3968:
22:
2584:
It is not known whether there is an algorithm to find all the rational points on an arbitrary curve of genus at least 2 over a number field. There is an algorithm that works in some cases. Its termination in general would follow from the conjectures that the Tate–Shafarevich group of an abelian
436:
3860:
2306:
in many examples, but it is not known whether there is an algorithm that always succeeds in computing this group. That would follow from the conjecture that the Tate–Shafarevich group is finite, or from the related
293:
3579:
3639:
2649:.) In dimension 1, this is exactly Faltings's theorem, since a curve is of general type if and only if it has genus at least 2. Lang also made finer conjectures relating finiteness of rational points to
789:
2708:. Not much is known about that case. The strongest known result on the Bombieri–Lang conjecture is Faltings's theorem on subvarieties of abelian varieties (generalizing the case of curves). Namely, if
298:
3269:
606:
2580:
2128:
530:
907:
1763:
1059:
2411:
2047:
1833:
945:
3914:
3621:
3300:
3154:
3058:
2992:
2932:
2880:
2824:
2693:
2517:
2442:
2159:
1874:
1106:
848:
1497:
1428:
1357:
3329:
2188:
1979:
1705:
1393:
730:
219:
3180:
3084:
3018:
2468:
2005:
1950:
1912:
1676:
1458:
664:
3187:
For hypersurfaces of smaller dimension (in terms of their degree), things can be more complicated. For example, the Hasse principle fails for the smooth cubic surface
3341:
has conjectured that the Brauer–Manin obstruction is the only obstruction to the Hasse principle for cubic surfaces. More generally, that should hold for every
1002:
4397:
1432:
is empty, because the square of any real number is nonnegative. On the other hand, in the terminology of algebraic geometry, the algebraic variety
431:{\displaystyle {\begin{aligned}&f_{1}(x_{1},\ldots ,x_{n})=0,\\&\qquad \quad \vdots \\&f_{r}(x_{1},\dots ,x_{n})=0.\end{aligned}}}
2787:. Frédéric Campana conjectured that a variety is potentially dense if and only if it has no rational fibration over a positive-dimensional
3855:{\displaystyle {\big |}|X(k)|-(q^{n-1}+\cdots +q+1){\big |}\leq {\bigg (}{\frac {(d-1)^{n+1}+(-1)^{n+1}(d-1)}{d}}{\bigg )}q^{(n-1)/2}.}
3494:
2058:
It is harder to determine whether a curve of genus 1 has a rational point. The Hasse principle fails in this case: for example, by
741:
4582:
4545:
4512:
4375:
4310:
4218:
2344:
Some of the great achievements of number theory amount to determining the rational points on particular curves. For example,
86:
2308:
58:
2945:
1773:
Much of number theory can be viewed as the study of rational points of algebraic varieties, a convenient setting being
65:
3190:
105:
4166:
2292:
1765:
the two problems are essentially equivalent, since every rational point can be scaled to become an integral point.
4606:
4194:
3991:
3338:
2884:) over a number field has potentially dense rational points. That is known only in special cases, for example if
543:
39:
2895:
One may ask when a variety has a rational point without extending the base field. In the case of a hypersurface
72:
2524:
2349:
254:
43:
2650:
4631:
4504:
3874:
3411:
is a more precise statement that would describe the asymptotics of the number of rational points of bounded
2065:
4440:
3945:
3342:
2586:
54:
483:
2949:
863:
1712:
1018:
4574:
2363:
2193:
2020:
1806:
1201:
918:
3890:
3597:
3276:
3130:
3034:
2968:
2908:
2856:
2800:
2669:
2493:
2418:
2135:
1850:
1082:
824:
1471:
1402:
1156:
1313:
4240:
4238:(2003), "Varieties over a finite field with trivial Chow group of 0-cycles have a rational point",
4197:; Kanevsky, Dimitri; Sansuc, Jean-Jacques (1987), "Arithmétique des surfaces cubiques diagonales",
3309:
2345:
2265:
2192:
but no rational point. The failure of the Hasse principle for curves of genus 1 is measured by the
2168:
1959:
1685:
1610:
1373:
689:
178:
161:
4429:
3163:
3067:
3001:
2451:
1988:
1933:
1895:
1659:
1441:
647:
32:
2059:
857:
2319:
2952:
says that the Hasse principle holds for quadric hypersurfaces over a number field (the case
79:
4592:
4555:
4522:
4489:
4448:
4418:
4385:
4351:
4320:
4279:
4259:
4228:
4187:
1644:
1118:
157:
8:
4562:
4529:
4359:
4235:
4158:
3986:
3981:
3941:
3384:
1782:
1520:
1126:
138:
4263:
3465:
4610:
4567:
4467:
4392:
4249:
3973:
3865:
There are also significant results about when a projective variety over a finite field
3100:
2960:
2889:
1138:
952:
810:
803:
276:
126:
2645:-rational points are contained in a finite union of lower-dimensional subvarieties of
4578:
4541:
4508:
4455:
4371:
4327:
4306:
4214:
3967:
3361:
3096:
1587:
134:
4477:
4406:
4339:
4298:
4267:
4206:
4175:
3428:
3408:
3334:
2833:
2728:
are contained in a finite union of translates of abelian subvarieties contained in
1509:
817:
4588:
4551:
4537:
4518:
4485:
4444:
4414:
4381:
4367:
4347:
4316:
4294:
4286:
4275:
4224:
4202:
4183:
3412:
3353:
2841:
2604:
2269:
2233:
1925:
1778:
1254:
669:
142:
4302:
4343:
3457:
3352:
has "many" rational points whenever it has one. For example, extending work of
2221:
1463:
265:
4481:
4271:
4625:
4410:
3369:
2792:
2634:
2261:
1774:
122:
4437:
Proceedings of the
International Congress of Mathematicians (Helsinki, 1978)
3453:
4496:
3438:
3416:
2832:
has potentially dense rational points, because (more strongly) it becomes
2657:
2618:
2353:
2010:
1954:
has a rational point if and only if it has a point over all completions of
1917:
172:
2844:
over a plane cubic curve). Campana's conjecture would also imply that a
1366:
680:
146:
2736:
contains no translated abelian subvarieties of positive dimension, then
4289:(2003), "Potential density of rational points on algebraic varieties",
4210:
3357:
2845:
2608:
2589:
is the only obstruction to the Hasse principle, in the case of curves.
150:
1924:
to determine whether a given conic has a rational point, based on the
4472:
4425:
4254:
4179:
1921:
156:
Understanding rational points is a central goal of number theory and
21:
2788:
2697:
over a number field does not have
Zariski dense rational points if
4291:
Higher
Dimensional Varieties and Rational Points (Budapest, 2001)
3029:. In higher dimensions, even more is true: every smooth cubic in
2656:
For example, the
Bombieri–Lang conjecture predicts that a smooth
2295:. Computer algebra programs can determine the Mordell–Weil group
1652:
1569:
3574:{\displaystyle {\big |}|X(k)|-(q+1){\big |}\leq 2g{\sqrt {q}}.}
1015:, not all zero, with the understanding that multiplying all of
4193:
3948:
variety, for example every Fano variety, over a finite field
2963:
proved the Hasse principle for smooth cubic hypersurfaces in
1307:
1189:
is an affine or projective variety (viewed as a scheme over
4612:
Local-global principles for rational points and zero-cycles
784:{\displaystyle \left({\frac {a}{c}},{\frac {b}{c}}\right),}
3460:
in any dimension, give strong estimates for the number of
2322:(formerly the Mordell conjecture) says that for any curve
1781:, the behavior of rational points depends strongly on the
1586:; this is the philosophy of identifying a scheme with its
4108:
Colliot-Thélène, Kanevsky & Sansuc (1987), section 7.
