4335:
4295:
4315:
4305:
4325:
2128:
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
2151:
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
1192:. 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
1196:. 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
1363:'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
1911:
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
781:. 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.
944:
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
1880:
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.
2491:
2124:
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
2768:
500:
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
1817:
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
904:
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
2321:
2415:
2368:
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
2233:, an analogue of the Birch–Swinnerton-Dyer conjecture (q.v.), leading quickly to a clarification of the latter and a recognition of its importance.
2494:, 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
2197:
was eventually proved. For abelian varieties, and in particular the Birch–Swinnerton-Dyer conjecture (q.v.), the
Tamagawa number approach to a
1849:
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.
3682:
1152:
1035:
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
594:
2486:, made public around 1949, on local zeta-functions. The proof was completed in 1973. Those being proved, there remain extensions of the
1116:
1059:
can become larger when linear terms reduce to 0 (the geometric formulation shows it is not the fault of a single set of coordinates).
4318:
1712:
states that there should be a uniform bound on the number of such points, depending only on the genus and the field of definition.
4018:
2550:
is an effective procedure for assigning a height function to any divisor on smooth projective variety over a number field (or to
1825:, or of a polynomial, regarded as a vector of coefficients, or of an algebraic number, from the height of its minimal polynomial.
2202:
2165:
3991:
3958:
3813:
3740:
3649:
3420:
3019:
2813:
2510:
decomposition of algebraic numbers in coordinates of points on algebraic varieties. It has remained somewhat under-developed.
1119:. The theory is subtle, in the sense that the freedom to change variables to try to improve matters is rather unobvious: see
3279:
Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Vol. I: Arithmetic
610:
489:
1978:, i.e. solubility guaranteed by a number of variables polynomial in the degree of an equation, grew out of studies of the
2194:
1284:
4369:
1193:
115:. Arithmetic geometry has also been defined as the application of the techniques of algebraic geometry to problems in
3924:
3882:
3848:
3313:
3115:
1721:
1781:
842:
723:
2866:
van der Geer, G.; Schoof, R. (2000). "Effectivity of
Arakelov divisors and the theta divisor of a number field".
1233:
1156:
866:
of an elliptic curve or abelian variety defined over a number field is a measure of its complexity introduced by
797:
432:
4049:
3677:
890:
4359:
3840:
2487:
2455:
1983:
1601:
is a geometrically irreducible
Zariski-closed subgroup of an affine torus (product of multiplicative groups).
1510:
flavour, and, on the other hand, implicated in some formulations of arithmetic conjectures. See for example
4308:
4095:
2353:
1741:
1662:
965:
364:
4364:
4090:
4075:
4011:
3300:
Roessler, Damian (2005). "A note on the Manin–Mumford conjecture". In van der Geer, Gerard; Moonen, Ben;
2946:
2463:
2369:
1364:
3805:
3732:
3669:
3412:
3400:
2805:
2438:
The initial idea, later somewhat modified, for proving the Weil conjectures (q.v.), was to construct a
2198:
1391:
1132:
1108:
719:
4270:
4229:
4108:
3274:
3141:
3136:
2720:
2394:
2218:
2093:
1818:
1511:
1217:
1136:
590:
497:
455:
123:
4114:
3200:
Igusa, Jun-Ichi (1974). "Complex powers and asymptotic expansions. I. Functions of certain types".
3063:
2955:
2153:
2069:
1761:
1124:
1048:
882:
598:
505:
501:
2535:
2201:
fails on a direct attempt, though it has had heuristic value over many years. Now a sophisticated
1858:
1846:
535:
4334:
4057:
4041:
2495:
1975:
1578:
1164:
1032:
917:
765:, defined by Vojta. The difference between the two may be compared to the difference between the
400:
392:
4298:
4118:
4067:
4004:
2419:
2190:
1637:
1515:
1387:
1304:
1288:
1140:
1044:
834:
805:
629:
621:
586:
509:
4338:
2303:. Algebraically closed fields are of Tsen rank zero. The Tsen rank is greater or equal to the
1814:
4275:
4204:
2304:
2097:
1991:
1709:
1641:
1056:
921:
849:, and was basic in the formulation of the Tate conjecture (q.v.) and numerous other theories.
447:
408:
3379:
1800:
is an algebraic variety which has only finitely many points in any finitely generated field.
1708:, and states that a curve of genus at least two has only finitely many rational points. The
4265:
4100:
3892:
3346:
3072:
2964:
2823:
2161:
1913:
1890:
1237:
661:
104:
78:
43:
25:
4328:
3968:
3934:
3900:
3858:
3823:
3713:
3626:
3525:
3430:
3323:
3286:
3281:. Progress in Mathematics (in French). Vol. 35. Birkhauser-Boston. pp. 327–352.
3231:
3170:
2992:
2905:
2831:
1063:
refers to the reduced variety having the same properties as the original, for example, an
8:
4234:
4143:
4138:
4132:
4124:
4085:
3832:
3613:(1936). "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper".
3202:
2730:
2725:
2527:
2459:
1749:
1614:
1610:
1570:
1329:
1321:
846:
677:
669:
372:
340:
100:
47:
4324:
3404:
3350:
3076:
2968:
2043:
and a vector of positive real numbers with components indexed by the infinite places of
1023:
of fractions are tricky, in that reduction modulo a prime in the denominator looks like
830:
633:
451:
4280:
4224:
4128:
4045:
3701:
3513:
3362:
3219:
3158:
3088:
2980:
2893:
2875:
2467:
1995:
1905:
1701:
1645:
1503:
1345:
1160:
1040:
1000:
972:
871:
640:
coefficients in this case. It is one of a number of theories deriving in some way from
554:
553:-adic analytic functions, is a special application but capable of proving cases of the
108:
33:
3942:
308:
93:
is something to be proved and studied as an extra topic, even knowing the geometry of
4194:
3987:
3954:
3920:
3878:
3844:
3809:
3736:
3645:
3517:
3496:(1990). "On the number of rational points of bounded height on algebraic varieties".
3416:
3366:
3309:
3223:
3132:
3111:
3092:
3015:
2984:
2809:
2451:
2447:
2439:
2427:
2398:
2073:
1797:
1407:
1403:
1395:
1368:
1341:
1280:
1104:
809:
778:
641:
566:
557:
for curves whose
Jacobian's rank is less than its dimension. It developed ideas from
459:
422:
is an approach to arithmetic geometry that explicitly includes the 'infinite primes'.
404:
336:
2897:
2798:(2008). "Computing Arakelov class groups". In Buhler, J.P.; P., Stevenhagen (eds.).
2748:
1542:(integral points case) and Piotr Blass have conjectured that algebraic varieties of
1367:(often called 'descents', when written out by equations); in more modern terms in a
4199:
4184:
3964:
3930:
3896:
3854:
3819:
3709:
3691:
3622:
3521:
3505:
3426:
3354:
3319:
3282:
3227:
3211:
3166:
3150:
3080:
2988:
2972:
2901:
2885:
2827:
2479:
2386:
2230:
2222:
2147:
2052:
2036:
1862:
1838:
1705:
1674:
1507:
1422:
1416:
1411:
1360:
1356:
1276:
1268:
1261:
1068:
1024:
976:
949:
937:
813:
804:
and other techniques that have not all been absorbed into general theories such as
766:
665:
625:
443:
368:
344:
112:
3696:
3215:
1252:
in
Diophantine geometry quantifies the size of solutions to Diophantine equations.
