161:
3091:
2999:
2983:
2912:
2896:
2838:
2490:
2397:
2241:
2233:
2158:
2063:
2047:
1873:
1849:
1811:
is an algebraic variety which has only finitely many points in any finitely generated field.
1719:, and states that a curve of genus at least two has only finitely many rational points. The
1716:
1685:
1518:
1433:
1427:
1422:
1371:
1367:
1287:
1279:
1272:
1079:
1035:
987:
960:
948:
824:
777:
676:
636:
454:
379:
355:
123:
3903:
2834:
4339:
4224:
4200:
4115:
3979:
3961:
3945:
3927:
3911:
3899:
3885:
3869:
3834:
3808:
3724:
3637:
3536:
3441:
3334:
3297:
3292:. Progress in Mathematics (in French). Vol. 35. Birkhauser-Boston. pp. 327–352.
3242:
3181:
3022:
3003:
2916:
2842:
2830:
2562:
2542:
2534:
2237:
2225:
2197:
2180:
2136:
1877:
1853:
1825:
1756:
1648:
1640:
1542:
1260:
1192:
1075:
1074:
refers to the reduced variety having the same properties as the original, for example, an
1055:
952:
912:
874:
804:
785:
573:
538:
504:
430:
407:
403:
3624:(1936). "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper".
3361:
3087:
2979:
2054:
and a vector of positive real numbers with components indexed by the infinite places of
1034:
of fractions are tricky, in that reduction modulo a prime in the denominator looks like
17:
4261:
4180:
4064:
3766:
3676:
3312:
3277:
2806:
2092:
1998:
1957:
1913:
1884:
1736:
1677:
1452:
1394:
1336:
1212:
1131:
1123:
1083:
1003:
964:
812:
781:
651:
coefficients in this case. It is one of a number of theories deriving in some way from
593:
569:
564:-adic analytic functions, is a special application but capable of proving cases of the
542:
391:
311:
85:
104:
is something to be proved and studied as an extra topic, even knowing the geometry of
4364:
4220:
4072:
4038:
3528:
3507:(1990). "On the number of rational points of bounded height on algebraic varieties".
3504:
3377:
3281:
3234:
3103:
3069:
2995:
2494:
2257:
1906:
1744:
1585:
1558:
1494:
1410:
1236:
924:
916:
878:
849:
800:
668:
568:
for curves whose
Jacobian's rank is less than its dimension. It developed ideas from
433:
is an approach to arithmetic geometry that explicitly includes the 'infinite primes'.
127:
40:
2908:
2809:(2008). "Computing Arakelov class groups". In Buhler, J.P.; P., Stevenhagen (eds.).
2759:
1553:(integral points case) and Piotr Blass have conjectured that algebraic varieties of
1378:(often called 'descents', when written out by equations); in more modern terms in a
4266:
4190:
4090:
2961:
2454:
2434:
2168:
2096:
1990:
1633:
1554:
1383:
1196:
956:
936:
897:
815:
and other techniques that have not all been absorbed into general theories such as
387:
315:
77:
73:
3707:
3226:
1263:
in
Diophantine geometry quantifies the size of solutions to Diophantine equations.
322:
attempts to state as much as possible about repeated prime factors in an equation
3994:
2810:
1219:) are at a general level connected with the limitations of the analytic approach.
3684:
2518:
2088:
1216:
1200:
1031:
1020:
840:
The search for a Weil cohomology (q.v.) was at least partially fulfilled in the
81:
69:
32:
3021:. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.).
2290:: that is, such that any system of polynomials with no constant term of degree
2144:
is the
Zariski closure of the images of all non-constant holomorphic maps from
1600:
is analytically hyperbolic if and only if all subvarieties are of general type.
1569:
a finitely-generated field. This circle of ideas includes the understanding of
4271:
4230:
4082:
3919:
3877:
3621:
3194:
2401:
2277:
2253:
2100:
1748:
1740:
1550:
1546:
1307:
1240:
1208:
1139:
1024:
684:
613:
48:
2273:
1298:
which shows the rational numbers are Hilbertian. Results are applied to the
2812:
Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography
2148:. Lang conjectured that the analytic and algebraic special sets are equal.
3391:
2 page exposition of the Mordell–Lang conjecture by B. Mazur, 3 Nov. 2005
3280:(1983). "Sous-variétés d'une variété abélienne et points de torsion". In
469:
could be regarded as Artin L-functions for the Galois representations on
3348:
McQuillan, Michael (1995). "Division points on semi-abelian varieties".
3319:. Progress in Mathematics. Vol. 239. Birkhäuser. pp. 311–318.
1883:
is a height function (q.v.) that is essentially intrinsic, and an exact
1832:. This may be used to define height on a point in projective space over
896:, the most celebrated conjecture of Diophantine geometry, was proved by
3716:
3653:
Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics
3520:
3369:
3173:
3095:
2987:
2900:
1691:
can only contain a finite number of points that are of finite order in
3072:(1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern".
2318:
but it is not known if they are equal except in the case of rank zero.
1887:, rather than approximately quadratic with respect to the addition on
1343:
counting numbers of points on an algebraic variety modulo high powers
963:
over the complex numbers, also, have some quite strict analogies with
406:
on a projective space over the field of algebraic numbers is a global
334:. For example 3 + 125 = 128 but the prime powers here are exceptional.
4170:
2891:
959:
should all be treated on the same basis. The idea goes further. Thus
823:
of local zeta-functions, the initial advance in the direction of the
3165:
2545:. They are used in the construction of the local components of the
2533:
on an algebraic variety is a real-valued function defined off some
2280:
who introduced their study in 1936, is the smallest natural number
3740:
Some Problems of Unlikely Intersections in Arithmetic and Geometry
2099:
obtained from reducing a given elliptic curve over the rationals.
920:
1243:
remain largely in the realm of conjecture, with the proof of the
4007:
3960:. Grundlehren der Mathematischen Wissenschaften. Vol. 322.
2763:
1647:. According to the Weil conjectures (q.v.) these functions, for
1573:
and the Lang conjectures on that, and the Vojta conjectures. An
1094:, assumed smooth, such that there is otherwise a smooth reduced
4011:
1023:. In the typical situation this presents little difficulty for
2568:
1775:
is a foundational result stating that for an abelian variety
1251:
is largely complementary to the theory of global L-functions.
655:, and has applications outside purely arithmetical questions.
143:
3017:
Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008).
