1219:
2208:
314:
Although
Mordell's theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve. The rank of certain elliptic curves can be calculated using numerical methods but (in the current state of knowledge) it is unknown if
1977:
836:
1461:
587:
2288:
310:
If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points. On the other hand, if the rank of the curve is greater than 0, then the curve has an infinite number of rational points.
2203:{\displaystyle \lim _{s\to 1}{\frac {L(A/\mathbb {Q} ,s)}{(s-1)^{r}}}={\frac {\#\mathrm {Sha} (A)\Omega _{A}R_{A}\prod _{p|N}c_{p}}{\#A(\mathbb {Q} )_{\text{tors}}\cdot \#{\hat {A}}(\mathbb {Q} )_{\text{tors}}}}.}
663:
3332:
1083:
At the time of the inception of the conjecture little was known, not even the well-definedness of the left side (referred to as analytic) or the right side (referred to as algebraic) of this equation.
1270:
1347:
1344:-axis is in a logarithmic scale, so the conjecture predicts that the data should tend to a line of slope equal to the rank of the curve, which is 1 in this case -that is, the quotient
2453:
1551:
1487:
940:
891:
2488:
4023:
2413:
101:
498:
1338:
2973:
2355:
1969:
1143:
1910:-functions. Admitting the BSD conjecture, these estimations correspond to information about the rank of families of elliptic curves in question. For example: suppose the
976:
2320:
1182:
2382:
1054:
1013:
2216:
1208:
1109:
284:. This means that for any elliptic curve there is a finite subset of the rational points on the curve, from which all further rational points may be generated.
142:, who developed the conjecture during the first half of the 1960s with the help of machine computation. Only special cases of the conjecture have been proven.
3745:
3022:
4033:
455:
642:
of the L-function is from some points of view a more natural object of study; on occasion, this means that one should consider poles rather than zeroes.)
54:
438:
is conceptually straightforward, as there are only a finite number of possibilities to check. However, for large primes it is computationally intensive.
2213:
All of the terms have the same meaning as for elliptic curves, except that the square of the order of the torsion needs to be replaced by the product
3942:
2509:
94:
75:
3628:
2755:
434:
Finding rational points on a general elliptic curve is a difficult problem. Finding the points on an elliptic curve modulo a given prime
3789:
3906:
3605:
4013:
2923:(2015). "Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0".
87:
831:{\displaystyle {\frac {L^{(r)}(E,1)}{r!}}={\frac {\#\mathrm {Sha} (E)\Omega _{E}R_{E}\prod _{p|N}c_{p}}{(\#E_{\mathrm {tor} })^{2}}}}
3972:
3069:
2664:
2555:
1758:
3870:
638:) was only established for curves with complex multiplication, which were also the main source of numerical examples. (NB that the
3738:
3911:
3896:
3013:
1911:
4028:
4008:
3932:
3836:
1705:-part of the TateâShafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes
1580:
1225:
3731:
3079:
3045:
2541:
2893:
Arthaud, Nicole (1978). "On Birch and
Swinnerton-Dyer's conjecture for elliptic curves with complex multiplication".