311:
attempts to state as much as possible about repeated prime factors in an equation
4213:
4189:
4104:
3983:
3950:
3916:
3888:
3874:
3797:
3011:
2819:
2799:
2551:
2531:
2523:
2226:
2214:
2186:
2169:
2125:
1866:
1842:
1745:
1629:
1531:
1249:
1208:) are at a general level connected with the limitations of the analytic approach.
1181:
1064:
941:
901:
863:
793:
774:
562:
527:
493:
419:
396:
829:
The search for a Weil cohomology (q.v.) was at least partially fulfilled in the
4250:
4169:
4053:
3755:
3665:
3266:
3010:. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.).
2279:: that is, such that any system of polynomials with no constant term of degree
2133:
is the
Zariski closure of the images of all non-constant holomorphic maps from
2081:
2005:
1987:
1946:
1902:
1873:
1725:
1666:
1589:
is analytically hyperbolic if and only if all subvarieties are of general type.
1558:
a finitely-generated field. This circle of ideas includes the understanding of
1441:
1383:
1325:
1201:
1112:
1072:
992:
953:
801:
770:
582:
558:
531:
380:
300:
74:
3301:
2795:
1120:
4353:
4209:
4061:
4027:
3493:
3270:
3058:
2246:
1895:
1733:
1574:
1547:
1483:
1399:
1225:
913:
905:
867:
838:
789:
657:
116:
29:
2483:
1287:
which shows the rational numbers are Hilbertian. Results are applied to the
4255:
4179:
4079:
2950:
2801:
Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography
2443:
2423:
2157:
2085:
1979:
1622:
1543:
1372:
1185:
945:
925:
886:
376:
304:
66:
62:
2137:. Lang conjectured that the analytic and algebraic special sets are equal.
3673:
3380:
2 page exposition of the Mordell–Lang conjecture by B. Mazur, 3 Nov. 2005
3269:(1983). "Sous-variétés d'une variété abélienne et points de torsion". In
2507:
2077:
1205:
1189:
1020:
1009:
458:
could be regarded as Artin L-functions for the Galois representations on
70:
58:
21:
3337:
McQuillan, Michael (1995). "Division points on semi-abelian varieties".
3308:. Progress in Mathematics. Vol. 239. Birkhäuser. pp. 311–318.
1872:
is a height function (q.v.) that is essentially intrinsic, and an exact
1821:. This may be used to define height on a point in projective space over
885:, the most celebrated conjecture of Diophantine geometry, was proved by
4260:
4219:
4071:
3908:
3866:
3705:
3642:
Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics
3610:
3509:
3358:
3183:
3162:
3084:
2976:
2889:
2390:
2266:
2242:
2089:
1737:
1729:
1680:
can only contain a finite number of points that are of finite order in
1539:
1535:
1296:
1229:
1197:
1128:
1013:
673:
602:
37:
3061:(1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern".
2557:
2307:
but it is not known if they are equal except in the case of rank zero.
1876:, rather than approximately quadratic with respect to the addition on
1332:
counting numbers of points on an algebraic variety modulo high powers
952:
over the complex numbers, also, have some quite strict analogies with
395:
on a projective space over the field of algebraic numbers is a global
323:. For example 3 + 125 = 128 but the prime powers here are exceptional.
132:
2880:
2262:
948:
should all be treated on the same basis. The idea goes further. Thus
812:
of local zeta-functions, the initial advance in the direction of the
3154:
2534:. They are used in the construction of the local components of the
2522:
on an algebraic variety is a real-valued function defined off some
2269:
who introduced their study in 1936, is the smallest natural number
3729:
Some Problems of Unlikely Intersections in Arithmetic and Geometry
2088:
obtained from reducing a given elliptic curve over the rationals.
1232:
remain largely in the realm of conjecture, with the proof of the
3996:
3949:. Grundlehren der Mathematischen Wissenschaften. Vol. 322.
1636:. According to the Weil conjectures (q.v.) these functions, for
1562:
and the Lang conjectures on that, and the Vojta conjectures. An
1083:, assumed smooth, such that there is otherwise a smooth reduced
4159:
1012:. In the typical situation this presents little difficulty for
1764:
is a foundational result stating that for an abelian variety
1240:
is largely complementary to the theory of global L-functions.
909:
644:, and has applications outside purely arithmetical questions.
3006:
Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008).
2500:
3105:
2752:
1490:-adic L-function earlier introduced by Kubota and Leopoldt.
2953:(1977). "On the conjecture of Birch and Swinnerton-Dyer".
702:: that is, such that any homogeneous polynomial of degree
2229:, but well within arithmetic geometry. It also gave, for
1916:
and it is conjectured that they are essentially the same.
1228:
formed from local zeta-functions. The properties of such
664:
to study from an arithmetic point of view (including the
3139:(November 1968). "Good reduction of abelian varieties".
1075:
remaining smooth. In general there will be a finite set
565:. (Other older methods for Diophantine problems include
40:, which can be related at various levels of generality.
3306:
Number fields and function fields — two parallel worlds
3005:
2506:
André Weil proposed a theory in the 1920s and 1930s on
2490:
congruence, which comes from an elementary method, and
1640:
varieties, exhibit properties closely analogous to the
979:
to varieties of dimension at least two is often called
482:
363:) on a global field is an extension of the concept of
4174:
4164:
1577:
to it exists, that is not constant. Examples include
1414:). In its early days in the late 1960s it was called
1200:
in small numbers of variables (and in particular for
3122:→ Contains an English translation of Faltings (1983)
2249:
introduced by John Tate to study bad reduction (see
3664:
1837:is a bimultiplicative pairing between divisors and
1466:) as Galois module. In the same way, Iwasawa added
936:It was realised in the nineteenth century that the
738:refers to two related concepts relative to a point
3760:Poids dans la cohomologie des variétés algébriques
2865:
1031:per fraction. With a little extra sophistication,
424:
2273:, if it exists, such that the field is of class T
2129:abelian subvarieties. For a complex variety, the
1617:for the number of points on an algebraic variety
696:, if it exists, such that the field of is class C
4351:
3731:. Annals of Mathematics Studies. Vol. 181.
2096:suggested it around 1960. It is a prototype for
940:of a number field has analogies with the affine
122:See also the glossary of number theory terms at
36:. Much of the theory is in the form of proposed
24:, areas growing out of the traditional study of
3830:
3203:Journal für die reine und angewandte Mathematik
2160:. The theorem may be used to obtain results on
1788:, but extends to all finitely-generated fields.
1163:solutions. The initial result of this type was
1145:
963:
3796:
1784:. This was proved initially for number fields
1506:is on one hand a quite general theory with an
609: = 1. This is a special case of the
103:can be more generally defined as the study of
4012:
3399:
2843:
2841:
2482:were three highly influential conjectures of
2450:at detecting topological structure, and have
1569:over the complex numbers is one such that no
908:has been considered the 'right' foundational
530:is a height function that is a distinguished
3804:. New Mathematical Monographs. Vol. 4.