1291:, meaning the set of solutions of the equations defining
809:
The concept also makes sense in more general settings. A
4159:"Orbifolds, special varieties and classification theory"
2268:
says that for an elliptic curve (or, more generally, an
118:
In algebraic geometry, a point with rational coordinates
2750:
2529:
145:
is generally understood. If the field is the field of
4293:, Bolyai Society Mathematical Studies, vol. 12,
3893:
3642:
3600:
3497:
3422:
3312:
3279:
3193:
3166:
3133:
3070:
3037:
3004:
2971:
2911:
2859:
2803:
2767:
rational points if there is a finite extension field
2672:
2597:
2527:
2496:
2454:
2421:
2366:
2356:) is equivalent to the statement that for an integer
2171:
2138:
2068:
2023:
1991:
1962:
1936:
1898:
1853:
1809:
1715:
1688:
1662:
1474:
1444:
1405:
1376:
1316:
1085:
1021:
955:
921:
866:
827:
744:
692:
650:
546:
486:
296:
181:
3963:
283:
of a collection of polynomials with coefficients in
2585:variety over a number field is finite and that the
1651:, meaning solutions of polynomial equations in the
46:. Unsourced material may be challenged and removed.
4566:
4199:Diophantine Approximation and Transcendence Theory
3908:
3854:
3615:
3573:
3323:
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3263:
3174:
3148:
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3012:
2986:
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2405:
2360:at least 3, the only rational points of the curve
2182:
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2122:
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1999:
1973:
1944:
1906:
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1757:
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1100:
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996:
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842:
783:
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658:
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524:
430:
213:
4458:(2002), "Unirationality of cubic hypersurfaces",
3814:
3727:
3399:infinite, unirationality implies that the set of
2936:over a number field, there are good results when
1185:. This agrees with the previous definitions when
1065:gives the same point in projective space. Then a
4623:
4460:Journal of the Mathematical Institute of Jussieu
4201:, Lecture Notes in Mathematics, vol. 1290,
4605:
4398:Journal fĂĽr die reine und angewandte Mathematik
4357:
4063:Hindry & Silverman (2000), Theorem F.1.1.1.
4009:Hindry & Silverman (2000), Theorem A.4.3.1.
3264:{\displaystyle 5x^{3}+9y^{3}+10z^{3}+12w^{3}=0}
2602:In higher dimensions, one unifying goal is the
1886:-rational points are completely understood. If
137:is a point whose coordinates belong to a given
4332:Proceedings of the London Mathematical Society
141:. If the field is not mentioned, the field of
4054:Hindry & Silverman (2000), section F.5.2.
4036:Hindry & Silverman (2000), Theorem E.0.1.
3717:
3645:
3547:
3500:
1801:is isomorphic to a conic (degree 2) curve in
1709:For homogeneous polynomial equations such as
149:, a rational point is more commonly called a
4561:
2791:of general type. A known case is that every
1777:projective varieties. For smooth projective
1768:
4326:
601:{\displaystyle f_{j}(a_{1},\dots ,a_{n})=0}
1845:-rational point, then it is isomorphic to
4528:
4471:
4253:
3896:
3603:
3314:
3282:
3168:
3136:
3072:
3040:
3006:
2974:
2914:
2862:
2806:
2675:
2575:{\displaystyle {\tfrac {(n-1)(n-2)}{2}}.}
2499:
2456:
2424:
2173:
2141:
2026:
1993:
1964:
1938:
1916:of rational numbers (or more generally a
1900:
1856:
1812:
1690:
1664:
1590:. Another formulation is that the scheme
1482:
1446:
1413:
1378:
1088:
924:
830:
652:
106:Learn how and when to remove this message
4330:(1983), "Cubic forms in ten variables",
3944:'s theorem that every smooth projective
2326:of genus at least 2 over a number field
679:For example, the rational points of the
672:, one says "rational point" instead of "
4285:
4234:
4156:
3103:says that for any odd positive integer
2472:are the obvious ones: and ; and for
1568:is determined up to isomorphism by the
1110:at which the given polynomials vanish.
4624:
4495:
4454:
4391:
3476:is a smooth projective curve of genus
2712:is a subvariety of an abelian variety
2123:{\displaystyle 3x^{3}+4y^{3}+5z^{3}=0}
1226:-rational points. For a general field
1159:of this morphism, that is, a morphism
4364:Diophantine Geometry: an Introduction
3633:, Deligne's theorem gives the bound:
3403:-rational points is Zariski dense in
3395:-rational point. (In particular, for
2851:(such as a smooth quartic surface in
2755:In the opposite direction, a variety
2245:as the zero element), and so the set
1241:gives only partial information about
4424:
4117:Colliot-Thélène (2015), section 6.1.
4027:Silverman (2009), Conjecture X.4.13.
2592:
2163:has a point over all completions of
525:{\displaystyle (a_{1},\dots ,a_{n})}
44:adding citations to reliable sources
15:
2751:Varieties with many rational points
2314:
2232:has the structure of a commutative
902:{\displaystyle x_{0},\dots ,x_{n}.}
13:
3873:-rational point. For example, the
3423:Counting points over finite fields
2598:Varieties with few rational points
1758:{\displaystyle x^{3}+y^{3}=z^{3},}
1054:{\displaystyle a_{0},\dots ,a_{n}}
856:can be defined by a collection of
735:are the pairs of rational numbers
441:These common zeros are called the
221:has no other rational points than
14:
4643:
4599:
4534:The Arithmetic of Elliptic Curves
4395:(1988), "On nonary cubic forms",
4045:Skorobogatov (2001), section 6,3.
3364:showed: for a cubic hypersurface
2484:(like any smooth curve of degree
2406:{\displaystyle x^{n}+y^{n}=z^{n}}
2042:{\displaystyle \mathbb {Q} _{p}.}
1828:{\displaystyle \mathbb {P} ^{2}.}
1647:traditionally meant the study of
1462:is not empty, because the set of
940:{\displaystyle \mathbb {P} ^{n},}
4072:Campana (2004), Conjecture 9.20.
4018:Silverman (2009), Remark X.4.11.