1902:
In a different direction, certain analytic methods allow for an estimation of the order of zero in the center of the
946:
2528:
258:
1456:{\displaystyle {\frac {\log \left(\prod _{p\leq X}{\frac {N_{p}}{p}}\right)}{\log C+r\log(\log X))}}\rightarrow 1}
369:
157:
2498:
1573:
1084:
842:
599:
Initially, this was based on somewhat tenuous trends in graphical plots; this induced a measure of skepticism in
281:
254:
35:
1642:
1016:
134:
and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians
3498:
3490:
3439:
3381:
3230:
3088:
639:
300:
184:
2418:
1218:
304:
2493:
The rank-one Birch-Swinnerton-Dyer conjecture for modular elliptic curves and modular abelian varieties of
1500:
4018:
3901:
3822:
3797:
3456:
3353:
2895:
1634:
1466:
902:
858:
420:
64:
3169:
2806:
3937:
3775:
3556:
3174:
2968:
2925:
2816:. No. 32. The Institute for Mathematical Sciences, National University of Singapore. pp. 32â36
2458:
447:
412:
357:
139:
626:
at this point. This was a far-sighted conjecture for the time, given that the analytic continuation of
582:{\displaystyle \prod _{p\leq x}{\frac {N_{p}}{p}}\approx C\log(x)^{r}{\mbox{ as }}x\rightarrow \infty }
1746:
4003:
3865:
3780:
3507:
3448:
3390:
3307:
3239:
3097:
2631:
2576:
2494:
2387:
2291:
1863:
292:
273:
3461:
3330:(1922). "On the rational solutions of the indeterminate equations of the third and fourth degrees".
3165:
3426:
1299:
1077:
145:
The modern formulation of the conjecture relates arithmetic data associated with an elliptic curve
59:
2325:
1952:
1891:
has a rational point of infinite order (thus, under the Birch and
Swinnerton-Dyer conjecture, its
1497:
as in the text. For comparison, a line of slope 1 in (log(log),log)-scale -that is, with equation
1126:
3891:
3845:
3581:
3531:
3486:
3474:
3406:
3263:
3209:
3183:
3153:
3113:
3049:
2989:
2952:
2934:
2718:
2692:
1789:
1720:
1120:
954:
646:
428:
361:
70:
2296:
1723:, which extends results #2 and #3 to all elliptic curves over the rationals, and shows that the
17:
3673:
3957:
3947:
3670:
3624:
3573:
3344:
3065:
3001:
2862:
2751:
2680:
2660:
2551:
2545:
1946:
1616:
606:
This in turn led them to make a general conjecture about the behavior of a curve's L-function
2283:{\displaystyle \#A(\mathbb {Q} )_{\text{tors}}\cdot \#{\hat {A}}(\mathbb {Q} )_{\text{tors}}}
841:
where the quantities on the right-hand side are invariants of the curve, studied by
Cassels,
3875:
3827:
3642:
3565:
3539:
3523:
3515:
3466:
3414:
3398:
3362:
3315:
3275:
3247:
3193:
3145:
3121:
3105:
3031:
2981:
2964:
2944:
2852:
2784:
2743:
2702:
2611:
1809:
1152:
1024:
846:
600:
219:
135:
49:
3720:(September 2016) given during the Clay Research Conference held at the University of Oxford
3638:
3593:
3259:
3205:
2908:
2874:
2737:
2714:
2360:
1032:
991:
3717:
3688:
3634:
3589:
3543:
3418:
3255:
3201:
3125:
3061:
2916:
2904:
2870:
2710:
1057:
645:
The conjecture was subsequently extended to include the prediction of the precise leading
1187:
3511:
3452:
3394:
3311:
3243:
3101:
27:
Clay problem about the set of rational solutions to equations defining an elliptic curve
3711:
3221:
2836:
1903:
1094:
365:
277:
127:
3997:
3806:
3761:
3535:
3478:
3430:
3410:
3327:
3267:
3157:
3117:
2993:
2742:. Graduate Texts in Mathematics. Vol. 201. New York, NY: Springer. p. 462.
2502:
894:
342:
188:
131:
3319:
2722:
3982:
3977:
3705:
3616:
3612:
3601:
3551:
3379:(1991). "The 'main conjectures' of Iwasawa theory for imaginary quadratic fields".
3213:
3083:
3009:
3005:
2956:
2920:
2652:
396:
346:
288:
150:
3036:
3017:
2706:
1111:
at a point where it is not at present known to be defined to the order of a group
2789:
2772:
1741:
proved that the average rank of the
MordellâWeil group of an elliptic curve over
3197:
3133:
2948:
1801:
1557:
The Birch and
Swinnerton-Dyer conjecture has been proved only in special cases:
115:
3723:
2659:. Graduate Texts in Mathematics. Vol. 97 (2nd ed.). Springer-Verlag.