3683:Journal of the American Mathematical Society
3560:in the volume (O. F. G. Schilling, editor),
3558:Algebraic cycles and poles of zeta functions
3491:
3411:. New Mathematical Monographs. Vol. 9.
3106:Cornell, Gary; Silverman, Joseph H. (1986).
1440:Picard variety), where the finite field has
876:
2945:
2767:Sutherland, Andrew V. (September 5, 2013).
1720:The Mordell–Lang conjecture, now proved by
1655:
4314:
4304:
4019:
4005:
3409:Logarithmic Forms and Diophantine Geometry
2838:
2766:
2290:variables has a non-trivial zero whenever
1968:
1714:
1188:is the same as solubility in all relevant
728:
710:variables has a non-trivial zero whenever
692:of a field is the smallest natural number
3695:
3336:
3131:
2879:
2502:Weil distributions on algebraic varieties
1564:analytically hyperbolic algebraic variety
1478:→ ∞, for his analogue, to a number field
930:
920:, the discovery of Grothendieck that the
615:
3941:
3469:
3467:
3299:
3243:
3241:
3057:
2358:
2314:
2245:is a particular elliptic curve over the
1371:group which is to be proved finite. See
1210:
1153:Grothendieck–Katz p-curvature conjecture
1027:, but that rules out only finitely many
924:are sheaves for it (i.e. a very general
682:
540:
3726:
3600:Bombieri & Gubler (2006) pp.176–230
3443:Bombieri & Gubler (2006) pp.301–314
3265:
3044:
3042:
3040:
2790:
2788:
2352:. The conjecture would follow from the
2348:-rational points on any curve of genus
2062:
1882:
1754:
571:
371:. It is a formal linear combination of
325:
57:that are finitely generated over their
4352:
3639:
2999:
2794:
2540:
2203:equivariant Tamagawa number conjecture
2102:
1603:
1444:added to make finite field extensions
1314:
4000:
3837:Diophantine Geometry: An Introduction
3789:Hindry & Silverman (2000) 184–185
3576:
3464:
3437:
3256:Bombieri & Gubler (2006) pp.82–93
3238:
3199:
2918:Bombieri & Gubler (2006) pp.66–67
2769:"Introduction to Arithmetic Geometry"
2442:applying to algebraic varieties over
1923:
1694:
1344:are now known, drawing on methods of
1295:) are in some sense analogous to the
783:
399:with local contributions coming from
3984:An invitation to arithmetic geometry
3907:
3865:
3774:
3762:, Actes ICM, Vancouver, 1974, 79–85.
3609:
3455:
3187:
3037:
3028:
2785:
2458:could be applied to the counting in
2221:, 1963) provided an analogue to the
1851:
1790:
1740:unifying the Mordell conjecture and
1254:
752:geometric (logarithmic) discriminant
611:Birch and Swinnerton-Dyer conjecture
496:postulates a connection between the
490:Birch and Swinnerton-Dyer conjecture
484:Birch and Swinnerton-Dyer conjecture
437:
379:having integer coefficients and the
3582:Hindry & Silverman (2000) p.480
3482:Hindry & Silverman (2000) p.488
3247:Hindry & Silverman (2000) p.479
2804:. MSRI Publications. Vol. 44.
2472:
2379:
2195:Weil conjecture on Tamagawa numbers
2179:
2166:Siegel's theorem on integral points
2139:
1845:used in Néron's formulation of the
1350:
1155:applies reduction modulo primes to
1107:, good reduction is connected with
823:
520:
349:
18:arithmetic and diophantine geometry
13:
3976:
2432:
2207:
1525:
1451:The local zeta-function (q.v.) of
1242:
1174:
895:
856:
413:
385:
61:—including as of special interest
14:
4381:
4026:
2526:which generalises the concept of
2332:> 2, there is a uniform bound
2324:states that for any number field
1455:can be recovered from the points
1377:
985:
724:quasi-algebraically closed fields
651:
624:is a p-adic cohomology theory in
293:
4333:
4323:
4313:
4303:
4294:
4293:
2512:
2021:
1782:finitely-generated abelian group
1470:-power roots of unity for fixed
1291:. Thin sets (the French word is
1285:Hilbert's irreducibility theorem
1157:algebraic differential equations
798:algebraic differential equations
722:are of Diophantine dimension 0;
636:which is deficient in using mod
470:
3871:Introduction to Arakelov theory
3802:Heights in Diophantine Geometry
3783:
3765:
3749:
3720:
3678:"Uniformity of rational points"
3658:
3633:
3603:
3594:
3585:
3567:
3562:Arithmetical Algebraic Geometry
3550:
3541:
3532:
3485:
3476:
3446:
3393:
3384:
3373:
3330:
3293:
3259:
3250:
3193:
3177:
3125:
3099:
3051:
2939:
2560:
2389:is a complex of conjectures by
2189:definition works well only for
1861:(also often referred to as the
1827:
1807:
1591:
1117:Néron–Ogg–Shafarevich criterion
433:arithmetic of abelian varieties
426:Arithmetic of abelian varieties
135:
4072:analytic theory of L-functions
4050:non-abelian class field theory
3913:Survey of Diophantine Geometry
3644:. Springer. pp. 109–126.
2930:
2921:
2912:
2859:
2850:
2760:
2742:
2454:acting in such a way that the
2446:that would both be as good as
2114:
2072:describes the distribution of
1585:> 1. Lang conjectured that
1283:. This is a geometric take on
1194:Hardy–Littlewood circle method
1079:of primes for a given variety
446:are defined for quite general
1:
3841:Graduate Texts in Mathematics
3697:10.1090/S0894-0347-97-00195-1
3216:10.1515/crll.1974.268-269.110
2736:
2456:Lefschetz fixed-point theorem
2235:
1184:states that solubility for a
995:in arithmetic problems is to
841:. It provided a proof of the
4096:Transcendental number theory
3556:It is mentioned in J. Tate,
2559:
2255:
2205:is a major research problem.
1986:. It stalled in the face of
1365:principal homogeneous spaces
1147:Grothendieck–Katz conjecture
966:Geometric class field theory
792:used distinctive methods of
746:defined over a number field
407:and the usual metric on the
134:
28:to encompass large parts of
7:
4319:List of recreational topics
4091:Computational number theory
4076:probabilistic number theory
3008:Cohomology of Number Fields
2714:
2492:improvements of Weil bounds
2464:motive (algebraic geometry)
2393:, making analogies between
1724:following work of Laurent,
1497:
1486:of class groups, finding a
1421:. The analogy was with the
1234:Taniyama–Shimura conjecture
1159:, to derive information on
720:Algebraically closed fields
680:is the most classical case.