3966:
3909:{\displaystyle \mathbb {P} ^{n}}
3616:{\displaystyle \mathbb {P} ^{n}}
3348:In some cases, it is known that
3295:{\displaystyle \mathbb {P} ^{3}}
3149:{\displaystyle \mathbb {P} ^{n}}
3053:{\displaystyle \mathbb {P} ^{n}}
2987:{\displaystyle \mathbb {P} ^{n}}
2927:{\displaystyle \mathbb {P} ^{n}}
2875:{\displaystyle \mathbb {P} ^{3}}
2819:{\displaystyle \mathbb {P} ^{3}}
2688:{\displaystyle \mathbb {P} ^{n}}
2512:{\displaystyle \mathbb {P} ^{2}}
2437:{\displaystyle \mathbb {P} ^{2}}
2309:Birch–Swinnerton-Dyer conjecture
2154:{\displaystyle \mathbb {P} ^{2}}
1869:{\displaystyle \mathbb {P} ^{1}}
1101:{\displaystyle \mathbb {P} ^{n}}
843:{\displaystyle \mathbb {P} ^{n}}
20:
4138:
4129:
4120:
4111:
4102:
4093:
4084:
3992:Functor represented by a scheme
3368:of dimension at least 2 over a
3121:, every hypersurface of degree
1492:{\displaystyle X(\mathbb {C} )}
1423:{\displaystyle X(\mathbb {R} )}
1245:. In particular, for a variety
1061:by the same nonzero element of
364:
363:
31:needs additional citations for
4144:Esnault (2003), Corollary 1.3.
4075:
4066:
4057:
4048:
4039:
4030:
4021:
4012:
4003:
3877:implies that any hypersurface
3836:
3824:
3803:
3791:
3776:
3766:
3748:
3735:
3712:
3675:
3668:
3664:
3658:
3651:
3542:
3530:
3523:
3519:
3513:
3506:
2946:Hardy–Littlewood circle method
2836:over some finite extension of
2559:
2547:
2544:
2532:
1793:Every smooth projective curve
1486:
1478:
1417:
1409:
1352:{\displaystyle x^{2}+y^{2}=-1}
988:
956:
589:
557:
519:
487:
415:
383:
344:
312:
255:algebraically closed extension
1:
4505:American Mathematical Society
4167:Annales de l'Institut Fourier
4150:
3324:{\displaystyle \mathbb {Q} ,}
2203:is a curve of genus 1 with a
2183:{\displaystyle \mathbb {Q} ,}
1974:{\displaystyle \mathbb {Q} ,}
1700:{\displaystyle \mathbb {Q} .}
1625:) can be identified with the
1504:More generally, for a scheme
1388:{\displaystyle \mathbb {R} .}
725:{\displaystyle x^{2}+y^{2}=1}
244:
214:{\displaystyle x^{n}+y^{n}=1}
4501:Rational Points on Varieties
4441:Academia Scientiarum Fennica
4430:"The work of Pierre Deligne"
4099:Heath-Brown (1983), Theorem.
4081:Hassett (2003), Theorem 6.4.
3343:rationally connected variety
3175:{\displaystyle \mathbb {Q} }
3079:{\displaystyle \mathbb {Q} }
3013:{\displaystyle \mathbb {Q} }
2463:{\displaystyle \mathbb {Q} }
2000:{\displaystyle \mathbb {R} }
1945:{\displaystyle \mathbb {Q} }
1907:{\displaystyle \mathbb {Q} }
1671:{\displaystyle \mathbb {Z} }
1453:{\displaystyle \mathbb {R} }
1397:Then the set of real points
659:{\displaystyle \mathbb {Q} }
7:
4607:Colliot-Thélène, Jean-Louis
4569:Torsors and Rational Points
4303:10.1007/978-3-662-05123-8_8
4195:Colliot-Thélène, Jean-Louis
4126:Kollár (2002), Theorem 1.1.
3959:
1797:of genus zero over a field
1545:means the set of morphisms
1207:, much of the structure of
10:
4648:
4575:Cambridge University Press
4157:Campana, Frédéric (2004),
3946:rationally chain connected
3584:For a smooth hypersurface
3426:
3339:Jean-Louis Colliot-Thélène
3088:has a rational point when
2053:
1788:
1680:rather than the rationals
1202:algebraically closed field
1173:such that the composition
1004:is given by a sequence of
634:Sometimes, when the field
4482:10.1017/S1474748002000117
4272:10.1007/s00222-002-0261-8
3940:, this also follows from
3875:Chevalley–Warning theorem
1769:Rational points on curves
1211:is determined by its set
4411:10.1515/crll.1988.386.32
4344:10.1112/plms/s3-47.2.225
4241:Inventiones Mathematicae
4135:Katz (1980), section II.
3997:
3464:-points in terms of the
2587:Brauer–Manin obstruction
1268:also determines the set
164:may be restated as: for
4090:Hooley (1988), Theorem.
3444:has only finitely many
2950:Hasse–Minkowski theorem
2651:Kobayashi hyperbolicity
2260:-rational points is an
860:equations in variables
638:is understood, or when
3910:
3856:
3617:
3575:
3488:(a prime power), then
3456:in dimension 1 and by
3448:-rational points. The
3325:
3296:
3265:
3184:has a rational point.
3176:
3150:
3107:, there is an integer
3080:
3054:
3014:
2988:
2928:
2876:
2820:
2689:
2613:that, for any variety
2576:
2513:
2464:
2438:
2407:
2194:Tate–Shafarevich group
2184:
2155:
2124:
2043:
2001:
1975:
1946:
1908:
1870:
1829:
1759:
1701:
1672:
1493:
1454:
1424:
1389:
1353:
1102:
1055:
998:
941:
903:
858:homogeneous polynomial
844:
785:
726:
660:
602:
526:
480:, that is, a sequence
432:
215:
3911:
3857:
3618:
3576:
3345:over a number field.
3326:
3297:
3266:
3177:
3151:
3081:
3055:
3015:
2989:
2944:, often based on the
2940:is much smaller than
2929:
2877:
2821:
2783:are Zariski dense in
2690:
2577:
2514:
2465:
2439:
2408:
2346:Fermat's Last Theorem
2185:
2156:
2125:
2044:
2002:
1976:
1947:
1909:
1871:
1830:
1760:
1702:
1673:
1645:Diophantine equations
1494:
1455:
1425:
1390:
1354:
1200:is a variety over an
1103:
1056:
999:
942:
904:
845:
786:
727:
661:
603:
527:
433:
275:is the set of common
216:
162:Fermat's Last Theorem
4632:Diophantine geometry
4563:Skorobogatov, Alexei
4530:Silverman, Joseph H.
4360:Silverman, Joseph H.
4297:, pp. 223–282,
3918:over a finite field
3891:
3640:
3598:
3495:
3310:
3277:
3191:
3164:
3131:
3068:
3035:
3002:
2969:
2909:
2857:
2828:over a number field
2801:
2779:-rational points of
2759:over a number field
2724:-rational points of
2716:over a number field
2670:
2664:in projective space
2629:-rational points of
2621:over a number field
2525:
2494:
2452:
2419:
2364:
2280:, the abelian group
2276:over a number field
2266:Mordell–Weil theorem
2169:
2136:
2066:
2021:
1989:
1960:
1934:
1896:
1851:
1807:
1713:
1686:
1660:
1598:determines a scheme
1515:and any commutative
1472:
1442:
1403:
1374:
1359:in the affine plane
1314:
1125:. This means that a
1113:More generally, let
1083:
1019:
953:
919:
864:
825:
742:
690:
648:
616:-rational points of
544:
484:
294:
179:
158:Diophantine geometry
40:improve this article
4393:Hooley, Christopher
4264:2003InMat.151..187E
3987:Birational geometry
3982:Arithmetic dynamics
3926:-rational point if
2948:. For example, the
1177:is the identity on
1127:morphism of schemes
4443:, pp. 47–52,
4328:Heath-Brown, D. R.