2322:. Elliptic curves as 1-dimensional abelian varieties are their own duals, i.e.
3754:
3693:
3527:
3470:
3376:
3367:
3348:
3225:
3053:
2747:
3701:
3577:
2985:
2866:
2384:
needs to be understood for the pairing between a basis for the free parts of
3678:
3136:(1941). "Die Typen der Multiplikatorenringe elliptischer Funktionenkörper".
1899:). The interest in this statement is that the condition is easily verified.
1777:
There are currently no proofs involving curves with a rank greater than 1.
1792:, this conjecture has multiple consequences, including the following two:
1685:
showed that for elliptic curves defined over an imaginary quadratic field
3585:
3519:
3402:
3251:
3149:
3109:
2615:
1780:
There is extensive numerical evidence for the truth of the conjecture.
451:
2357:, which simplifies the statement of the BSD conjecture. The regulator
603:(Birch's Ph.D. advisor). Over time the numerical evidence stacked up.
473:
on elliptic curves whose rank was known. From these numerical results
3188:
2857:
2697:
261:, which has offered a $ 1,000,000 prize for the first correct proof.
3569:
3018:"On the Modularity of Elliptic Curves over Q: Wild 3-Adic Exercises"
1149:
is known when additionally the analytic rank is at most 1, i.e., if
1145:
the left side is now known to be well-defined and the finiteness of
423:. It was subsequently shown to be true for all elliptic curves over
3138:
2773:"The BirchâSwinnerton-Dyer conjecture and Heegner points: a survey"
1765:, they conclude that a positive proportion of elliptic curves over
126:) describes the set of rational solutions to equations defining an
2939:
1217:
3491:"A classical Diophantine problem and modular forms of weight 3/2"
3172:(2010). "On the BirchâSwinnerton-Dyer quotients modulo squares".
2632:"Numerical evidence for the Birch and Swinnerton-Dyer Conjecture"
2577:"Numerical evidence for the Birch and Swinnerton-Dyer Conjecture"
1721:
all elliptic curves defined over the rational numbers are modular
2807:"Shou-Wu Zhang: Number Theory and Arithmetic Algebraic Geometry"
415:
to the whole complex plane. This conjecture was first proved by
3727:
2599:
1808:
is the area of a right triangle with rational side lengths (a
1091:
This remarkable conjecture relates the behavior of a function
183: = 1. More specifically, it is conjectured that the
3554:(1995). "Modular elliptic curves and Fermat's last theorem".
1914:
and the BSD conjecture, the average rank of curves given by
1804:
integer. Assuming the Birch and
Swinnerton-Dyer conjecture,
3086:(1977). "On the conjecture of Birch and Swinnerton-Dyer".
1663:, 1) is not zero has rank 0, and a modular elliptic curve
1949:
over number fields. A version for abelian varieties over
1712:
1611:) is a finite group. This was extended to the case where
238:= 1 is given by more refined arithmetic data attached to
2683:(2004). "The Average Analytic Rank of Elliptic Curves".
1874:
is a congruent number if and only if the elliptic curve
1641:= 1 then it has a rational point of infinite order; see
3228:(1986). "Heegner points and derivatives of L-series".
564:
2639:
2584:
2461:
2421:
2390:
2363:
2328:
2299:
2219:
1980:
1955:
1812:) if and only if the number of triplets of integers (
1503:
1469:
1350:
1302:
1228:
1190:
1155:
1129:
1097:
1035:
994:
957:
905:
861:
666:
501:
291:
then some point in a finite basis must have infinite
2547:
Visions of Infinity: The Great Mathematical Problems
1754:
982:
multiplied by the number of connected components of
474:
3956:
3920:
3884:
3858:
3815:
3768:
1773:, satisfy the Birch and Swinnerton-Dyer conjecture.