526:The canonical height on an
85:the existence of points of
46:in general is the study of
10:
4386:
3806:Cambridge University Press
3733:Princeton University Press
3413:Cambridge University Press
2806:Cambridge University Press
2408:
1236:being a breakthrough. The
1133:semistable abelian variety
4370:Glossaries of mathematics
4289:
4271:Diophantine approximation
4243:
4230:Chinese remainder theorem
4152:
4034:
3800:; Gubler, Walter (2006).
3727:Zannier, Umberto (2012).
3142:The Annals of Mathematics
2721:Glossary of number theory
2554:on non-smooth varieties).
2488:Chevalley–Warning theorem
2395:Diophantine approximation
2035:is a formal product of a
1984:Chevalley–Warning theorem
1819:lowest common denominator
1137:semistable elliptic curve
916:goes back to the fact of
672:are computed in terms of
660:are some of the simplest
591:imaginary quadratic field
498:rank of an elliptic curve
383:having real coefficients.
124:Glossary of number theory
4115:Arithmetic combinatorics
3064:Inventiones Mathematicae
2956:Inventiones Mathematicae
2572:
2567:
2496:Algebraic geometry codes
2462:. For later history see
2354:Bombieri–Lang conjecture
1742:Manin–Mumford conjecture
1663:Manin–Mumford conjecture
1657:Manin–Mumford conjecture
1579:compact Riemann surfaces
1419:analogue of the Jacobian
1336:of a fixed prime number
1125:potential good reduction
742:on an algebraic variety
632:to fill the gap left by
454:in the 1960s meant that
147:
142:
107:of finite type over the
4086:Geometric number theory
4042:Algebraic number theory
3982:Dino Lorenzini (1996),
3947:Algebraic Number Theory
2370:Mordell–Lang conjecture
2191:linear algebraic groups
2131:holomorphic special set
2011:a prime number or ideal
1976:quasi-algebraic closure
1970:Quasi-algebraic closure
1716:Mordell–Lang conjecture
1392:Stickelberger's theorem
1033:homogeneous coordinates
918:faithfully-flat descent
763:arithmetic discriminant
736:discriminant of a point
730:Discriminant of a point
335:is the analogue of the
4205:Transcendental numbers
4119:additive number theory
4068:Analytic number theory
3780:Lang (1997) pp.164,212
3640:Lorenz, Falko (2008).
3564:, pages 93–110 (1965).
3538:Lang (1997) pp.161–162
3110:. New York: Springer.
2927:Lang (1988) pp.156–157
2702:
2697:
2692:
2687:
2682:
2677:
2672:
2667:
2662:
2657:
2652:
2647:
2642:
2637:
2632:
2627:
2622:
2617:
2612:
2607:
2602:
2597:
2592:
2587:
2582:
2577:
2420:Alexander Grothendieck
2265:of a field, named for
2199:local–global principle
2098:Galois representations
1669:, states that a curve
1560:analytic hyperbolicity
1554:-rational points, for
1516:Lichtenbaum conjecture
1388:analytic number theory
1305:Baire category theorem
1289:inverse Galois problem
932:Function field analogy
922:representable functors
835:Alexander Grothendieck
808:. He first proved the
806:crystalline cohomology
630:Alexander Grothendieck
622:Crystalline cohomology
617:Crystalline cohomology
587:complex multiplication
456:Hasse–Weil L-functions
450:. The introduction of
448:Galois representations
409:non-Archimedean fields
277:
272:
267:
262:
257:
252:
247:
242:
237:
232:
227:
222:
217:
212:
207:
202:
197:
192:
187:
182:
177:
172:
167:
162:
157:
152:
16:This is a glossary of
4276:Irrationality measure
4266:Diophantine equations
4109:Hodge–Arakelov theory
3547:Neukirch (1999) p.185
2847:Neukirch (1999) p.189
2422:of analogies between
2366:unlikely intersection
2360:Unlikely intersection
2322:uniformity conjecture
2316:Uniformity conjecture
2305:Diophantine dimension
2162:Diophantine equations
1736:, is a conjecture of
1710:Uniformity conjecture
1642:Riemann zeta-function
1512:Birch–Tate conjecture
1482:, and considered the
1267:is one for which the
1220:, sometimes called a
1218:Hasse–Weil L-function
1212:Hasse–Weil L-function
1057:Zariski tangent space
883:Fermat's Last Theorem
878:Fermat's Last Theorem
690:Diophantine dimension
684:Diophantine dimension
73:. Of those, only the
26:Diophantine equations
4360:Diophantine geometry
4235:Arithmetic functions
4101:Diophantine geometry
3833:Silverman, Joseph H.
3615:J. Chinese Math. Soc
3573:Lang (1997) pp.17–23
3452:Lang (1988) pp.66–69
3210:(268–269): 110–130.
2936:Lang (1997) pp.91–96
2856:Lang (1988) pp.74–75
2808:. pp. 447–495.
2460:local zeta-functions
2418:is a formulation by
2404:
2375:
2310:
2175:
2168:and solution of the
2092:and, independently,
2070:Sato–Tate conjecture
2064:Sato–Tate conjecture
2058:
2001:
1964:
1953:and in addition the
1919:
1914:height zeta function
1891:Nevanlinna invariant
1884:Nevanlinna invariant
1803:
1768:over a number field
1762:Mordell–Weil theorem
1756:Mordell–Weil theorem
1651:
1521:
1493:
1432:over a finite field
1342:rationality theorems
1310:
1238:Langlands philosophy
1170:
1165:Eisenstein's theorem
1051:on reduction modulo
1008:or, more generally,
959:
870:in his proof of the
852:
847:local zeta-functions
819:
670:local zeta-functions
662:projective varieties
647:
579:Coates–Wiles theorem
573:Coates–Wiles theorem
516:
506:Gross–Zagier theorem
502:Coates–Wiles theorem
465:
401:Fubini–Study metrics
333:Arakelov class group
327:Arakelov class group
289:
89:with coordinates in
79:algebraically closed
44:Diophantine geometry
4281:Continued fractions
4144:Arithmetic dynamics
4139:Arithmetic topology
4133:P-adic Hodge theory
4125:Arithmetic geometry
4058:Iwasawa–Tate theory
3351:1995InMat.120..143M
3108:Arithmetic geometry
3077:1983InMat..73..349F
2969:1977InMat..39..223C
2868:Selecta Mathematica
2749:Arithmetic geometry
2731:Arithmetic dynamics
2726:Arithmetic topology
2561:Contents:
2548:Weil height machine
2542:Weil height machine
2344:) on the number of
1929:An Abelian variety
1750:semiabelian variety
1615:generating function
1611:local zeta-function
1605:Local zeta-function
1571:holomorphic mapping
1386:builds up from the
1330:generating function
1322:Igusa zeta-function
1316:Igusa zeta-function
1047:point may become a
983:class field theory.
956:over number fields.