4211:10.1007/BFb0078705
4205:, pp. 1–108,
3974:Mathematics portal
3906:
3852:
3613:
3571:
3472:. For example, if
3321:
3292:
3261:
3172:
3146:
3111:such that for all
3099:. More generally,
3076:
3050:
3010:
2984:
2961:Christopher Hooley
2924:
2890:elliptic fibration
2872:
2840:(unless it is the
2816:
2685:
2572:
2567:
2509:
2460:
2434:
2403:
2320:Faltings's theorem
2293:finitely generated
2180:
2151:
2120:
2062:, the cubic curve
2039:
1997:
1971:
1942:
1904:
1866:
1825:
1755:
1697:
1668:
1489:
1450:
1420:
1385:
1349:
1098:
1051:
994:
937:
899:
840:
811:projective variety
804:Pythagorean triple
781:
722:
676:-rational point".
656:
598:
522:
428:
426:
211:
127:algebraic geometry
4584:978-0-521-80237-6
4547:978-0-387-96203-0
4514:978-1-4704-3773-2
4377:978-0-387-98981-5
4312:978-3-642-05644-4
4220:978-3-540-18597-0
3956:-rational point.
3869:has at least one
3810:
3566:
3337:and Richard Guy.
3097:Roger Heath-Brown
2765:potentially dense
2593:Higher dimensions
2566:
1588:functor of points
1147:is given. Then a
997:{\displaystyle ,}
771:
758:
620:is often denoted
135:algebraic variety
116:
115:
108:
90:
4639:
4618:
4617:
4595:
4572:
4558:
4536:(2nd ed.),
4525:
4492:
4475:
4451:
4434:
4421:
4388:
4354:
4323:
4287:Hassett, Brendan
4282:
4257:
4231:
4190:
4180:10.5802/aif.2027
4163:
4145:
4142:
4136:
4133:
4127:
4124:
4118:
4115:
4109:
4106:
4100:
4097:
4091:
4088:
4082:
4079:
4073:
4070:
4064:
4061:
4055:
4052:
4046:
4043:
4037:
4034:
4028:
4025:
4019:
4016:
4010:
4007:
3976:
3971:
3970:
3955:
3951:
3939:
3935:
3925:
3921:
3917:
3915:
3913:
3912:
3907:
3905:
3904:
3899:
3884:
3880:
3872:
3868:
3861:
3859:
3858:
3853:
3848:
3847:
3843:
3818:
3817:
3811:
3806:
3790:
3789:
3762:
3761:
3733:
3731:
3730:
3721:
3720:
3693:
3692:
3671:
3654:
3649:
3648:
3632:
3628:
3624:
3622:
3620:
3619:
3614:
3612:
3611:
3606:
3591:
3587:
3580:
3578:
3577:
3572:
3567:
3562:
3551:
3550:
3526:
3509:
3504:
3503:
3487:
3483:
3479:
3475:
3471:
3463:
3450:Weil conjectures
3447:
3443:
3436:
3429:Weil conjectures
3409:Manin conjecture
3406:
3402:
3398:
3394:
3390:
3382:
3378:
3374:
3367:
3351:
3332:
3330:
3328:
3327:
3322:
3317:
3303:
3301:
3299:
3298:
3293:
3291:
3290:
3285:
3270:
3268:
3267:
3262:
3254:
3253:
3238:
3237:
3222:
3221:
3206:
3205:
3183:
3181:
3179:
3178:
3173:
3171:
3157:
3155:
3153:
3152:
3147:
3145:
3144:
3139:
3124:
3120:
3110:
3106:
3094:
3087:
3085:
3083:
3082:
3077:
3075:
3061:
3059:
3057:
3056:
3051:
3049:
3048:
3043:
3028:
3021:
3019:
3017:
3016:
3011:
3009:
2995:
2993:
2991:
2990:
2985:
2983:
2982:
2977:
2958:
2943:
2939:
2935:
2933:
2931:
2930:
2925:
2923:
2922:
2917:
2902:
2898:
2887:
2883:
2881:
2879:
2878:
2873:
2871:
2870:
2865:
2850:
2839:
2831:
2827:
2825:
2823:
2822:
2817:
2815:
2814:
2809:
2786:
2782:
2778:
2774:
2770:
2763:is said to have
2762:
2758:
2746:
2735:
2731:
2727:
2723:
2719:
2715:
2711:
2707:
2696:
2694:
2692:
2691:
2686:
2684:
2683:
2678:
2663:
2648:
2644:
2641:. (That is, the
2640:
2632:
2628:
2624:
2616:
2581:
2579:
2578:
2573:
2568:
2562:
2530:
2520:
2518:
2516:
2515:
2510:
2508:
2507:
2502:
2487:
2483:
2479:
2475:
2471:
2469:
2467:
2466:
2461:
2459:
2445:
2443:
2441:
2440:
2435:
2433:
2432:
2427:
2412:
2410:
2409:
2404:
2402:
2401:
2389:
2388:
2376:
2375:
2359:
2340:
2329:
2325:
2315:Genus at least 2
2305:
2290:
2279:
2275:
2259:
2255:
2244:
2231:
2228:. In this case,
2227:
2219:
2215:
2207:-rational point
2206:
2202:
2191:
2189:
2187:
2186:
2181:
2176:
2162:
2160:
2158:
2157:
2152:
2150:
2149:
2144:
2129:
2127:
2126:
2121:
2113:
2112:
2097:
2096:
2081:
2080:
2050:
2048:
2046:
2045:
2040:
2035:
2034:
2029:
2008:
2006:
2004:
2003:
1998:
1996:
1982:
1980:
1978:
1977:
1972:
1967:
1953:
1951:
1949:
1948:
1943:
1941:
1915:
1913:
1911:
1910:
1905:
1903:
1889:
1885:
1881:
1877:
1875:
1873:
1872:
1867:
1865:
1864:
1859:
1844:
1840:
1836:
1834:
1832:
1831:
1826:
1821:
1820:
1815:
1800:
1796:
1764:
1762:
1761:
1756:
1751:
1750:
1738:
1737:
1725:
1724:
1708:
1706:
1704:
1703:
1698:
1693:
1679:
1677:
1675:
1674:
1669:
1667:
1639:
1635:
1628:
1624:
1620:
1616:
1608:
1604:
1597:
1593:
1585:
1567:
1563:
1555:
1544:
1540:
1536:
1525:
1518:
1514:
1510:commutative ring
1507:
1500:
1498:
1496:
1495:
1490:
1485:
1461:
1459:
1457:
1456:
1451:
1449:
1435:
1431:
1429:
1427:
1426:
1421:
1416:
1396:
1394:
1392:
1391:
1386:
1381:
1364:
1358:
1356:
1355:
1350:
1339:
1338:
1326:
1325:
1305:
1298:
1294:
1290:
1282:
1278:
1267:
1263:
1259:
1252:
1248:
1244:
1240:
1229:
1225:
1221:
1210:
1206:
1199:
1192:
1188:
1184:
1176:
1172:
1154:
1150:
1146:
1124:
1116:
1109:
1107:
1105:
1104:
1099:
1097:
1096:
1091:
1076:
1072:
1068:
1064:
1060:
1058:
1057:
1052:
1050:
1049:
1031:
1030:
1014:
1010:
1003:
1001:
1000:
995:
987:
986:
968:
967:
948:
946:
944:
943:
938:
933:
932:
927:
912:
908:
906:
905:
900:
895:
894:
876:
875:
855:
851:
849:
847:
846:
841:
839:
838:
833:
818:projective space
815:
801:
790:
788:
787:
782:
777:
773:
772:
764:
759:
751:
731:
729:
728:
723:
715:
714:
702:
701:
675:
670:rational numbers
667:
665:
663:
662:
657:
655:
641:
637:
630:
619:
615:
611:
607:
605:
604:
599:
588:
587:
569:
568:
556:
555:
539:
535:
531:
529:
528:
523:
518:
517:
499:
498:
479:
476:that belongs to
475:
471:
463:
455:
448:
437:
435:
434:
429:
427:
414:
413:
395:
394:
382:
381:
371:
359:
343:
342:
324:
323:
311:
310:
300:
286:
282:
274:
270:
263:
259:
252:
240:
236:
232:
228:
224:
220:
218:
217:
212:
204:
203:
191:
190:
170:
143:rational numbers
111:
104:
100:
97:
91:
89:
55:"Rational point"
48:
24:
16:
4647:
4646:
4642:
4641:
4640:
4638:
4637:
4636:
4622:
4621:
4615:
4602:
4585:
4548:
4538:Springer Nature
4515:
4432:
4378:
4368:Springer Nature
4313:
4295:Springer Nature
4236:Esnault, Hélène
4221:
4203:Springer Nature
4161:
4153:
4148:
4143:
4139:
4134:
4130:
4125:
4121:
4116:
4112:
4107:
4103:
4098:
4094:
4089:
4085:
4080:
4076:
4071:
4067:
4062:
4058:
4053:
4049:
4044:
4040:
4035:
4031:
4026:
4022:
4017:
4013:
4008:
4004:
4000:
3972:
3965:
3962:
3953:
3949:
3937:
3927:
3923:
3919:
3900:
3895:
3894:
3892:
3889:
3888:
3886:
3882:
3878:
3870:
3866:
3839:
3823:
3819:
3813:
3812:
3779:
3775:
3751:
3747:
3734:
3732:
3726:
3725:
3716:
3715:
3682:
3678:
3667:
3650:
3644:
3643:
3641:
3638:
3637:
3630:
3626:
3607:
3602:
3601:
3599:
3596:
3595:
3593:
3589:
3585:
3561:
3546:
3545:
3522:
3505:
3499:
3498:
3496:
3493:
3492:
3485:
3481:
3477:
3473:
3469:
3461:
3445:
3441:
3434:
3431:
3425:
3404:
3400:
3396:
3392:
3388:
3380:
3376:
3372:
3365:
3354:Beniamino Segre
3349:
3313:
3311:
3308:
3307:
3305:
3286:
3281:
3280:
3278:
3275:
3274:
3272:
3249:
3245:
3233:
3229:
3217:
3213:
3201:
3197:
3192:
3189:
3188:
3167:
3165:
3162:
3161:
3159:
3140:
3135:
3134:
3132:
3129:
3128:
3126:
3122:
3112:
3108:
3104:
3101:Birch's theorem
3089:
3071:
3069:
3066:
3065:
3063:
3044:
3039:
3038:
3036:
3033:
3032:
3030:
3023:
3005:
3003:
3000:
2999:
2997:
2978:
2973:
2972:
2970:
2967:
2966:
2964:
2953:
2941:
2937:
2918:
2913:
2912:
2910:
2907:
2906:
2904:
2900:
2896:
2885:
2866:
2861:
2860:
2858:
2855:
2854:
2852:
2848:
2837:
2829:
2810:
2805:
2804:
2802:
2799:
2798:
2796:
2784:
2780:
2776:
2772:
2768:
2760:
2756:
2753:
2737:
2733:
2729:
2725:
2721:
2717:
2713:
2709:
2698:
2679:
2674:
2673:
2671:
2668:
2667:
2665:
2661:
2646:
2642:
2638:
2630:
2626:
2622:
2614:
2600:
2595:
2531:
2528:
2526:
2523:
2522:
2503:
2498:
2497:
2495:
2492:
2491:
2489:
2485:
2481:
2480:odd. The curve
2477:
2476:even; and for
2473:
2455:
2453:
2450:
2449:
2447:
2428:
2423:
2422:
2420:
2417:
2416:
2414:
2397:
2393:
2384:
2380:
2371:
2367:
2365:
2362:
2361:
2357:
2331:
2327:
2323:
2317:
2296:
2281:
2277:
2273:
2270:abelian variety
2257:
2246:
2243:
2237:
2234:algebraic group
2229:
2225:
2217:
2214:
2208:
2204:
2200:
2172:
2170:
2167:
2166:
2164:
2145:
2140:
2139:
2137:
2134:
2133:
2131:
2108:
2104:
2092:
2088:
2076:
2072:
2067:
2064:
2063:
2056:
2030:
2025:
2024:
2022:
2019:
2018:
2016:
1992:
1990:
1987:
1986:
1984:
1963:
1961:
1958:
1957:
1955:
1937:
1935:
1932:
1931:
1929:
1928:: a conic over
1926:Hasse principle
1920:), there is an
1899:
1897:
1894:
1893:
1891:
1887:
1883:
1879:
1860:
1855:
1854:
1852:
1849:
1848:
1846:
1842:
1838:
1816:
1811:
1810:
1808:
1805:
1804:
1802:
1798:
1794:
1791:
1771:
1746:
1742:
1733:
1729:
1720:
1716:
1714:
1711:
1710:
1689:
1687:
1684:
1683:
1681:
1663:
1661:
1658:
1657:
1655:
1649:integral points
1637:
1634:
1630:
1626:
1622:
1618:
1614:
1606:
1603:
1599:
1595:
1591:
1572:
1565:
1557:
1546:
1542:
1538:
1527:
1523:
1516:
1512:
1505:
1481:
1473:
1470:
1469:
1467:
1445:
1443:
1440:
1439:
1437:
1433:
1412:
1404:
1401:
1400:
1398:
1377:
1375:
1372:
1371:
1369:
1360:
1334:
1330:
1321:
1317:
1315:
1312:
1311:
1303:
1296:
1295:with values in
1292:
1288:
1285:rational points
1280:
1269:
1265:
1261:
1257:
1255:field extension
1250:
1246:
1242:
1231:
1227:
1223:
1212:
1208:
1204:
1197:
1190:
1186:
1178:
1174:
1160:
1152:
1148:
1129:
1122:
1114:
1092:
1087:
1086:
1084:
1081:
1080:
1078:
1074:
1070:
1066:
1062:
1045:
1041:
1026:
1022:
1020:
1017:
1016:
1012:
1005:
982:
978:
963:
959:
954:
951:
950:
928:
923:
922:
920:
917:
916:
914:
910:
890:
886:
871:
867:
865:
862:
861:
853:
834:
829:
828:
826:
823:
822:
820:
813:
795:
763:
750:
749:
745:
743:
740:
739:
710:
706:
697:
693:
691:
688:
687:
673:
651:
649:
646:
645:
643:
639:
635:
621:
617:
613:
609:
583:
579:
564:
560:
551:
547:
545:
542:
541:
537:
533:
513:
509:
494:
490:
485:
482:
481:
477:
473:
469:
461:
453:
446:
425:
424:
409:
405:
390:
386:
377:
373:
369:
368:
357:
356:
338:
334:
319:
315:
306:
302:
297:
295:
292:
291:
284:
280:
272:
268:
261:
257:
250:
247:
238:
234:
230:
226:
222:
199:
195:
186:
182:
180:
177:
176:
165:
160:. For example,
119:
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
4645:
4635:
4634:
4620:
4619:
4601:
4600:External links
4598:
4597:
4596:
4583:
4559:
4546:
4526:
4513:
4493:
4466:(3): 467–476,
4452:
4422:
4405:(386): 32–98,
4389:
4376:
4358:Hindry, Marc;
4355:
4338:(2): 225–257,
4324:
4311:
4283:
4248:(1): 187–191,
4232:
4219:
4191:
4174:(3): 499–630,
4152:
4149:
4147:
4146:
4137:
4128:
4119:
4110:
4101:
4092:
4083:
4074:
4065:
4056:
4047:
4038:
4029:
4020:
4011:
4001:
3999:
3996:
3995:
3994:
3989:
3984:
3978:
3977:
3961:
3958:
3942:Hélène Esnault
3903:
3898:
3863:
3862:
3851:
3846:
3842:
3838:
3835:
3832:
3829:
3826:
3822:
3816:
3809:
3805:
3802:
3799:
3796:
3793:
3788:
3785:
3782:
3778:
3774:
3771:
3768:
3765:
3760:
3757:
3754:
3750:
3746:
3743:
3740:
3737:
3729:
3724:
3719:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3691:
3688:
3685:
3681:
3677:
3674:
3670:
3666:
3663:
3660:
3657:
3653:
3647:
3610:
3605:
3582:
3581:
3570:
3565:
3560:
3557:
3554:
3549:
3544:
3541:
3538:
3535:
3532:
3529:
3525:
3521:
3518:
3515:
3512:
3508:
3502:
3458:Pierre Deligne
3427:Main article:
3424:
3421:
3320:
3316:
3289:
3284:
3260:
3257:
3252:
3248:
3244:
3241:
3236:
3232:
3228:
3225:
3220:
3216:
3212:
3209:
3204:
3200:
3196:
3170:
3143:
3138:
3074:
3047:
3042:
3008:
2981:
2976:
2921:
2916:
2869:
2864:
2813:
2808:
2775:such that the
2752:
2749:
2682:
2677:
2599:
2596:
2594:
2591:
2571:
2565:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2506:
2501:
2458:
2431:
2426:
2400:
2396:
2392:
2387:
2383:
2379:
2374:
2370:
2350:Richard Taylor
2316:
2313:
2241:
2222:elliptic curve
2212:
2179:
2175:
2148:
2143:
2119:
2116:
2111:
2107:
2103:
2100:
2095:
2091:
2087:
2084:
2079:
2075:
2071:
2055:
2052:
2038:
2033:
2028:
1995:
1983:that is, over
1970:
1966:
1940:
1902:
1863:
1858:
1824:
1819:
1814:
1790:
1787:
1785:of the curve.
1770:
1767:
1754:
1749:
1745:
1741:
1736:
1732:
1728:
1723:
1719:
1696:
1692:
1666:
1643:The theory of
1632:
1601:
1501:is not empty.
1488:
1484:
1480:
1477:
1448:
1419:
1415:
1411:
1408:
1384:
1380:
1348:
1345:
1342:
1337:
1333:
1329:
1324:
1320:
1095:
1090:
1048:
1044:
1040:
1037:
1034:
1029:
1025:
993:
990:
985:
981:
977:
974:
971:
966:
962:
958:
936:
931:
926:
898:
893:
889:
885:
882:
879:
874:
870:
837:
832:
792:
791:
780:
776:
770:
767:
762:
757:
754:
748:
733:
732:
721:
718:
713:
709:
705:
700:
696:
654:
597:
594:
591:
586:
582:
578:
575:
572:
567:
563:
559:
554:
550:
521:
516:
512:
508:
505:
502:
497:
493:
489:
472:is a point of
458:rational point
439:
438:
423:
420:
417:
412:
408:
404:
401:
398:
393:
389:
385:
380:
376:
372:
370:
367:
362:
360:
358:
355:
352:
349:
346:
341:
337:
333:
330:
327:
322:
318:
314:
309:
305:
301:
299:
266:affine variety
249:Given a field
246:
243:
210:
207:
202:
198:
194:
189:
185:
131:rational point
117:
114:
113:
28:
26:
19:
9:
6:
4:
3:
2:
4644:
4633:
4630:
4629:
4627:
4614:
4613:
4608:
4604:
4603:
4594:
4590:
4586:
4580:
4576:
4571:
4570:
4564:
4560:
4557:
4553:
4549:
4543:
4539:
4535:
4531:
4527:
4524:
4520:
4516:
4510:
4506:
4502:
4498:
4497:Poonen, Bjorn
4494:
4491:
4487:
4483:
4479:
4474:
4469:
4465:
4461:
4457:
4456:Kollár, János
4453:
4450:
4446:
4442:
4438:
4431:
4427:
4423:
4420:
4416:
4412:
4408:
4404:
4400:
4399:
4394:
4390:
4387:
4383:
4379:
4373:
4369:
4365:
4361:
4356:
4353:
4349:
4345:
4341:
4337:
4333:
4329:
4325:
4322:
4318:
4314:
4308:
4304:
4300:
4296:
4292:
4288:
4284:
4281:
4277:
4273:
4269:
4265:
4261:
4256:
4251:
4247:
4243:
4242:
4237:
4233:
4230:
4226:
4222:
4216:
4212:
4208:
4204:
4200:
4196:
4192:
4189:
4185:
4181:
4177:
4173:
4169:
4168:
4160:
4155:
4154:
4141:
4132:
4123:
4114:
4105:
4096:
4087:
4078:
4069:
4060:
4051:
4042:
4033:
4024:
4015:
4006:
4002:
3993:
3990:
3988:
3985:
3983:
3980:
3979:
3975:
3969:
3964:
3957:
3947:
3943:
3936:. For smooth
3934:
3930:
3901:
3876:
3849:
3844:
3840:
3833:
3830:
3827:
3820:
3807:
3800:
3797:
3794:
3786:
3783:
3780:
3772:
3769:
3763:
3758:
3755:
3752:
3744:
3741:
3738:
3722:
3709:
3706:
3703:
3700:
3697:
3694:
3689:
3686:
3683:
3679:
3672:
3661:
3655:
3636:
3635:
3634:
3625:over a field
3608:
3568:
3563:
3558:
3555:
3552:
3539:
3536:
3533:
3527:
3516:
3510:
3491:
3490:
3489:
3480:over a field
3467:
3466:Betti numbers
3459:
3455:
3451:
3440:
3430:
3420:
3418:
3414:
3410:
3386:
3371:
3370:perfect field
3363:
3359:
3355:
3346:
3344:
3340:
3336:
3318:
3287:
3258:
3255:
3250:
3246:
3242:
3239:
3234:
3230:
3226:
3223:
3218:
3214:
3210:
3207:
3202:
3198:
3194:
3185:
3141:
3119:
3115:
3102:
3098:
3092:
3045:
3026:
2979:
2962:
2956:
2951:
2947:
2919:
2893:
2891:
2867:
2847:
2843:
2835:
2811:
2794:
2793:cubic surface
2790:
2766:
2748:
2744:
2740:
2705:
2701:
2680:
2659:
2654:
2652:
2636:
2635:Zariski dense
2625:, the set of
2620:
2612:
2610:
2606:
2590:
2588:
2582:
2569:
2563:
2556:
2553:
2550:
2541:
2538:
2535:
2504:
2429:
2398:
2394:
2390:
2385:
2381:
2377:
2372:
2368:
2355:
2351:
2347:
2342:
2338:
2334:
2321:
2312:
2310:
2303:
2299:
2294:
2288:
2284:
2271:
2267:
2263:
2262:abelian group
2253:
2249:
2240:
2235:
2223:
2220:is called an
2211:
2197:
2195:
2177:
2146:
2117:
2114:
2109:
2105:
2101:
2098:
2093:
2089:
2085:
2082:
2077:
2073:
2069:
2061:
2051:
2036:
2031:
2015:
2013:
1968:
1927:
1923:
1919:
1890:is the field
1882:, and so its
1861:
1822:
1817:
1786:
1784:
1780:
1776:
1766:
1752:
1747:
1743:
1739:
1734:
1730:
1726:
1721:
1717:
1694:
1654:
1650:
1646:
1641:
1612:
1589:
1583:
1579:
1575:
1571:
1564:. The scheme
1561:
1554:
1550:
1534:
1530:
1522:
1511:
1502:
1475:
1465:
1406:
1382:
1368:
1363:
1346:
1343:
1340:
1335:
1331:
1327:
1322:
1318:
1309:
1302:Example: Let
1300:
1286:
1276:
1272:
1256:
1249:over a field
1238:
1234:
1219:
1215:
1203:
1194:
1182:
1171:
1167:
1163:
1158:
1144:
1140:
1136:
1132:
1128:
1121:over a field
1120:
1111:
1093:
1046:
1042:
1038:
1035:
1032:
1027:
1023:
1008:
991:
983:
979:
975:
972:
969:
964:
960:
934:
929:
896:
891:
887:
883:
880:
877:
872:
868:
859:
852:over a field
835:
819:
812:
807:
805:
799:
778:
774:
768:
765:
760:
755:
752:
746:
738:
737:
736:
719:
716:
711:
707:
703:
698:
694:
686:
685:
684:
682:
677:
671:
642:is the field
632:
628:
624:
612:. The set of
595:
592:
584:
580:
576:
573:
570:
565:
561:
552:
548:
514:
510:
506:
503:
500:
495:
491:
467:
459:
450:
444:
421:
418:
410:
406:
402:
399:
396:
391:
387:
378:
374:
365:
361:
353:
350:
347:
339:
335:
331:
328:
325:
320:
316:
307:
303:
290:
289:
288:
278:
267:
256:
242:
208:
205:
200:
196:
192:
187:
183:
174:
168:
163:
159:
154:
152:
148:
144:
140:
136:
132:
128:
124:
123:number theory
110:
107:
99:
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
4611:
4568:
4533:
4500:
4473:math/0005146
4463:
4459:
4439:, Helsinki:
4436:
4402:
4396:
4363:
4335:
4331:
4290:
4255:math/0207022
4245:
4239:
4198:
4171:
4165:
4140:
4131:
4122:
4113:
4104:
4095:
4086:
4077:
4068:
4059:
4050:
4041:
4032:
4023:
4014:
4005:
3932:
3928:
3864:
3583:
3452:, proved by
3449:
3439:finite field
3432:
3417:Fano variety
3391:if it has a
3379:not a cone,
3362:János Kollár
3347:
3186:
3117:
3113:
3090:
3024:
2954:
2894:
2764:
2754:
2747:is finite.)
2742:
2738:
2703:
2699:
2658:hypersurface
2655:
2619:general type
2603:
2601:
2583:
2521:) has genus
2354:Andrew Wiles
2343:
2336:
2332:
2318:
2301:
2297:
2286:
2282:
2251:
2247:
2238:
2209:
2198:
2060:Ernst Selmer
2057:
2014:-adic fields
2011:
1918:number field
1792:
1772:
1648:
1642:
1581:
1577:
1573:
1559:
1552:
1548:
1532:
1528:
1503:
1367:real numbers
1361:
1301:
1284:
1274:
1270:
1236:
1232:
1217:
1213:
1195:
1180:
1169:
1165:
1161:
1142:
1134:
1130:
1112:
1011:elements of
1006:
808:
797:
793:
734:
683:of equation
678:
633:
626:
622:
536:elements of
465:
457:
451:
442:
440:
248:
175:of equation
173:Fermat curve
166:
155:
147:real numbers
130:
120:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
4426:Katz, N. M.
3385:unirational
3335:Ian Cassels
2720:, then all
2348:(proved by
2341:is finite.