345:from the number of points on the curve modulo each
2482:
2447:
2407:
2376:
2349:
2314:
2282:
2202:
1963:
1945:There is a version of this conjecture for general
1545:
1481:
1455:
1332:
1265:{\displaystyle \prod _{p\leq X}{\frac {N_{p}}{p}}}
1264:
1202:
1176:
1137:
1103:
1048:
1007:
970:
934:
885:
830:
581:
3713:What is the Birch and Swinnerton-Dyer Conjecture?
3623:. American Mathematical Society. pp. 31â44.
2657:Introduction to Elliptic Curves and Modular Forms
1745:is bounded above by 7/6. Combining this with the
3060:. Lecture Notes in Mathematics. Vol. 1716.
2837:"The BlochâKato conjecture on special values of
1982:
2455:relative to the Poincare bundle on the product
1738:
622:= 1, namely that it would have a zero of order
299:basis points with infinite order is called the
287:If the number of rational points on a curve is
1697:-series of the elliptic curve was not zero at
253:The conjecture was chosen as one of the seven
3739:
657: = 1. It is conjecturally given by
95:
8:
3433:(2014). "The Iwasawa main conjectures for GL
3349:"On the parity of ranks of Selmer groups IV"
3023:Journal of the American Mathematical Society
1762:
4024:University of Cambridge Computer Laboratory
2736:Hindry, Marc; Silverman, Joseph H. (2000).
1843:is twice the number of triplets satisfying
1630:
1561:
456:University of Cambridge Computer Laboratory
218:= 1. The first non-zero coefficient in the
3746:
3732:
3724:
3606:"The Birch and Swinnerton-Dyer conjecture"
2845:Journal de théorie des nombres de Bordeaux
1087:expressed this in 1974 in a famous quote.
102:
88:
31:
3460:
3366:
3187:
3035:
2971:(1965). "Notes on Elliptic Curves (II)".
2938:
2856:
2788:
2696:
2469:
2468:
2460:
2438:
2437:
2423:
2422:
2420:
2398:
2397:
2389:
2368:
2362:
2330:
2329:
2327:
2301:
2300:
2298:
2274:
2266:
2265:
2251:
2250:
2238:
2230:
2229:
2218:
2188:
2180:
2179:
2165:
2164:
2152:
2144:
2143:
2126:
2112:
2108:
2098:
2088:
2064:
2058:
2046:
2015:
2014:
2009:
1997:
1985:
1979:
1957:
1956:
1954:
1770:
1648:
1502:
1468:
1387:
1381:
1369:
1351:
1349:
1301:
1288:varies over the first 100000 primes. The
1251:
1245:
1233:
1227:
1189:
1154:
1131:
1130:
1128:
1096:
1040:
1034:
999:
993:
962:
956:
909:
904:
870:
869:
860:
819:
802:
801:
783:
769:
765:
755:
745:
721:
715:
674:
667:
665:
563:
557:
524:
518:
506:
500:
458:to calculate the number of points modulo
3702:The Birch and Swinnerton-Dyer Conjecture
2448:{\displaystyle {\hat {A}}(\mathbb {Q} )}
1123:proved in 2001 for elliptic curves over
130:. It is an open problem in the field of
2521:
2508:Another generalization is given by the
1867:
1769:have analytic rank zero, and hence, by
1750:
1727:-functions of all elliptic curves over
1624:
416:
269:
34:
3689:"Birch and Swinnerton-Dyer Conjecture"
2841:-functions. A survey of known results"
55:NavierâStokes existence and smoothness
2739:Diophantine Geometry: An Introduction
2562:Cassels was highly skeptical at first
1716:
1682:
1651:showed that a modular elliptic curve
1546:{\displaystyle \log y=a+\log(\log x)}
1296:is drawn at distance proportional to
850:
337:can be defined for an elliptic curve
247:
7:
3928:Birch and Swinnerton-Dyer conjecture
3058:Arithmetic Theory of Elliptic Curves
2529:Birch and Swinnerton-Dyer Conjecture
1482:{\displaystyle X\rightarrow \infty }
935:{\displaystyle \#\mathrm {Sha} (E)=}
886:{\displaystyle \#E_{\mathrm {tor} }}
120:Birch and Swinnerton-Dyer conjecture
45:Birch and Swinnerton-Dyer conjecture
2805:Leong, Y. K. (JulyâDecember 2018).