843:functional equation
510:Kolyvagin's theorem
341:divisor class group
136:Contents:
101:Arithmetic geometry
48:algebraic varieties
4365:Algebraic geometry
4225:Modular arithmetic
4195:Irrational numbers
4129:anabelian geometry
4046:class field theory
3771:Lang (1988) pp.1–9
3510:10.1007/bf01453564
3359:10.1007/BF01241125
3133:Serre, Jean-Pierre
3085:10.1007/BF01388432
2977:10.1007/BF01402975
2890:10.1007/PL00001393
2468:motivic cohomology
2452:Frobenius mappings
2074:Frobenius elements
2031:in a number field
1996:mathematical logic
1957:-torsion has rank
1939:ordinary reduction
1925:Ordinary reduction
1906:projective variety
1702:Mordell conjecture
1696:Mordell conjecture
1646:Riemann hypothesis
1628:, over the finite
1504:Algebraic K-theory
1404:p-adic L-functions
1396:ideal class groups
1346:mathematical logic
1224:L-function, is an
1161:algebraic function
1141:Serre–Tate theorem
1041:singularity theory
1004:all prime numbers
975:-style results on
973:class field theory
872:Mordell conjecture
555:Mordell conjecture
405:Archimedean fields
375:of the field with
34:algebraic geometry
4347:
4346:
4244:Advanced concepts
4200:Algebraic numbers
4185:Composite numbers
3992:978-0-8218-0267-0
3986:, AMS Bookstore,
3960:978-3-540-65399-8
3843:. Vol. 201.
3815:978-0-521-71229-3
3742:978-0-691-15371-1
3651:978-0-387-72487-4
3591:Lang (1997) p.179
3461:Lang (1997) p.212
3422:978-0-521-88268-2
3405:Wüstholz, Gisbert
3048:Lang (1997) p.171
3034:Lang (1997) p.146
3021:978-3-540-37888-4
2815:978-0-521-20833-8
2536:Néron–Tate height
2448:singular homology
2440:cohomology theory
2428:l-adic cohomology
2399:Nevanlinna theory
2231:elliptic surfaces
2110:Chabauty's method
1992:Ax–Kochen theorem
1859:Néron–Tate height
1853:Néron–Tate height
1847:Néron–Tate height
1798:Mordellic variety
1792:Mordellic variety
1412:Bernoulli numbers
1408:Kummer congruence
1369:Galois cohomology
1281:Jean-Pierre Serre
1269:projective spaces
1256:Hilbertian fields
1105:abelian varieties
977:abelian coverings
971:The extension of
950:elliptic surfaces
779:desingularisation
561:'s method for an
547:Chabauty's method
542:Chabauty's method
536:Néron–Tate height
460:l-adic cohomology
444:Artin L-functions
439:Artin L-functions
431:See main article
345:Arakelov divisors
337:ideal class group
81:; over any other
4377:
4337:
4327:
4317:
4316:
4307:
4306:
4297:
4296:
4190:Rational numbers
4021:
4014:
4007:
3998:
3997:
3972:
3943:Neukirch, Jürgen
3938:
3904:
3862:
3827:
3798:Bombieri, Enrico
3790:
3787:
3781:
3778:
3772:
3769:
3763:
3753:
3747:
3746:
3724:
3718:
3717:
3699:
3662:
3656:
3655:
3637:
3631:
3630:
3607:
3601:
3598:
3592:
3589:
3583:
3580:
3574:
3571:
3565:
3554:
3548:
3545:
3539:
3536:
3530:
3529:
3489:
3483:
3480:
3474:
3473:Lang (1988) p.77
3471:
3462:
3459:
3453:
3450:
3444:
3441:
3435:
3434:
3397:
3391:
3390:Lang (1997) p.15
3388:
3382:
3377:
3371:
3370:
3334:
3328:
3327:
3297:
3291:
3290:
3263:
3257:
3254:
3248:
3245:
3236:
3235:
3197:
3191:
3181:
3175:
3174:
3129:
3123:
3121:
3103:
3097:
3096:
3055:
3049:
3046:
3035:
3032:
3026:
3025:
3003:
2997:
2996:
2943:
2937:
2934:
2928:
2925:
2919:
2916:
2910:
2909:
2883:
2863:
2857:
2854:
2848:
2845:
2836:
2835:
2792:
2783:
2782:
2780:
2778:
2773:
2764:
2758:
2746:
2562:
2552:Cartier divisors
2528:Green's function
2480:Weil conjectures
2474:Weil conjectures
2387:Vojta conjecture
2381:Vojta conjecture
2227:algebraic cycles
2223:Hodge conjecture
2181:Tamagawa numbers
2148:subspace theorem
2141:Subspace theorem
2053:Arakelov divisor
2037:fractional ideal
1863:canonical height
1839:algebraic cycles
1706:Faltings theorem
1675:Jacobian variety
1665:, now proved by
1644:, including the
1630:field extensions
1508:abstract algebra
1464:
1449:
1423:Jacobian variety
1361:Pierre de Fermat
1357:Infinite descent
1352:Infinite descent
1279:in the sense of
1262:Hilbertian field
1111:in the field of
1067:having the same
1025:division by zero
938:ring of integers
831:étale cohomology
825:Étale cohomology
814:Weil conjectures
802:Koszul complexes
767:arithmetic genus
678:Waring's problem
666:Fermat varieties
634:étale cohomology
628:, introduced by
626:characteristic p
522:Canonical height
452:étale cohomology
369:fractional ideal
357:Arakelov divisor
351:Arakelov divisor
137:
113:ring of integers
4385:
4384:
4380:
4379:
4378:
4376:
4375:
4374:
4350:
4349:
4348:
4343:
4285:
4251:Quadratic forms
4239:
4214:P-adic analysis
4170:Natural numbers
4148:
4105:Arakelov theory
4030:
4025:
3979:
3977:Further reading
3961:
3951:Springer-Verlag
3927:
3917:Springer-Verlag
3885:
3875:Springer-Verlag
3851:
3816:
3793:
3788:
3784:
3779:
3775:
3770:
3766:
3754:
3750:
3743:
3725:
3721:
3666:Caporaso, Lucia
3663:
3659:
3652:
3638:
3634:
3608:
3604:
3599:
3595:
3590:
3586:
3581:
3577:
3572:
3568:
3555:
3551:
3546:
3542:
3537:
3533:
3492:Batyrev, V.V.;
3490:
3486:
3481:
3477:
3472:
3465:
3460:
3456:
3451:
3447:
3442:
3438:
3423:
3398:
3394:
3389:
3385:
3378:
3374:
3335:
3331:
3316:
3298:
3294:
3267:Raynaud, Michel
3264:
3260:
3255:
3251:
3246:
3239:
3198:
3194:
3182:
3178:
3155:10.2307/1970722
3130:
3126:
3118:
3104:
3100:
3056:
3052:
3047:
3038:
3033:
3029:
3022:
3014:. p. 361.