1501:-adic L-function earlier introduced by Kubota and Leopoldt.
2964:(1977). "On the conjecture of Birch and Swinnerton-Dyer".
713:: that is, such that any homogeneous polynomial of degree
4185:
4175:
2240:, but well within arithmetic geometry. It also gave, for
1927:
and it is conjectured that they are essentially the same.
1239:
formed from local zeta-functions. The properties of such
675:
to study from an arithmetic point of view (including the
3150:(November 1968). "Good reduction of abelian varieties".
1086:
remaining smooth. In general there will be a finite set
576:. (Other older methods for Diophantine problems include
51:, which can be related at various levels of generality.
3317:
Number fields and function fields — two parallel worlds
2517:
André Weil proposed a theory in the 1920s and 1930s on
2501:
congruence, which comes from an elementary method, and
1651:
varieties, exhibit properties closely analogous to the
990:
to varieties of dimension at least two is often called
374:) on a global field is an extension of the concept of
1588:
to it exists, that is not constant. Examples include
1425:). In its early days in the late 1960s it was called
1211:
in small numbers of variables (and in particular for
3133:→ Contains an English translation of Faltings (1983)
2260:
introduced by John Tate to study bad reduction (see
4254:
4163:
4045:
1848:is a bimultiplicative pairing between divisors and
1477:) as Galois module. In the same way, Iwasawa added
947:It was realised in the nineteenth century that the
749:refers to two related concepts relative to a point
3771:Poids dans la cohomologie des variétés algébriques
1042:per fraction. With a little extra sophistication,
2284:, if it exists, such that the field is of class T
2140:abelian subvarieties. For a complex variety, the
1628:for the number of points on an algebraic variety
707:, if it exists, such that the field of is class C
3742:. Annals of Mathematics Studies. Vol. 181.
2107:suggested it around 1960. It is a prototype for
951:of a number field has analogies with the affine
133:See also the glossary of number theory terms at
47:. Much of the theory is in the form of proposed
35:, areas growing out of the traditional study of
3214:Journal für die reine und angewandte Mathematik
2171:. The theorem may be used to obtain results on
1799:, but extends to all finitely-generated fields.
1174:solutions. The initial result of this type was
1795:. This was proved initially for number fields
1517:is on one hand a quite general theory with an
620: = 1. This is a special case of the
114:can be more generally defined as the study of
4023:
2854:
2852:
2493:were three highly influential conjectures of
2461:at detecting topological structure, and have
1580:over the complex numbers is one such that no
919:has been considered the 'right' foundational
541:is a height function that is a distinguished
8:
3815:. New Mathematical Monographs. Vol. 4.
3694:Journal of the American Mathematical Society
3571:in the volume (O. F. G. Schilling, editor),
3569:Algebraic cycles and poles of zeta functions
3422:. New Mathematical Monographs. Vol. 9.
3117:Cornell, Gary; Silverman, Joseph H. (1986).
1451:Picard variety), where the finite field has
2778:Sutherland, Andrew V. (September 5, 2013).
1731:The Mordell–Lang conjecture, now proved by
4325:
4315:
4030:
4016:
4008:
3420:Logarithmic Forms and Diophantine Geometry
2301:variables has a non-trivial zero whenever
1199:is the same as solubility in all relevant
721:variables has a non-trivial zero whenever
703:of a field is the smallest natural number
3706:
2890:
2513:Weil distributions on algebraic varieties
1575:analytically hyperbolic algebraic variety
1489:→ ∞, for his analogue, to a number field
931:, the discovery of Grothendieck that the
3480:
3478:
3254:
3252:
2256:is a particular elliptic curve over the
1382:group which is to be proved finite. See
1164:Grothendieck–Katz p-curvature conjecture
1038:, but that rules out only finitely many
935:are sheaves for it (i.e. a very general
3611:Bombieri & Gubler (2006) pp.176–230
3454:Bombieri & Gubler (2006) pp.301–314
3055:
3053:
3051:
2801:
2799:
2752:
2363:. The conjecture would follow from the
2359:-rational points on any curve of genus
382:. It is a formal linear combination of
68:that are finitely generated over their
2214:equivariant Tamagawa number conjecture
1455:added to make finite field extensions
3848:Diophantine Geometry: An Introduction
3800:Hindry & Silverman (2000) 184–185
3267:Bombieri & Gubler (2006) pp.82–93
2929:Bombieri & Gubler (2006) pp.66–67
2780:"Introduction to Arithmetic Geometry"
2453:applying to algebraic varieties over
1355:are now known, drawing on methods of
1306:) are in some sense analogous to the
410:with local contributions coming from
7:
3995:An invitation to arithmetic geometry
3773:, Actes ICM, Vancouver, 1974, 79–85.
3198:
2469:could be applied to the counting in
2232:, 1963) provided an analogue to the
1751:unifying the Mordell conjecture and
763:geometric (logarithmic) discriminant
622:Birch and Swinnerton-Dyer conjecture
507:postulates a connection between the
501:Birch and Swinnerton-Dyer conjecture
495:Birch and Swinnerton-Dyer conjecture
480:
390:having integer coefficients and the
3593:Hindry & Silverman (2000) p.480
3493:Hindry & Silverman (2000) p.488
3258:Hindry & Silverman (2000) p.479
2815:. MSRI Publications. Vol. 44.
2206:Weil conjecture on Tamagawa numbers
2177:Siegel's theorem on integral points
1856:used in Néron's formulation of the
1166:applies reduction modulo primes to
1118:, good reduction is connected with
29:arithmetic and diophantine geometry
1462:The local zeta-function (q.v.) of
72:—including as of special interest
25:
2537:which generalises the concept of
2343:> 2, there is a uniform bound
2335:states that for any number field
1466:can be recovered from the points
735:quasi-algebraically closed fields
635:is a p-adic cohomology theory in
4344:
4334:
4324:
4314:
4305:
4304:
1793:finitely-generated abelian group
1481:-power roots of unity for fixed
1302:. Thin sets (the French word is
1296:Hilbert's irreducibility theorem
1168:algebraic differential equations
809:algebraic differential equations
733:are of Diophantine dimension 0;
647:which is deficient in using mod
3882:Introduction to Arakelov theory
3813:Heights in Diophantine Geometry
3689:"Uniformity of rational points"
3573:Arithmetical Algebraic Geometry
2400:is a complex of conjectures by
2200:definition works well only for
1872:(also often referred to as the
1128:Néron–Ogg–Shafarevich criterion
444:arithmetic of abelian varieties
437:Arithmetic of abelian varieties
4083:analytic theory of L-functions
4061:non-abelian class field theory
3924:Survey of Diophantine Geometry
3655:. Springer. pp. 109–126.