2777:Current Developments in Mathematics
2600:"The arithmetic of elliptic curves"
4034:Unsolved problems in number theory
2483:{\displaystyle A\times {\hat {A}}}
2247:
2220:
2161:
2134:
2085:
2071:
2068:
2065:
2061:
1755:Dokchitser & Dokchitser (2010)
1572:with complex multiplication by an
1476:
959:
916:
913:
910:
906:
877:
874:
871:
862:
809:
806:
803:
794:
742:
728:
725:
722:
718:
576:
475:Birch & Swinnerton-Dyer (1965)
303:of the curve, and is an important
280:on an elliptic curve has a finite
25:
3973:Main conjecture of Iwasawa theory
1759:main conjecture of Iwasawa theory
1115:which is not known to be finite!
315:these methods handle all curves.
76:YangâMills existence and mass gap
124:BirchâSwinnerton-Dyer conjecture
18:Birch-Swinnerton-Dyer conjecture
3320:10.1070/im1989v032n03abeh000779
3298:) for a class of Weil curves".
2408:{\displaystyle A(\mathbb {Q} )}
1870:), is related to the fact that
1689:with complex multiplication by
1675:, 1) has a first-order zero at
1568:is a curve over a number field
1184:vanishes at most to order 1 at
1027:of a basis of rational points,
469:) for a large number of primes
387:) only converges for values of
307:property of an elliptic curve.
3907:RamanujanâPetersson conjecture
3897:Generalized Riemann hypothesis
3793:-functions of Hecke characters
3704:: An Interview with Professor
2474:
2442:
2434:
2428:
2402:
2394:
2335:
2306:
2271:
2262:
2256:
2235:
2226:
2185:
2176:
2170:
2149:
2140:
2113:
2081:
2075:
2043:
2030:
2025:
2003:
1989:
1912:generalized Riemann hypothesis
1553:- is drawn in red in the plot.
1540:
1528:
1473:
1447:
1441:
1438:
1426:
1327:
1324:
1318:
1309:
1171:
1159:
926:
920:
816:
791:
770:
738:
732:
698:
686:
681:
675:
573:
554:
547:
356:-function is analogous to the
1:
3866:Analytic class number formula
3621:The Millennium prize problems
3037:10.1090/S0894-0347-01-00370-8
2707:10.1215/S0012-7094-04-12235-3
2531:at Clay Mathematics Institute
1739:Bhargava & Shankar (2015)
1333:{\displaystyle \log(\log(X))}
391:in the complex plane with Re(
364:that is defined for a binary
3871:Riemannâvon Mangoldt formula
3674:"Swinnerton-Dyer Conjecture"
2790:10.4310/CDM.2013.v2013.n1.a3
2550:, Basic Books, p. 253,
2350:{\displaystyle {\hat {E}}=E}
1964:{\displaystyle \mathbb {Q} }
1292:-axis is in log(log) scale -
1138:{\displaystyle \mathbb {Q} }
368:. It is a special case of a
202:is the order of the zero of
3198:10.4007/annals.2010.172.567
2949:10.4007/annals.2015.181.2.4
971:{\displaystyle \Omega _{E}}
4050:
2499:totally real number fields
2315:{\displaystyle {\hat {A}}}
1763:Skinner & Urban (2014)
1757:and with the proof of the
1637:has a first-order zero at
1210:. Both parts remain open.