3012:Springer-Verlag
3004:
3000:
2944:
2940:
2935:
2931:
2926:
2922:
2917:
2913:
2864:
2860:
2855:
2851:
2846:
2839:
2816:
2793:
2786:
2776:
2774:
2771:
2765:
2761:
2747:
2743:
2739:
2717:
2712:
2711:
2710:
2709:
2563:
2543:
2532:Arakelov theory
2524:Cartier divisor
2515:
2503:
2475:
2435:
2434:Weil cohomology
2416:yoga of weights
2411:
2407:
2382:
2378:
2361:
2317:
2313:
2302:
2284:
2278:
2258:
2238:
2215:Tate conjecture
2210:
2209:Tate conjecture
2187:Tamagawa number
2182:
2178:
2170:S-unit equation
2154:absolute values
2142:
2126:Zariski closure
2117:
2105:
2104:Skolem's method
2082:elliptic curves
2065:
2061:
2049:replete divisor
2024:
2012:
2004:
1988:counterexamples
1971:
1967:
1926:
1922:
1885:
1867:abelian variety
1854:
1843:Abelian variety
1830:
1810:
1806:
1793:
1757:
1746:abelian variety
1717:
1697:
1658:
1654:
1606:
1594:
1573:from the whole
1534:(dimension 2),
1532:Enrico Bombieri
1528:
1527:Lang conjecture
1524:
1500:
1496:
1462:
1447:
1406:(with roots in
1394:as a theory of
1380:
1353:
1317:
1313:
1257:
1250:height function
1245:
1244:Height function
1213:
1202:elliptic curves
1182:Hasse principle
1177:
1176:Hasse principle
1173:
1148:
1113:division points
1091:
1065:algebraic curve
991:Fundamental to
988:
968:
962:
954:elliptic curves
942:coordinate ring
933:
902:Flat cohomology
898:
897:Flat cohomology
879:
864:Faltings height
859:
858:Faltings height
855:
826:
822:
794:p-adic analysis
786:
775:geometric genus
731:
726:of dimension 1.
701:
685:
654:
650:
618:
605:with a zero at
597:1 and positive
581:states that an
574:
563:algebraic torus
543:
528:abelian variety
523:
519:
494:elliptic curves
485:
473:
468:
440:
427:
420:Arakelov theory
416:
415:Arakelov theory
397:height function
393:Arakelov height
388:
387:Arakelov height
381:infinite places
361:replete divisor
352:
328:
296:
292:
287:
286:
285:
284:
138:
129:
75:complex numbers
12:
11:
5:
4383:
4373:
4372:
4367:
4362:
4345:
4344:
4342:
4341:
4331:
4321:
4311:
4309:List of topics
4301:
4290:
4287:
4286:
4284:
4283:
4278:
4273:
4268:
4263:
4258:
4253:
4247:
4245:
4241:
4240:
4238:
4237:
4232:
4227:
4222:
4217:
4210:P-adic numbers
4207:
4202:
4197:
4192:
4187:
4182:
4177:
4172:
4167:
4162:
4156:
4154:
4150:
4149:
4147:
4146:
4141:
4136:
4122:
4112:
4098:
4093:
4088:
4083:
4065:
4054:Iwasawa theory
4038:
4036:
4032:
4031:
4024:
4023:
4016:
4009:
4001:
3995:
3994:
3978:
3975:
3974:
3973:
3959:
3939:
3925:
3905:
3883:
3863:
3849:
3831:Hindry, Marc;
3828:
3814:
3792:
3791:
3782:
3773:
3764:
3756:Pierre Deligne
3748:
3741:
3719:
3657:
3650:
3632:
3602:
3593:
3584:
3575:
3566:
3549:
3540:
3531:
3484:
3475:
3463:
3454:
3445:
3436:
3421:
3392:
3383:
3372:
3345:(1): 143–159.
3329:
3314:
3292:
3271:Artin, Michael
3258:
3249:
3237:
3192:
3176:
3149:(3): 492–517.
3124:
3116:
3098:
3071:(3): 349–366.
3059:Faltings, Gerd
3050:
3036:
3027:
3020:
2998:
2963:(3): 223–251.
2938:
2929:
2920:
2911:
2874:(4): 377–398.
2870:. New Series.
2858:
2849:
2837:
2814:
2784:
2759:
2740:
2738:
2735:
2734:
2733:
2728:
2723:
2716:
2713:
2706:
2705:
2700:
2695:
2690:
2685:
2680:
2675:
2670:
2665:
2660:
2655:
2650:
2645:
2640:
2635:
2630:
2625:
2620:
2615:
2610:
2605:
2600:
2595:
2590:
2585:
2580:
2575:
2570:
2564:
2558:
2556:
2555:
2544:
2541:
2539:
2516:
2513:
2511:
2504:
2501:
2499:
2476:
2473:
2471:
2436:
2433:
2431:
2412:
2409:
2406:
2403:
2402:
2383:
2380:
2377:
2374:
2373:
2362:
2359:
2357:
2318:
2315:
2312:
2309:
2308:
2298:
2282:
2274:
2259:
2256:
2254:
2251:good reduction
2247:p-adic numbers
2239:
2236:
2234:
2211:
2208:
2206:
2183:
2180:
2177:
2174:
2173:
2143:
2140:
2138:
2118:
2115:
2113:
2106:
2103:
2101:
2066:
2063:
2060:
2057:
2056:
2025:
2022:
2020:
2017:good reduction
2013:
2006:
2003:
2000:
1999:
1972:
1969:
1966:
1963:
1962:
1947:good reduction
1927:
1924:
1921:
1918:
1917:
1886:
1883:
1881:
1874:quadratic form
1855:
1852:
1850:
1831:
1828:
1826:
1811:
1808:
1805:
1802:
1801:
1794:
1791:
1789:
1758:
1755:
1753:
1718:
1715:
1713:
1698:
1695:
1693:
1667:Michel Raynaud
1659:
1656:
1653:
1650:
1649:
1607:
1604:
1602:
1595:
1592:
1590:
1529:
1526:
1523:
1520:
1519:
1501:
1498:
1495:
1492:
1491:
1442:roots of unity
1400:Galois modules
1384:Iwasawa theory
1381:
1379:Iwasawa theory
1378:
1376:
1354:
1351:
1349:
1326:Jun-ichi Igusa
1318:
1315:
1312:
1309:
1308:
1258:
1255:
1253:
1246:
1243:
1241:
1214:
1211:
1209:
1178:
1175:
1172:
1169:
1168:
1149:
1146:
1144:
1087:
1073:smooth variety
1061:Good reduction
1055:, because the
1049:singular point
1019:; for example
993:local analysis
989:
987:Good reduction
986:
984:
969:
964:
961:
958:
957:
934:
931:
929:
899:
896:
894:
891:Richard Taylor
880:
877:
875:
860:
857:
854:
851:
850:
827:
824:
821:
818:
817:
787:
785:Dwork's method
784:
782:
771:singular curve
732:
729:
727:
697:
686:
683:
681:
658:Diagonal forms
655:
653:Diagonal forms
652:
649:
646:
645:
642:Dwork's method
619:
616:
614:
583:elliptic curve
575:
572:
570:
567:Runge's method
559:Thoralf Skolem
544:
541:
539:
532:quadratic form
524:
521:
518:
515:
514:
513:
486:
483:
481:
478:good reduction
474:
471:
467:
464:
463:
441:
438:
436:
428:
425:
423:
417:
414:
412:
389:
386:
384:
353:
350:
348:
329:
326:
324:
301:abc conjecture
297:
295:abc conjecture
294:
291:
288:
281:
280:
275:
270:
265:
260:
255:
250:
245:
240:
235:
230:
225:
220:
215:
210:
205:
200:
195:
190:
185:
180:
175:
170:
165:
160:
155:
150:
145:
139:
133:
131:
9:
6:
4:
3:
2:
4382:
4371:
4368:
4366:
4363:
4361:
4358:
4357:
4355:
4340:
4336:
4332:
4330:
4326:
4322:
4320:
4312:
4310:
4302:
4300:
4292:
4291:
4288:
4282:
4279:
4277:
4274:
4272:
4269:
4267:
4264:
4262:
4259:
4257:
4256:Modular forms
4254:
4252:
4249:
4248:
4246:
4242:
4236:
4233:
4231:
4228:
4226:
4223:
4221:
4218:
4215:
4211:
4208:
4206:
4203:
4201:
4198:
4196:
4193:
4191:
4188:
4186:
4183:
4181:
4180:Prime numbers
4178:
4176:
4173:
4171:
4168:
4166:
4163:
4161:
4158:
4157:
4155:
4151:
4145:
4142:
4140:
4137:
4134:
4130:
4126:
4123:
4120:
4116:
4113:
4110:
4106:
4102:
4099:
4097:
4094:
4092:
4089:
4087:
4084:
4081:
4077:
4073:
4069:
4066:
4063:
4062:Kummer theory
4059:
4055:
4051:
4047:
4043:
4040:
4039:
4037:
4033:
4029:
4028:Number theory
4022:
4017:
4015:
4010:
4008:
4003:
4002:
3999:
3993:
3989:
3985:
3981:
3980:
3970:
3966:
3962:
3956:
3952:
3948:
3944:
3940:
3936:
3932:
3928:
3926:3-540-61223-8
3922:
3918:
3914:
3910:
3906:
3902:
3898:
3894:
3890:
3886:
3884:0-387-96793-1
3880:
3876:
3872:
3868:
3864:
3860:
3856:
3852:
3850:0-387-98981-1
3846:
3842:
3838:
3834:
3829:
3825:
3821:
3817:
3811:
3807:
3803:
3799:
3795:
3794:
3786:
3777:
3768:
3761:
3757:
3752:
3744:
3738:
3734:
3730:
3723:
3715:
3711:
3707:
3703:
3698:
3693:
3689:
3685:
3684:
3679:
3675:
3671:
3667:
3661:
3653:
3647:
3643:
3636:
3628:
3624:
3620:
3616:
3612:
3606:
3597:
3588:
3579:
3570:
3563:
3559:
3553:
3544:
3535:
3527:
3523:
3519:
3515:
3511:
3507:
3503:
3499:
3495:
3488:
3479:
3470:
3468:
3458:
3449:
3440:
3432:
3428:
3424:
3418:
3415:. p. 3.
3414:
3410:
3406:
3402:
3396:
3387:
3381:
3376:
3368:
3364:
3360:
3356:
3352:
3348:
3344:
3340:
3333:
3325:
3321:
3317:
3315:0-8176-4397-4
3311:
3307:
3303:
3296:
3288:
3284:
3280:
3276:
3272:
3268:
3262:
3253:
3244:
3242:
3233:
3229:
3225:
3221:
3217:
3213:
3209:
3205:
3204:
3196:
3189:
3185:
3180:
3172:
3168:
3164:
3160:
3156:
3152:
3148:
3144:
3143:
3138:
3134:
3128:
3119:
3117:0-387-96311-1
3113:
3109:
3102:
3094:
3090:
3086:
3082:
3078:
3074:
3070:
3066:
3065:
3060:
3054:
3045:
3043:
3041:
3031:
3023:
3017:
3013:
3009:
3002:
2994:
2990:
2986:
2982:
2978:
2974:
2970:
2966:
2962:
2958:
2957:
2952:
2948:
2942:
2933:
2924:
2915:
2907:
2903:
2899:
2895:
2891:
2887:
2882:
2877:
2873:
2869:
2862:
2853:
2844:
2842:
2833:
2829:
2825:
2821:
2817:
2811:
2807:
2803:
2802:
2797:
2791:
2789:
2770:
2763:
2757:
2755:
2750:
2745:
2741:
2732:
2729:
2727:
2724:
2722:
2719:
2718:
2708:
2704:
2701:
2699:
2696:
2694:
2691:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2669:
2666:
2664:
2661:
2659:
2656:
2654:
2651:
2649:
2646:
2644:
2641:
2639:
2636:
2634:
2631:
2629:
2626:
2624:
2621:
2619:
2616:
2614:
2611:
2609:
2606:
2604:
2601:
2599:
2596:
2594:
2591:
2589:
2586:
2584:
2581:
2579:
2576:
2574:
2571:
2569:
2566:
2565:
2553:
2549:
2545:
2537:
2533:
2529:
2525:
2521:
2520:Weil function
2517:
2514:Weil function
2509:
2505:
2497:
2493:
2489:
2485:
2481:
2477:
2469:
2465:
2461:
2457:
2453:
2449:
2445:
2444:finite fields
2441:
2437:
2429:
2425:
2421:
2417:
2413:
2400:
2396:
2392:
2388:
2384:
2371:
2367:
2363:
2355:
2351:
2347:
2343:
2339:
2335:
2331:
2327:
2323:
2319:
2306:
2301:
2297:
2293:
2289:
2285:
2277:
2272:
2268:
2264:
2260:
2252:
2248:
2244:
2240:
2232:
2228:
2224:
2220:
2216:
2212:
2204:
2200:
2196:
2192:
2188:
2184:
2171:
2167:
2163:
2159:
2158:number fields
2155:
2150:
2149:
2144:
2136:
2132:
2127:
2123:
2119:
2111:
2107:
2099:
2095:
2091:
2087:
2086:finite fields
2083:
2079:
2075:
2071:
2067:
2054:
2050:
2046:
2042:
2038:
2034:
2030:
2029:replete ideal
2026:
2023:Replete ideal
2018:
2014:
2010:
1997:
1993:
1989:
1985:
1981:
1977:
1974:The topic of
1973:
1960:
1956:
1952:
1948:
1944:
1940:
1936:
1933:of dimension
1932:
1928:
1915:
1910:
1907:
1904:
1900:
1897:
1896:ample divisor
1893:
1892:
1887:
1879:
1875:
1871:
1868:
1864:
1860:
1856:
1848:
1844:
1840:
1836:
1832:
1824:
1820:
1816:
1812:
1799:
1795:
1787:
1783:
1779:
1775:
1771:
1767:
1763:
1759:
1751:
1747:
1743:
1739:
1735:
1731:
1727:
1723:
1719:
1711:
1707:
1703:
1699:
1691:
1687:
1683:
1679:
1676:
1672:
1668:
1664:
1660:
1647:
1643:
1639:
1635:
1631:
1627:
1624:
1620:
1616:
1612:
1608:
1600:
1596:
1588:
1584:
1580:
1576:
1575:complex plane
1572:
1568:
1565:
1561:
1557:
1553:
1549:
1548:Zariski dense
1545:
1541:
1537:
1533:
1530:
1517:
1513:
1509:
1505:
1502:
1489:
1485:
1484:inverse limit
1481:
1477:
1473:
1469:
1465:
1458:
1454:
1450:
1443:
1439:
1435:
1431:
1427:
1424:
1420:
1418:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1382:
1374:
1370:
1366:
1362:
1358:
1355:
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1319:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1263:
1259:
1251:
1247:
1239:
1235:
1231:
1227:
1226:Euler product
1223:
1219:
1215:
1207:
1203:
1199:
1195:
1191:
1187:
1183:
1179:
1166:
1162:
1158:
1154:
1150:
1142:
1138:
1134:
1130:
1126:
1122:
1118:
1114:
1110:
1106:
1102:
1099:
1095:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1054:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1015:
1011:
1007:
1003:
1002:
998:
994:
990:
982:
978:
974:
970:
967:
955:
951:
947:
946:global fields
943:
939:
935:
927:
923:
919:
915:
914:scheme theory
911:
907:
906:flat topology
903:
900:
892:
888:
884:
881:
873:
869:
865:
861:
848:
844:
840:
839:Michael Artin
836:
832:
828:
815:
811:
807:
803:
799:
795:
791:
790:Bernard Dwork
788:
780:
776:
772:
768:
764:
760:
756:
753:
749:
745:
741:
737:
733:
725:
721:
717:
713:
709:
705:
700:
695:
691:
687:
679:
675:
671:
667:
663:
659:
656:
643:
639:
635:
631:
627:
623:
620:
612:
608:
604:
600:
596:
592:
588:
584:
580:
576:
568:
564:
560:
556:
552:
548:
545:
537:
533:
529:
525:
511:
507:
503:
499:
495:
491:
487:
479:
475:
472:Bad reduction
469:
461:
457:
453:
449:
445:
442:
435:
434:
429:
421:
418:
410:
406:
402:
398:
394:
390:
382:
378:
377:finite places
374:
370:
366:
362:
358:
354:
346:
342:
338:
334:
330:
322:
318:
314:
310:
306:
302:
298:
283:
279:
276:
274:
271:
269:
266:
264:
261:
259:
256:
254:
251:
249:
246:
244:
241:
239:
236:
234:
231:
229:
226:
224:
221:
219:
216:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
179:
176:
174:
171:
169:
166:
164:
161:
159:
156:
154:
151:
149:
146:
144:
141:
140:
130:
127:
125:
120:
118:
117:number theory
114:
110:
106:
102:
98:
96:
92:
88:
84:
80:
76:
72:
68:
67:finite fields
64:
63:number fields
60:
56:
52:
49:
45:
41:
39:
35:
31:
30:number theory
27:
23:
19:
4153:Key concepts
4080:sieve theory
3946:
3912:
3873:. New York:
3870:
3836:
3801:
3785:
3776:
3767:
3759:
3751:
3728:
3722:
3687:
3681:
3674:Mazur, Barry
3660:
3641:
3635:
3618:
3614:
3605:
3596:
3587:
3578:
3569:
3561:
3557:
3552:
3543:
3534:
3501:
3497:
3494:Manin, Yu.I.
3487:
3478:
3457:
3448:
3439:
3408:
3395:
3386:
3375:
3342:
3339:Invent. Math
3338:
3332:
3305:
3302:Schoof, René
3295:
3278:
3261:
3252:
3207:
3201:
3195:
3179:
3146:
3140:
3127:
3107:
3101:
3068:
3062:
3053:
3030:
3007:
3001:
2960:
2954:
2941:
2932:
2923:
2914:
2881:math/9802121
2871:
2867:
2861:
2852:
2800:
2796:Schoof, René
2775:. Retrieved
2762:
2753:
2744:
2707:
2547:
2519:
2424:Hodge theory
2365:
2349:
2345:
2341:
2337:
2333:
2329:
2325:
2299:
2295:
2291:
2287:
2280:
2275:
2270:
2250:
2193:. There the
2146:
2134:
2130:
2121:
2109:
2078:Tate modules
2048:
2044:
2040:
2032:
2028:
2016:
2008:
1980:Brauer group
1958:
1954:
1950:
1942:
1938:
1934:
1930:
1908:
1898:
1889:
1877:
1869:
1835:Néron symbol
1834:
1829:Néron symbol
1822:
1815:naive height
1809:Naive height
1785:
1777:
1773:
1769:
1765:
1689:
1685:
1681:
1677:
1670:
1638:non-singular
1633:
1625:
1623:finite field
1618:
1599:linear torus
1598:
1593:Linear torus
1586:
1582:
1566:
1563:
1559:
1555:
1551:
1546:do not have
1544:general type
1487:
1479:
1475:
1471:
1467:
1460:
1456:
1452:
1445:
1437:
1433:
1429:
1425:
1415:
1373:Selmer group
1337:
1333:
1324:, named for
1300:
1292:
1272:
1264:
1221:
1206:cubic curves
1190:local fields
1186:global field
1109:ramification
1100:
1097:
1093:
1088:
1084:
1080:
1076:
1060:
1052:
1045:non-singular
1036:
1028:
1021:denominators
1016:
1010:prime ideals
1005:
999:
996:
980:
926:gluing axiom
887:Andrew Wiles
762:
758:
754:
751:
747:
743:
739:
735:
715:
711:
707:
703:
698:
693:
689:
637:
606:
595:class number
578:
550:
546:
477:
430:
360:
356:
332:
320:
316:
312:
282:
128:
121:
99:
94:
90:
86:
82:
71:local fields
59:prime fields
54:
53:over fields
50:
42:
17:
15:
4339:Wikiversity
4261:L-functions
3909:Lang, Serge
3867:Lang, Serge
3690:(1): 1–35.
3670:Harris, Joe
3401:Baker, Alan
2508:prime ideal
2185:The direct
2122:special set
2116:Special set
2100:in general.
1941:at a prime
1704:is now the
1550:subsets of
1428:of a curve
1297:meagre sets
1230:L-functions
1198:cubic forms
1121:Néron model
810:rationality
674:Jacobi sums
549:, based on
38:conjectures
22:mathematics
4354:Categories
4220:Arithmetic
3969:0956.11021
3935:0869.11051
3901:0667.14001
3859:0948.11023
3824:1130.11034
3714:0872.14017
3627:0015.38803
3526:0679.14008
3431:1145.11004
3324:1098.14030
3287:0581.14031
3275:Tate, John
3232:0287.43007
3171:0172.46101
3145:. Second.
3137:Tate, John
2993:0359.14009
2947:Coates, J.
2906:1030.11063
2832:1188.11076
2737:References
2484:André Weil
2391:Paul Vojta
2267:C. C. Tsen
2243:Tate curve
2237:Tate curve
2225:, also on
2145:Schmidt's
2090:Mikio Sato
2007:Reduction
1990:; but see
1945:if it has
1772:the group
1728:, Hindry,
1540:Paul Vojta
1536:Serge Lang
1340:. General
1129:Tate curve
1043:enters: a
1039:. However
1014:almost all
833:theory of
761:) and the
603:L-function
69:—and over
3621:: 81–92.
3518:119945673
3504:: 27–43.
3498:Math. Ann
3367:120053132
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