2465:acting in such a way that the
2457:that would both be as good as
2083:describes the distribution of
1596:> 1. Lang conjectured that
1294:. This is a geometric take on
1205:Hardy–Littlewood circle method
1090:of primes for a given variety
457:are defined for quite general
1:
3852:Graduate Texts in Mathematics
3708:10.1090/S0894-0347-97-00195-1
3227:10.1515/crll.1974.268-269.110
2467:Lefschetz fixed-point theorem
1195:states that solubility for a
1006:in arithmetic problems is to
852:. It provided a proof of the
4107:Transcendental number theory
3567:It is mentioned in J. Tate,
2216:is a major research problem.
1997:. It stalled in the face of
1376:principal homogeneous spaces
1158:Grothendieck–Katz conjecture
977:Geometric class field theory
803:used distinctive methods of
757:defined over a number field
418:and the usual metric on the
39:to encompass large parts of
4330:List of recreational topics
4102:Computational number theory
4087:probabilistic number theory
3019:Cohomology of Number Fields
2503:improvements of Weil bounds
2475:motive (algebraic geometry)
2404:, making analogies between
1735:following work of Laurent,
1497:of class groups, finding a
1432:. The analogy was with the
1245:Taniyama–Shimura conjecture
1170:, to derive information on
731:Algebraically closed fields
691:is the most classical case.
537:The canonical height on an
96:the existence of points of
57:in general is the study of
4397:
3817:Cambridge University Press
3744:Princeton University Press
3424:Cambridge University Press
2817:Cambridge University Press
1247:being a breakthrough. The
1144:semistable abelian variety
4381:Glossaries of mathematics
4300:
4282:Diophantine approximation
4241:Chinese remainder theorem
3811:; Gubler, Walter (2006).
3738:Zannier, Umberto (2012).
3153:The Annals of Mathematics
2732:Glossary of number theory
2565:on non-smooth varieties).
2499:Chevalley–Warning theorem
2406:Diophantine approximation
2046:is a formal product of a
1995:Chevalley–Warning theorem
1830:lowest common denominator
1148:semistable elliptic curve
927:goes back to the fact of
683:are computed in terms of
671:are some of the simplest
602:imaginary quadratic field
509:rank of an elliptic curve
394:having real coefficients.
135:Glossary of number theory
4126:Arithmetic combinatorics
3075:Inventiones Mathematicae
2967:Inventiones Mathematicae
2583:
2578:
2507:Algebraic geometry codes
2473:. For later history see
2365:Bombieri–Lang conjecture
1753:Manin–Mumford conjecture
1674:Manin–Mumford conjecture
1668:Manin–Mumford conjecture
1590:compact Riemann surfaces
1430:analogue of the Jacobian
1347:of a fixed prime number
1136:potential good reduction
753:on an algebraic variety
643:to fill the gap left by
465:in the 1960s meant that
158:
153:
118:of finite type over the
18:Reduction modulo a prime
4097:Geometric number theory
4053:Algebraic number theory
3993:Dino Lorenzini (1996),
3958:Algebraic Number Theory
2381:Mordell–Lang conjecture
2202:linear algebraic groups
2142:holomorphic special set
2022:a prime number or ideal
1987:quasi-algebraic closure
1981:Quasi-algebraic closure
1727:Mordell–Lang conjecture
1403:Stickelberger's theorem
1044:homogeneous coordinates
929:faithfully-flat descent
774:arithmetic discriminant
747:discriminant of a point
741:Discriminant of a point
346:is the analogue of the
4216:Transcendental numbers
4130:additive number theory
4079:Analytic number theory
3791:Lang (1997) pp.164,212
3651:Lorenz, Falko (2008).
3575:, pages 93–110 (1965).
3549:Lang (1997) pp.161–162
3121:. New York: Springer.
2938:Lang (1988) pp.156–157
2713:
2708:
2703:
2698:
2693:
2688:
2683:
2678:
2673:
2668:
2663:
2658:
2653:
2648:
2643:
2638:
2633:
2628:
2623:
2618:
2613:
2608:
2603:
2598:
2593:
2588:
2431:Alexander Grothendieck
2276:of a field, named for
2210:local–global principle
2109:Galois representations
1680:, states that a curve
1571:analytic hyperbolicity
1565:-rational points, for
1527:Lichtenbaum conjecture
1399:analytic number theory
1316:Baire category theorem
1300:inverse Galois problem
943:Function field analogy
933:representable functors
846:Alexander Grothendieck
819:. He first proved the
817:crystalline cohomology
641:Alexander Grothendieck
633:Crystalline cohomology
628:Crystalline cohomology
598:complex multiplication
467:Hasse–Weil L-functions
461:. The introduction of
459:Galois representations
420:non-Archimedean fields
288:
283:
278:
273:
268:
263:
258:
253:
248:
243:
238:
233:
228:
223:
218:
213:
208:
203:
198:
193:
188:
183:
178:
173:
168:
163:
27:This is a glossary of
4287:Irrationality measure
4277:Diophantine equations
4120:Hodge–Arakelov theory
3558:Neukirch (1999) p.185
2858:Neukirch (1999) p.189
2433:of analogies between
2377:unlikely intersection
2371:Unlikely intersection
2333:uniformity conjecture
2327:Uniformity conjecture
2316:Diophantine dimension
2173:Diophantine equations
1747:, is a conjecture of
1721:Uniformity conjecture
1653:Riemann zeta-function
1523:Birch–Tate conjecture
1493:, and considered the
1278:is one for which the
1231:, sometimes called a
1229:Hasse–Weil L-function
1223:Hasse–Weil L-function
1068:Zariski tangent space
894:Fermat's Last Theorem
889:Fermat's Last Theorem
701:Diophantine dimension
695:Diophantine dimension
84:. Of those, only the
37:Diophantine equations
4371:Diophantine geometry
4246:Arithmetic functions
4112:Diophantine geometry
3844:Silverman, Joseph H.
3626:J. Chinese Math. Soc
3584:Lang (1997) pp.17–23
3463:Lang (1988) pp.66–69
3221:(268–269): 110–130.