427:, as a consequence of the
375:The natural definition of
259:Clay Mathematics Institute
4014:Millennium Prize Problems
3471:10.1007/s00222-013-0448-1
3368:10.1112/S0010437X09003959
2748:10.1007/978-1-4612-1210-2
2685:Duke Mathematical Journal
1862:. This statement, due to
1631:Gross & Zagier (1986)
1574:imaginary quadratic field
1562:Coates & Wiles (1977)
1023:which is defined via the
419:for elliptic curves with
255:Millennium Prize Problems
36:Millennium Prize Problems
3499:Inventiones Mathematicae
3440:Inventiones Mathematicae
3382:Inventiones Mathematicae
3231:Inventiones Mathematicae
3089:Inventiones Mathematicae
2986:10.1515/crll.1965.218.79
1895:-function has a zero at
492:obeys an asymptotic law
3823:Dedekind zeta functions
3278:(1989). "Finiteness of
1068:dividing the conductor
411:) could be extended by
156:to the behavior of the
3354:Compositio Mathematica
3333:Proc. Camb. Phil. Soc.
2969:Swinnerton-Dyer, Peter
2896:Compositio Mathematica
2630:Cremona, John (2011).
2598:Tate, John T. (1974).
2575:Cremona, John (2011).
2484:
2449:
2409:
2378:
2351:
2316:
2284:
2204:
1965:
1635:modular elliptic curve
1554:
1547:
1483:
1457:
1334:
1266:
1204:
1178:
1177:{\displaystyle L(E,s)}
1139:
1117:
1105:
1050:
1009:
978:is the real period of
972:
947:TateâShafarevich group
936:
887:
832:
583:
421:complex multiplication
3943:BlochâKato conjecture
3938:Beilinson conjectures
3921:Algebraic conjectures
3776:Riemann zeta function
3611:. In Carlson, James;
3557:Annals of Mathematics
3175:Annals of Mathematics
2974:J. Reine Angew. Math.
2926:Annals of Mathematics
2835:Kings, Guido (2003).
2510:Bloch-Kato conjecture
2485:
2450:
2410:
2379:
2377:{\displaystyle R_{A}}
2352:
2317:
2285:
2205:
1966:
1548:
1484:
1458:
1335:
1267:
1221:
1205:
1179:
1140:
1106:
1089:
1076:. It can be found by
1051:
1049:{\displaystyle c_{p}}
1010:
1008:{\displaystyle R_{E}}
973:
937:
888:
833:
584:
448:Peter Swinnerton-Dyer
413:analytic continuation
370:HasseâWeil L-function
358:Riemann zeta function
140:Peter Swinnerton-Dyer
4029:Zeta and L-functions
4009:Diophantine geometry
3948:Langlands conjecture
3933:Deligne's conjecture
3885:Analytic conjectures
3427:Skinner, Christopher
3170:Dokchitser, Vladimir
2459:
2419:
2388:
2361:
2326:
2297:
2292:dual abelian variety
2217:
1978:
1953:
1715:, extending work of
1713:Breuil et al. (2001)
1643:GrossâZagier theorem
1603:, 1) is not 0 then
1501:
1467:
1348:
1300:
1226:
1222:A plot, in blue, of
1188:
1153:
1127:
1095:
1033:
992:
955:
945:is the order of the
903:
893:is the order of the
859:
664:
499:
3902:Lindelöf hypothesis
3708:by Agnes F. Beaudry
3512:1983InMat..72..323T
3487:Tunnell, Jerrold B.
3453:2014InMat.195....1S
3395:1991InMat.103...25R
3312:1989IzMat..32..523K
3244:1986InMat..84..225G
3102:1977InMat..39..223C
2771:Zhang, Wei (2013).