1629:-points of
1617:-points of
1611:base change
1541:-points of
1230:, however,
681:unit circle
4151:References
3881:of degree
3588:of degree
3454:André Weil
3433:A variety
3358:Yuri Manin
2899:of degree
2846:K3 surface
2660:of degree
2611:conjecture
2330:, the set
1613:, and the
1526:, the set
1151:-point of
1077:-point of
1069:-point of
913:-point of
540:such that
245:Definition
229:, and, if
151:real point
96:April 2019
66:newspapers
4532:(2009) ,
3831:−
3798:−
3770:−
3742:−
3723:≤
3698:⋯
3687:−
3673:−
3629:of order
3553:≤
3528:−
3484:of order
2732:. (So if
2554:−
2539:−
1922:algorithm
1365:over the
1344:−
1036:…
973:…
881:…
574:…
504:…
400:…
366:⋮
329:…
253:, and an
233:is even,
4626:Category
4609:(2015),
4565:(2001),
4499:(2017),
4428:(1980),
4362:(2000),
3960:See also
2834:rational
2789:orbifold
2605:Bombieri
2009:and all
1653:integers
1253:and any
1155:means a
1073:means a
949:written
608:for all
4593:1845760
4556:2514094
4523:3729254
4490:1956057
4449:0562594
4419:0936992
4386:1745599
4352:0703978
4321:2011748
4280:1943746
4260:Bibcode
4229:0927558
4188:2097416
3916:
3887:
3623:
3594:
3437:over a
3407:.) The
3331:
3306:
3302:
3273:
3182:
3160:
3156:
3127:
3086:
3064:
3060:
3031:
3020:
2998:
2994:
2965:
2934:
2905:
2888:has an
2882:
2853:
2826:
2797:
2695:
2666:
2633:is not
2519:
2490:
2470:
2448:
2444:
2415:
2216:, then
2190:
2165:
2161:
2132:
2054:Genus 1
2049:
2017:
2007:
1985:
1981:
1956:
1952:
1930:
1914:
1892:
1876:
1847:
1835:
1803:
1789:Genus 0
1707:
1682:
1678:
1656:
1570:functor
1521:algebra
1508:over a
1499:
1468:
1466:points
1464:complex
1460:
1438:
1430:
1399:
1395:
1370:
1306:be the
1164:: Spec(
1157:section
1108:
1079:
947:
915:
850:
821:
798:a, b, c
666:
644:
239:(0, –1)
235:(–1, 0)
80:scholar
4591:
4581:
4554:
4544:
4521:
4511:
4488:
4447:
4417:
4384:
4374:
4350:
4319:
4309:
4278:
4227:
4217:
4186:
3952:has a
3922:has a
3413:height
2264:. The
2236:(with
1841:has a
1779:curves
1775:smooth
1636:(over
1621:(over
1310:curve
1119:scheme
794:where
443:points
227:(0, 1)
223:(1, 0)
171:, the
169:> 2
133:of an
82:
75:
68:
61:
53:
4616:(PDF)
4468:arXiv
4433:(PDF)
4250:arXiv
4162:(PDF)
3998:Notes
3415:on a
3387:over
3375:with
3304:over
3158:over
3095:, by
3062:over
3022:when
2996:over
2446:over
2224:over
1878:over
1783:genus
1605:over
1594:over
1558:Spec(
1556:over
1547:Spec(
1436:over
1308:conic
1196:When
1179:Spec(
1117:be a
802:is a
468:) of
466:point
277:zeros
271:over
264:, an
139:field
87:JSTOR
73:books
4579:ISBN
4542:ISBN
4509:ISBN
4403:1988
4372:ISBN
4307:ISBN
4215:ISBN
3356:and
2842:cone
2609:Lang
2352:and
1551:) →
1168:) →
1139:Spec
460:(or
237:and
129:, a
125:and
59:news
4478:doi
4407:doi
4340:doi
4299:doi
4268:doi
4246:151
4207:doi
4176:doi
3885:in
3592:in
3468:of
3383:is
3333:by
3271:in
3125:in
3093:≥ 9
3027:≥ 8
2959:).
2957:= 2
2903:in
2795:in
2771:of
2706:+ 2
2637:in
2617:of
2488:in
2413:in
2291:is
2256:of
2199:If
2130:in
1837:If
1640:).
1609:by
1537:of
1287:of
1279:of
1260:of
1222:of
1193:).
1009:+ 1
816:in
668:of
532:of
445:of
279:in
260:of
153:.
121:In
42:by
4628::
4589:MR
4587:,
4577:,
4573:,
4552:MR
4550:,
4540:,
4519:MR
4517:,
4507:,
4503:,
4486:MR
4484:,
4476:,
4462:,
4445:MR
4435:,
4415:MR
4413:,
4401:,
4382:MR
4380:,
4370:,
4366:,
4348:MR
4346:,
4336:47
4334:,
4317:MR
4315:,
4305:,
4276:MR
4274:,
4266:,
4258:,
4244:,
4225:MR
4223:,
4213:,
4184:MR
4182:,
4172:54
4170:,
4164:,
3931:≤
3419:.
3360:,
3243:12
3227:10
3116:≥
2892:.
2702:≥
2653:.
2311:.
2272:)
2196:.
1576:↦
1299:.
1264:,
1175:fa
1137:→
1133::
909:A
806:.
631:.
452:A
449:.
422:0.
287::
241:.
225:,
4480::
4470::
4464:1
4409::
4342::
4301::
4270::
4262::
4252::
4209::
4178::
3954:k
3950:k
3938:X
3933:n
3929:d
3924:k
3920:k
3902:n
3897:P
3883:d
3879:X
3871:k
3867:k
3850:.
3845:2
3841:/
3837:)
3834:1
3828:n
3825:(
3821:q
3815:)
3808:d
3804:)
3801:1
3795:d
3792:(
3787:1
3784:+
3781:n
3777:)
3773:1
3767:(
3764:+
3759:1
3756:+
3753:n
3749:)
3745:1
3739:d
3736:(
3728:(
3718:|
3713:)
3710:1
3707:+
3704:q
3701:+
3695:+
3690:1
3684:n
3680:q
3676:(
3669:|
3665:)
3662:k
3659:(
3656:X
3652:|
3646:|
3631:q
3627:k
3609:n
3604:P
3590:d
3586:X
3569:.
3564:q
3559:g
3556:2
3548:|
3543:)
3540:1
3537:+
3534:q
3531:(
3524:|
3520:)
3517:k
3514:(
3511:X
3507:|
3501:|
3486:q
3482:k
3478:g
3474:X
3470:X
3462:k
3446:k
3442:k
3435:X
3405:X
3401:k
3397:k
3393:k
3389:k
3381:X
3377:X
3373:k
3366:X
3350:X
3319:,
3315:Q
3288:3
3283:P
3259:0
3256:=
3251:3
3247:w
3240:+
3235:3
3231:z
3224:+
3219:3
3215:y
3211:9
3208:+
3203:3
3199:x
3195:5
3169:Q
3142:n
3137:P
3123:d
3118:N
3114:n
3109:N
3105:d
3091:n
3073:Q
3046:n
3041:P
3025:n
3007:Q
2980:n
2975:P
2955:d
2942:n
2938:d
2920:n
2915:P
2901:d
2897:X
2886:X
2868:3
2863:P
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2838:k
2830:k
2812:3
2807:P
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2781:X
2777:E
2773:k
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2761:k
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2745:)
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2741:(
2739:X
2734:X
2730:X
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2722:k
2718:k
2714:A
2710:X
2704:n
2700:d
2681:n
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2662:d
2647:X
2643:k
2639:X
2631:X
2627:k
2623:k
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2607:–
2570:.
2564:2
2560:)
2557:2
2551:n
2548:(
2545:)
2542:1
2536:n
2533:(
2505:2
2500:P
2486:n
2482:X
2478:n
2474:n
2457:Q
2430:2
2425:P
2399:n
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2391:=
2386:n
2382:y
2378:+
2373:n
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2358:n
2339:)
2337:k
2335:(
2333:X
2328:k
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2283:X
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2252:k
2250:(
2248:X
2242:0
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2178:,
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2115:=
2110:3
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2102:5
2099:+
2094:3
2090:y
2086:4
2083:+
2078:3
2074:x
2070:3
2037:.
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2027:Q
2012:p
1994:R
1969:,
1965:Q
1939:Q
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1862:1
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1823:.
1818:2
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1735:3
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1695:.
1691:Q
1665:Z
1638:S
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1531:(
1529:X
1524:S
1519:-
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1341:=
1336:2
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1024:a
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1007:n
992:,
989:]
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961:a
957:[
935:,
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593:=
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585:n
581:a
577:,
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308:1
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285:k
281:K
273:k
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231:n
209:1
206:=
201:n
197:y
193:+
188:n
184:x
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103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
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