2947:Lang (1997) pp.91–96
2867:Lang (1988) pp.74–75
2819:. pp. 447–495.
2471:local zeta-functions
2429:is a formulation by
2179:and solution of the
2103:and, independently,
2081:Sato–Tate conjecture
2075:Sato–Tate conjecture
1964:and in addition the
1925:height zeta function
1902:Nevanlinna invariant
1895:Nevanlinna invariant
1779:over a number field
1773:Mordell–Weil theorem
1767:Mordell–Weil theorem
1443:over a finite field
1353:rationality theorems
1249:Langlands philosophy
1176:Eisenstein's theorem
1062:on reduction modulo
1019:or, more generally,
881:in his proof of the
858:local zeta-functions
681:local zeta-functions
673:projective varieties
590:Coates–Wiles theorem
584:Coates–Wiles theorem
517:Gross–Zagier theorem
513:Coates–Wiles theorem
412:Fubini–Study metrics
344:Arakelov class group
338:Arakelov class group
100:with coordinates in
90:algebraically closed
55:Diophantine geometry
4292:Continued fractions
4155:Arithmetic dynamics
4150:Arithmetic topology
4144:P-adic Hodge theory
4136:Arithmetic geometry
4069:Iwasawa–Tate theory
3362:1995InMat.120..143M
3119:Arithmetic geometry
3088:1983InMat..73..349F
2980:1977InMat..39..223C
2879:Selecta Mathematica
2760:Arithmetic geometry
2742:Arithmetic dynamics
2737:Arithmetic topology
2572:Contents:
2559:Weil height machine
2553:Weil height machine
2355:) on the number of
1940:An Abelian variety
1761:semiabelian variety
1626:generating function
1622:local zeta-function
1616:Local zeta-function
1582:holomorphic mapping
1397:builds up from the
1341:generating function
1333:Igusa zeta-function
1327:Igusa zeta-function
1058:point may become a
994:class field theory.
967:over number fields.
854:functional equation
521:Kolyvagin's theorem
352:divisor class group
147:Contents:
112:Arithmetic geometry
59:algebraic varieties
4376:Algebraic geometry
4236:Modular arithmetic
4206:Irrational numbers
4140:anabelian geometry
4057:class field theory
3782:Lang (1988) pp.1–9
3521:10.1007/bf01453564
3370:10.1007/BF01241125
3144:Serre, Jean-Pierre
3096:10.1007/BF01388432
2988:10.1007/BF01402975
2901:10.1007/PL00001393
2479:motivic cohomology
2463:Frobenius mappings
2085:Frobenius elements
2042:in a number field
2007:mathematical logic
1968:-torsion has rank
1950:ordinary reduction
1936:Ordinary reduction
1917:projective variety
1713:Mordell conjecture
1707:Mordell conjecture
1657:Riemann hypothesis
1639:, over the finite
1515:Algebraic K-theory
1415:p-adic L-functions
1407:ideal class groups
1357:mathematical logic
1235:L-function, is an
1172:algebraic function
1152:Serre–Tate theorem
1052:singularity theory
1015:all prime numbers
986:-style results on
984:class field theory
883:Mordell conjecture
566:Mordell conjecture
416:Archimedean fields
386:of the field with
45:algebraic geometry
4358:
4357:
4255:Advanced concepts
4211:Algebraic numbers
4196:Composite numbers
4003:978-0-8218-0267-0
3997:, AMS Bookstore,
3971:978-3-540-65399-8
3854:. Vol. 201.
3826:978-0-521-71229-3
3753:978-0-691-15371-1
3662:978-0-387-72487-4
3602:Lang (1997) p.179
3472:Lang (1997) p.212
3433:978-0-521-88268-2
3416:Wüstholz, Gisbert
3059:Lang (1997) p.171
3045:Lang (1997) p.146
3032:978-3-540-37888-4
2826:978-0-521-20833-8
2547:Néron–Tate height
2459:singular homology
2451:cohomology theory
2439:l-adic cohomology
2410:Nevanlinna theory
2242:elliptic surfaces
2121:Chabauty's method
2003:Ax–Kochen theorem
1870:Néron–Tate height
1864:Néron–Tate height
1858:Néron–Tate height
1809:Mordellic variety
1803:Mordellic variety
1423:Bernoulli numbers
1419:Kummer congruence
1380:Galois cohomology
1292:Jean-Pierre Serre
1280:projective spaces
1267:Hilbertian fields
1116:abelian varieties
988:abelian coverings
982:The extension of
961:elliptic surfaces
790:desingularisation
572:'s method for an
558:Chabauty's method
553:Chabauty's method
547:Néron–Tate height
471:l-adic cohomology
455:Artin L-functions
450:Artin L-functions
442:See main article
356:Arakelov divisors
348:ideal class group
92:; over any other
16:(Redirected from
4388:
4348:
4338:
4328:
4327:
4318:
4317:
4308:
4307:
4201:Rational numbers
4032:
4025:
4018:
4009:
3983:
3954:Neukirch, Jürgen
3949:
3915:
3873:
3838:
3809:Bombieri, Enrico
3801:
3798:
3792:
3789:
3783:
3780:
3774:
3764:
3758:
3757:
3735:
3729:
3728:
3710:
3673:
3667:
3666:
3648:
3642:
3641:
3618:
3612:
3609:
3603:
3600:
3594:
3591:
3585:
3582:
3576:
3565:
3559:
3556:
3550:
3547:
3541:
3540:
3500:
3494:
3491:
3485:
3484:Lang (1988) p.77
3482:
3473:
3470:
3464:
3461:
3455:
3452:
3446:
3445:
3408:
3402:
3401:Lang (1997) p.15
3399:
3393:
3388:
3382:
3381:
3345:
3339:
3338:
3308:
3302:
3301:
3274:
3268:
3265:
3259:
3256:
3247:
3246:
3208:
3202:
3192:
3186:
3185:
3140:
3134:
3132:
3114:
3108:
3107:
3066:
3060:
3057:
3046:
3043:
3037:
3036:
3014:
3008:
3007:
2954:
2948:
2945:
2939:
2936:
2930:
2927:
2921:
2920:
2894:
2874:
2868:
2865:
2859:
2856:
2847:
2846:
2803:
2794:
2793:
2791:
2789:
2784:
2775:
2769:
2757:
2573:
2563:Cartier divisors
2539:Green's function
2491:Weil conjectures
2485:Weil conjectures
2398:Vojta conjecture
2392:Vojta conjecture
2238:algebraic cycles
2234:Hodge conjecture
2192:Tamagawa numbers
2159:subspace theorem
2152:Subspace theorem
2064:Arakelov divisor
2048:fractional ideal
1874:canonical height
1850:algebraic cycles
1717:Faltings theorem
1686:Jacobian variety
1676:, now proved by
1655:, including the
1641:field extensions
1519:abstract algebra
1475:
1460:
1434:Jacobian variety
1372:Pierre de Fermat
1368:Infinite descent
1363:Infinite descent
1290:in the sense of
1273:Hilbertian field
1122:in the field of
1078:having the same
1036:division by zero
949:ring of integers
842:étale cohomology
836:Étale cohomology
825:Weil conjectures
813:Koszul complexes
778:arithmetic genus
689:Waring's problem
677:Fermat varieties
645:étale cohomology
639:, introduced by
637:characteristic p
533:Canonical height
463:étale cohomology
380:fractional ideal
368:Arakelov divisor
362:Arakelov divisor
148:
124:ring of integers
21:
4396:
4395:
4391:
4390:
4389:
4387:
4386:
4385:
4361:
4360:
4359:
4354:
4296:
4262:Quadratic forms
4250:
4225:P-adic analysis
4181:Natural numbers
4159:
4116:Arakelov theory
4041:
4036:
3990:
3988:Further reading
3972:
3962:Springer-Verlag
3952:
3938:
3928:Springer-Verlag
3918:
3896:
3886:Springer-Verlag
3876:
3862:
3841:
3827:
3807:
3804:
3799:
3795:
3790:
3786:
3781:
3777:
3765:
3761:
3754:
3737:
3736:
3732:
3677:Caporaso, Lucia
3675:
3674:
3670:
3663:
3650:
3649:
3645:
3620:
3619:
3615:
3610:
3606:
3601:
3597:
3592:
3588:
3583:
3579:
3566:
3562:
3557:
3553:
3548:
3544:
3503:Batyrev, V.V.;
3502:
3501:
3497:
3492:
3488:
3483:
3476:
3471:
3467:
3462:
3458:
3453:
3449:
3434:
3410:
3409:
3405:
3400:
3396:
3389:
3385:
3347:
3346:
3342:
3327:
3310:
3309:
3305:
3278:Raynaud, Michel
3276:
3275:
3271:
3266:
3262:
3257:
3250:
3210:
3209:
3205:
3193:
3189:
3166:10.2307/1970722
3142:
3141:
3137:
3129:
3116:
3115:
3111:
3068:
3067:
3063:
3058:
3049:
3044:
3040:
3033:
3025:. p. 361.
3023:Springer-Verlag
3016:
3015:
3011:
2956:
2955:
2951:
2946:
2942:
2937:
2933:
2928:
2924:
2876:
2875:
2871:
2866:
2862:
2857:
2850:
2827:
2805:
2804:
2797:
2787:
2785:
2782:
2777:
2776:
2772:
2758:
2754:
2750:
2728:
2723:
2722:
2721:
2720:
2574:
2571:
2554:
2543:Arakelov theory
2535:Cartier divisor
2526:
2514:
2486:
2446:
2445:Weil cohomology
2427:yoga of weights
2422:
2418:
2393:
2389:
2372:
2328:
2324:
2313:
2295:
2289:
2269:
2249:
2226:Tate conjecture
2221:
2220:Tate conjecture
2198:Tamagawa number
2193:
2189:
2181:S-unit equation
2165:absolute values
2153:
2137:Zariski closure
2128:
2116:
2115:Skolem's method
2093:elliptic curves
2076:
2072:
2060:replete divisor
2035:
2023:
2015:
1999:counterexamples
1982:
1978:
1937:
1933:
1896:
1878:abelian variety
1865:
1854:Abelian variety
1841:
1821:
1817:
1804:
1768:
1757:abelian variety
1728:
1708:
1669:
1665:
1617:
1605:
1584:from the whole
1545:(dimension 2),
1543:Enrico Bombieri
1539:
1538:Lang conjecture
1535:
1511:
1507:
1473:
1458:
1417:(with roots in
1405:as a theory of
1391:
1364:
1328:
1324:
1268:
1261:height function
1256:
1255:Height function
1224:
1213:elliptic curves
1193:Hasse principle
1188:
1187:Hasse principle
1184:
1159:
1124:division points
1102:
1076:algebraic curve
1002:Fundamental to
999:
979:
973:
965:elliptic curves
953:coordinate ring
944:
913:Flat cohomology
909:
908:Flat cohomology
890:
875:Faltings height
870:
869:Faltings height
866:
837:
833:
805:p-adic analysis
797:
786:geometric genus
742:
737:of dimension 1.
712:
696:
665:
661:
629:
616:with a zero at
608:1 and positive
592:states that an
585:
574:algebraic torus
554:
539:abelian variety
534:
530:
505:elliptic curves
496:
484:
479:
451:
438:
431:Arakelov theory
427:
426:Arakelov theory
408:height function
404:Arakelov height
399:
398:Arakelov height
392:infinite places
372:replete divisor
363:
339:
307:
303:
298:
297:
296:
295:
149:
146:
140:
86:complex numbers
23:
22:
15:
12:
11:
5:
4394:
4392:
4384:
4383:
4378:
4373:
4363:
4362:
4356:
4355:
4353:
4352:
4342:
4332:
4322:
4320:List of topics
4312:
4301:
4298:
4297:
4295:
4294:
4289:
4284:
4279:
4274:
4269:
4264:
4258:
4256:
4252:
4251:
4249:
4248:
4243:
4238:
4233:
4228:
4221:P-adic numbers
4218:
4213:
4208:
4203:
4198:
4193:
4188:
4183:
4178:
4173:
4167:
4165:
4161:
4160:
4158:
4157:
4152:
4147:
4133:
4123:
4109:
4104:
4099:
4094:
4076:
4065:Iwasawa theory
4049:
4047:
4043:
4042:
4037:
4035:
4034:
4027:
4020:
4012:
4006:
4005:
3989:
3986:
3985:
3984:
3970:
3950:
3936:
3916:
3894:
3874:
3860:
3842:Hindry, Marc;
3839:
3825:
3803:
3802:
3793:
3784:
3775:
3767:Pierre Deligne
3759:
3752:
3730:
3668:
3661:
3643:
3613:
3604:
3595:
3586:
3577:
3560:
3551:
3542:
3495:
3486:
3474:
3465:
3456:
3447:
3432:
3403:
3394:
3383:
3356:(1): 143–159.
3340:
3325:
3303:
3282:Artin, Michael
3269:
3260:
3248:
3203:
3187:
3160:(3): 492–517.
3135:
3127:
3109:
3082:(3): 349–366.
3070:Faltings, Gerd
3061:
3047:
3038:
3031:
3009:
2974:(3): 223–251.
2949:
2940:
2931:
2922:
2885:(4): 377–398.
2881:. New Series.
2869:
2860:
2848:
2825:
2795:
2770:
2751:
2749:
2746:
2745:
2744:
2739:
2734:
2727:
2724:
2717:
2716:
2711:
2706:
2701:
2696:
2691:
2686:
2681:
2676:
2671:
2666:
2661:
2656:
2651:
2646:
2641:
2636:
2631:
2626:
2621:
2616:
2611:
2606:
2601:
2596:
2591:
2586:
2581:
2575:
2570:
2569:
2567:
2566:
2555:
2552:
2550:
2527:
2524:
2522:
2515:
2512:
2510:
2487:
2484:
2482:
2447:
2444:
2442:
2423:
2420:
2417:
2414:
2413:
2394:
2391:
2388:
2385:
2384:
2373:
2370:
2368:
2329:
2326:
2323:
2320:
2319:
2309:
2293:
2285:
2270:
2267:
2265:
2262:good reduction
2258:p-adic numbers
2250:
2247:
2245:
2222:
2219:
2217:
2194:
2191:
2188:
2185:
2184:
2154:
2151:
2149:
2129:
2126:
2124:
2117:
2114:
2112:
2077:
2074:
2071:
2068:
2067:
2036:
2033:
2031:
2028:good reduction
2024:
2017:
2014:
2011:
2010:
1983:
1980:
1977:
1974:
1973:
1958:good reduction
1938:
1935:
1932:
1929:
1928:
1897:
1894:
1892:
1885:quadratic form
1866:
1863:
1861:
1842:
1839:
1837:
1822:
1819:
1816:
1813:
1812:
1805:
1802:
1800:
1769:
1766:
1764:
1729:
1726:
1724:
1709:
1706:
1704:
1678:Michel Raynaud
1670:
1667:
1664:
1661:
1660:
1618:
1615:
1613:
1606:
1603:
1601:
1540:
1537:
1534:
1531:
1530:
1512:
1509:
1506:
1503:
1502:
1453:roots of unity
1411:Galois modules
1395:Iwasawa theory
1392:
1390:Iwasawa theory
1389:
1387:
1365:
1362:
1360:
1337:Jun-ichi Igusa
1329:
1326:
1323:
1320:
1319:
1269:
1266:
1264:
1257:
1254:
1252:
1225:
1222:
1220:
1189:
1186:
1183:
1180:
1179:
1160:
1157:
1155:
1098:
1084:smooth variety
1072:Good reduction
1066:, because the
1060:singular point
1030:; for example
1004:local analysis
1000:
998:Good reduction
997:
995:
980:
975:
972:
969:
968:
945:
942:
940:
910:
907:
905:
902:Richard Taylor
891:
888:
886:
871:
868:
865:
862:
861:
838:
835:
832:
829:
828:
798:
796:Dwork's method
795:
793:
782:singular curve
743:
740:
738:
708:
697:
694:
692:
669:Diagonal forms
666:
664:Diagonal forms
663:
660:
657:
656:
653:Dwork's method
630:
627:
625:
594:elliptic curve
586:
583:
581:
578:Runge's method
570:Thoralf Skolem
555:
552:
550:
543:quadratic form
535:
532:
529:
526:
525:
524:
497:
494:
492:
489:good reduction
485:
482:
478:
475:
474:
452:
449:
447:
439:
436:
434:
428:
425:
423:
400:
397:
395:
364:
361:
359:
340:
337:
335:
312:abc conjecture
308:
306:abc conjecture
305:
302:
299:
292:
291:
286:
281:
276:
271:
266:
261:
256:
251:
246:
241:
236:
231:
226:
221:
216:
211:
206:
201:
196:
191:
186:
181:
176:
171:
166:
161:
156:
150:
145:
144:
142:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4393:
4382:
4379:
4377:
4374:
4372:
4369:
4368:
4366:
4351:
4347:
4343:
4341:
4337:
4333:
4331:
4323:
4321:
4313:
4311:
4303:
4302:
4299:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4267:Modular forms
4265:
4263:
4260:
4259:
4257:
4253:
4247:
4244:
4242:
4239:
4237:
4234:
4232:
4229:
4226:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4202:
4199:
4197:
4194:
4192:
4191:Prime numbers
4189:
4187:
4184:
4182:
4179:
4177:
4174:
4172:
4169:
4168:
4166:
4162:
4156:
4153:
4151:
4148:
4145:
4141:
4137:
4134:
4131:
4127:
4124:
4121:
4117:
4113:
4110:
4108:
4105:
4103:
4100:
4098:
4095:
4092:
4088:
4084:
4080:
4077:
4074:
4073:Kummer theory
4070:
4066:
4062:
4058:
4054:
4051:
4050:
4048:
4044:
4040:
4039:Number theory
4033:
4028:
4026:
4021:
4019:
4014:
4013:
4010:
4004:
4000:
3996:
3992:
3991:
3987:
3981:
3977:
3973:
3967:
3963:
3959:
3955:
3951:
3947:
3943:
3939:
3937:3-540-61223-8
3933:
3929:
3925:
3921:
3917:
3913:
3909:
3905:
3901:
3897:
3895:0-387-96793-1
3891:
3887:
3883:
3879:
3875:
3871:
3867:
3863:
3861:0-387-98981-1
3857:
3853:
3849:
3845:
3840:
3836:
3832:
3828:
3822:
3818:
3814:
3810:
3806:
3805:
3797:
3794:
3788:
3785:
3779:
3776:
3772:
3768:
3763:
3760:
3755:
3749:
3745:
3741:
3734:
3731:
3726:
3722:
3718:
3714:
3709:
3704:
3700:
3696:
3695:
3690:
3686:
3682:
3678:
3672:
3669:
3664:
3658:
3654:
3647:
3644:
3639:
3635:
3631:
3627:
3623:
3617:
3614:
3608:
3605:
3599:
3596:
3590:
3587:
3581:
3578:
3574:
3570:
3564:
3561:
3555:
3552:
3546:
3543:
3538:
3534:
3530:
3526:
3522:
3518:
3514:
3510:
3506:
3499:
3496:
3490:
3487:
3481:
3479:
3475:
3469:
3466:
3460:
3457:
3451:
3448:
3443:
3439:
3435:
3429:
3426:. p. 3.
3425:
3421:
3417:
3413:
3407:
3404:
3398:
3395:
3392:
3387:
3384:
3379:
3375:
3371:
3367:
3363:
3359:
3355:
3351:
3344:
3341:
3336:
3332:
3328:
3326:0-8176-4397-4
3322:
3318:
3314:
3307:
3304:
3299:
3295:
3291:
3287:
3283:
3279:
3273:
3270:
3264:
3261:
3255:
3253:
3249:
3244:
3240:
3236:
3232:
3228:
3224:
3220:
3216:
3215:
3207:
3204:
3200:
3196:
3191:
3188:
3183:
3179:
3175:
3171:
3167:
3163:
3159:
3155:
3154:
3149:
3145:
3139:
3136:
3130:
3128:0-387-96311-1
3124:
3120:
3113:
3110:
3105:
3101:
3097:
3093:
3089:
3085:
3081:
3077:
3076:
3071:
3065:
3062:
3056:
3054:
3052:
3048:
3042:
3039:
3034:
3028:
3024:
3020:
3013:
3010:
3005:
3001:
2997:
2993:
2989:
2985:
2981:
2977:
2973:
2969:
2968:
2963:
2959:
2953:
2950:
2944:
2941:
2935:
2932:
2926:
2923:
2918:
2914:
2910:
2906:
2902:
2898:
2893:
2888:
2884:
2880:
2873:
2870:
2864:
2861:
2855:
2853:
2849:
2844:
2840:
2836:
2832:
2828:
2822:
2818:
2814:
2813:
2808:
2802:
2800:
2796:
2781:
2774:
2771:
2768:
2766:
2761:
2756:
2753:
2747:
2743:
2740:
2738:
2735:
2733:
2730:
2729:
2725:
2719:
2715:
2712:
2710:
2707:
2705:
2702:
2700:
2697:
2695:
2692:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2665:
2662:
2660:
2657:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2620:
2617:
2615:
2612:
2610:
2607:
2605:
2602:
2600:
2597:
2595:
2592:
2590:
2587:
2585:
2582:
2580:
2577:
2576:
2564:
2560:
2556:
2551:
2548:
2544:
2540:
2536:
2532:
2531:Weil function
2528:
2525:Weil function
2523:
2520:
2516:
2511:
2508:
2504:
2500:
2496:
2492:
2488:
2483:
2480:
2476:
2472:
2468:
2464:
2460:
2456:
2455:finite fields
2452:
2448:
2443:
2440:
2436:
2432:
2428:
2424:
2419:
2415:
2411:
2407:
2403:
2399:
2395:
2390:
2386:
2382:
2378:
2374:
2369:
2366:
2362:
2358:
2354:
2350:
2346:
2342:
2338:
2334:
2330:
2325:
2321:
2317:
2312:
2308:
2304:
2300:
2296:
2288:
2283:
2279:
2275:
2271:
2266:
2263:
2259:
2255:
2251:
2246:
2243:
2239:
2235:
2231:
2227:
2223:
2218:
2215:
2211:
2207:
2203:
2199:
2195:
2190:
2186:
2182:
2178:
2174:
2170:
2169:number fields
2166:
2161:
2160:
2155:
2150:
2147:
2143:
2138:
2134:
2130:
2125:
2122:
2118:
2113:
2110:
2106:
2102:
2098:
2097:finite fields
2094:
2090:
2086:
2082:
2078:
2073:
2069:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2040:replete ideal
2037:
2034:Replete ideal
2032:
2029:
2025:
2021:
2016:
2012:
2008:
2004:
2000:
1996:
1992:
1988:
1985:The topic of
1984:
1979:
1975:
1971:
1967:
1963:
1959:
1955:
1951:
1947:
1944:of dimension
1943:
1939:
1934:
1930:
1926:
1921:
1918:
1915:
1911:
1908:
1907:ample divisor
1904:
1903:
1898:
1893:
1890:
1886:
1882:
1879:
1875:
1871:
1867:
1862:
1859:
1855:
1851:
1847:
1843:
1838:
1835:
1831:
1827:
1823:
1818:
1814:
1810:
1806:
1801:
1798:
1794:
1790:
1786:
1782:
1778:
1774:
1770:
1765:
1762:
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1730:
1725:
1722:
1718:
1714:
1710:
1705:
1702:
1698:
1694:
1690:
1687:
1683:
1679:
1675:
1671:
1666:
1662:
1658:
1654:
1650:
1646:
1642:
1638:
1635:
1631:
1627:
1623:
1619:
1614:
1611:
1607:
1602:
1599:
1595:
1591:
1587:
1586:complex plane
1583:
1579:
1576:
1572:
1568:
1564:
1560:
1559:Zariski dense
1556:
1552:
1548:
1544:
1541:
1536:
1532:
1528:
1524:
1520:
1516:
1513:
1508:
1504:
1500:
1496:
1495:inverse limit
1492:
1488:
1484:
1480:
1476:
1469:
1465:
1461:
1454:
1450:
1446:
1442:
1438:
1435:
1431:
1429:
1424:
1420:
1416:
1412:
1408:
1404:
1400:
1396:
1393:
1388:
1385:
1381:
1377:
1373:
1369:
1366:
1361:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1325:
1321:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1285:
1281:
1277:
1274:
1270:
1265:
1262:
1258:
1253:
1250:
1246:
1242:
1238:
1237:Euler product
1234:
1230:
1226:
1221:
1218:
1214:
1210:
1206:
1202:
1198:
1194:
1190:
1185:
1181:
1177:
1173:
1169:
1165:
1161:
1156:
1153:
1149:
1145:
1141:
1137:
1133:
1129:
1125:
1121:
1117:
1113:
1110:
1106:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1065:
1061:
1057:
1053:
1049:
1045:
1041:
1037:
1033:
1029:
1026:
1022:
1018:
1014:
1013:
1009:
1005:
1001:
996:
993:
989:
985:
981:
978:
974:
970:
966:
962:
958:
957:global fields
954:
950:
946:
941:
938:
934:
930:
926:
925:scheme theory
922:
918:
917:flat topology
914:
911:
906:
903:
899:
895:
892:
887:
884:
880:
876:
872:
867:
863:
859:
855:
851:
850:Michael Artin
847:
843:
839:
834:
830:
826:
822:
818:
814:
810:
806:
802:
801:Bernard Dwork
799:
794:
791:
787:
783:
779:
775:
771:
767:
764:
760:
756:
752:
748:
744:
739:
736:
732:
728:
724:
720:
716:
711:
706:
702:
698:
693:
690:
686:
682:
678:
674:
670:
667:
662:
658:
654:
650:
646:
642:
638:
634:
631:
626:
623:
619:
615:
611:
607:
603:
599:
595:
591:
587:
582:
579:
575:
571:
567:
563:
559:
556:
551:
548:
544:
540:
536:
531:
527:
522:
518:
514:
510:
506:
502:
498:
493:
490:
486:
483:Bad reduction
481:
476:
472:
468:
464:
460:
456:
453:
448:
446:
445:
440:
435:
432:
429:
424:
421:
417:
413:
409:
405:
401:
396:
393:
389:
388:finite places
385:
381:
377:
373:
369:
365:
360:
357:
353:
349:
345:
341:
336:
333:
329:
325:
321:
317:
313:
309:
304:
300:
294:
290:
287:
285:
282:
280:
277:
275:
272:
270:
267:
265:
262:
260:
257:
255:
252:
250:
247:
245:
242:
240:
237:
235:
232:
230:
227:
225:
222:
220:
217:
215:
212:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
185:
182:
180:
177:
175:
172:
170:
167:
165:
162:
160:
157:
155:
152:
151:
141:
138:
136:
131:
129:
128:number theory
125:
121:
117:
113:
109:
107:
103:
99:
95:
91:
87:
83:
79:
78:finite fields
75:
74:number fields
71:
67:
63:
60:
56:
52:
50:
46:
42:
41:number theory
38:
34:
30:
19:
4164:Key concepts
4091:sieve theory
3957:
3923:
3884:. New York:
3881:
3847:
3812:
3796:
3787:
3778:
3770:
3762:
3739:
3733:
3698:
3692:
3685:Mazur, Barry
3671:
3652:
3646:
3629:
3625:
3616:
3607:
3598:
3589:
3580:
3572:
3568:
3563:
3554:
3545:
3512:
3508:
3505:Manin, Yu.I.
3498:
3489:
3468:
3459:
3450:
3419:
3406:
3397:
3386:
3353:
3350:Invent. Math
3349:
3343:
3316:
3313:Schoof, René
3306:
3289:
3272:
3263:
3218:
3212:
3206:
3190:
3157:
3151:
3138:
3118:
3112:
3079:
3073:
3064:
3041:
3018:
3012:
2971:
2965:
2952:
2943:
2934:
2925:
2892:math/9802121
2882:
2878:
2872:
2863:
2811:
2807:Schoof, René
2786:. Retrieved
2773:
2764:
2755:
2718:
2558:
2530:
2435:Hodge theory
2376:
2360:
2356:
2352:
2348:
2344:
2340:
2336:
2310:
2306:
2302:
2298:
2291:
2286:
2281:
2261:
2204:. There the
2157:
2145:
2141:
2132:
2120:
2089:Tate modules
2059:
2055:
2051:
2043:
2039:
2027:
2019:
1991:Brauer group
1969:
1965:
1961:
1953:
1949:
1945:
1941:
1919:
1909:
1900:
1888:
1880:
1846:Néron symbol
1845:
1840:Néron symbol
1833:
1826:naive height
1820:Naive height
1796:
1788:
1784:
1780:
1776:
1700:
1696:
1692:
1688:
1681:
1649:non-singular
1644:
1636:
1634:finite field
1629:
1610:linear torus
1609:
1604:Linear torus
1597:
1593:
1577:
1574:
1570:
1566:
1562:
1557:do not have
1555:general type
1498:
1490:
1486:
1482:
1478:
1471:
1467:
1463:
1456:
1448:
1444:
1440:
1436:
1426:
1384:Selmer group
1348:
1344:
1335:, named for
1311:
1303:
1283:
1275:
1232:
1217:cubic curves
1201:local fields
1197:global field
1120:ramification
1111:
1108:
1104:
1099:
1095:
1091:
1087:
1071:
1063:
1056:non-singular
1047:
1039:
1032:denominators
1027:
1021:prime ideals
1016:
1010:
1007:
991:
937:gluing axiom
898:Andrew Wiles
773:
769:
765:
762:
758:
754:
750:
746:
726:
722:
718:
714:
709:
704:
700:
648:
617:
606:class number
589:
561:
557:
488:
441:
371:
367:
343:
331:
327:
323:
293:
139:
132:
110:
105:
101:
97:
93:
82:local fields
70:prime fields
65:
64:over fields
61:
53:
28:
26:
4350:Wikiversity
4272:L-functions
3920:Lang, Serge
3878:Lang, Serge
3701:(1): 1–35.
3681:Harris, Joe
3412:Baker, Alan
2519:prime ideal
2196:The direct
2133:special set
2127:Special set
2111:in general.
1952:at a prime
1715:is now the
1561:subsets of
1439:of a curve
1308:meagre sets
1241:L-functions
1209:cubic forms
1132:Néron model
821:rationality
685:Jacobi sums
560:, based on
49:conjectures
33:mathematics
4365:Categories
4231:Arithmetic
3980:0956.11021
3946:0869.11051
3912:0667.14001
3870:0948.11023
3835:1130.11034
3725:0872.14017
3638:0015.38803
3537:0679.14008
3442:1145.11004
3335:1098.14030
3298:0581.14031
3286:Tate, John
3243:0287.43007
3182:0172.46101
3156:. Second.
3148:Tate, John
3004:0359.14009
2958:Coates, J.
2917:1030.11063
2843:1188.11076
2748:References
2495:André Weil
2402:Paul Vojta
2278:C. C. Tsen
2254:Tate curve
2248:Tate curve
2236:, also on
2156:Schmidt's
2101:Mikio Sato
2018:Reduction
2001:; but see
1956:if it has
1783:the group
1739:, Hindry,
1551:Paul Vojta
1547:Serge Lang
1351:. General
1140:Tate curve
1054:enters: a
1050:. However
1025:almost all
844:theory of
772:) and the
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