1203:{\displaystyle s=1}
446:In the early 1960s
341:by constructing an
65:Poincaré conjecture
60:P versus NP problem
3892:Riemann hypothesis
3816:Algebraic examples
3671:Weisstein, Eric W.
3528:10338.dmlcz/137483
3520:10.1007/BF01389327
3403:10.1007/BF01239508
3252:10.1007/BF01388809
3222:Gross, Benedict H.
3150:10.1007/BF02940746
3110:10.1007/BF01402975
3002:Breuil, Christophe
2681:Heath-Brown, D. R.
2616:10.1007/BF01389745
2480:
2445:
2405:
2374:
2347:
2312:
2280:
2200:
2121:
1996:
1971:is the following:
1961:
1790:Riemann hypothesis
1555:
1543:
1479:
1453:
1380:
1330:
1262:
1244:
1200:
1174:
1135:
1121:modularity theorem
1101:
1046:
1005:
968:
932:
883:
828:
778:
647:Taylor coefficient
579:
568:
517:
429:modularity theorem
362:Dirichlet L-series
122:(often called the
71:Riemann hypothesis
3991:
3990:
3769:Analytic examples
3630:978-0-8218-3679-8
3560:. Second Series.
3276:Kolyvagin, Victor
3048:; Greenberg, R.;
2757:978-0-387-98975-4
2477:
2431:
2338:
2309:
2277:
2259:
2241:
2195:
2191:
2173:
2155:
2104:
2053:
1981:
1947:abelian varieties
1864:Tunnell's theorem
1633:showed that if a
1617:abelian extension
1445:
1396:
1365:
1260:
1229:
1104:{\displaystyle L}
1025:canonical heights
826:
761:
710:
567:
533:
502:
477:conjectured that
399:conjectured that
274:Mordell's theorem
112:
111:
16:(Redirected from
4041:
3912:Artin conjecture
3876:Weil conjectures
3748:
3741:
3734:
3725:
3698:
3684:
3683:
3657:
3655:
3653:
3648:on 29 March 2018
3647:
3641:. Archived from
3610:
3597:
3547:
3495:
3482:
3464:
3422:
3372:
3370:
3361:(6): 1351â1359.
3340:
3323:
3271:
3217:
3191:
3161:
3129:
3075:
3041:
3039:
2997:
2960:
2942:
2917:Bhargava, Manjul
2912:
2879:
2878:
2860:
2858:10.5802/jtnb.396
2832:
2826:
2825:
2823:
2821:
2811:
2802:
2796:
2794:
2792:
2768:
2762:
2761:
2733:
2727:
2726:
2700:
2677:
2671:
2670:
2649:
2643:
2642:
2636:
2627:
2621:
2619:
2595:
2589:
2587:
2581:
2572:
2566:
2564:
2538:
2532:
2526:
2489:
2487:
2486:
2481:
2479:
2478:
2470:
2454:
2452:
2451:
2446:
2441:
2433:
2432:
2424:
2414:
2412:
2411:
2406:
2401:
2383:
2381:
2380:
2375:
2373:
2372:
2356:
2354:
2353:
2348:
2340:
2339:
2331:
2321:
2319:
2318:
2313:
2311:
2310:
2302:
2289:
2287:
2286:
2281:
2279:
2278:
2275:
2269:
2261:
2260:
2252:
2243:
2242:
2239:
2233:
2209:
2207:
2206:
2201:
2196:
2194:
2193:
2192:
2189:
2183:
2175:
2174:
2166:
2157:
2156:
2153:
2147:
2132:
2131:
2130:
2120:
2116:
2103:
2102:
2093:
2092:
2074:
2059:
2054:
2052:
2051:
2050:
2028:
2018:
2013:
1998:
1995:
1970:
1968:
1967:
1962:
1960:
1935:
1932:is smaller than
1931:
1898:
1894:
1890:
1861:
1842:
1823:
1819:
1815:
1810:congruent number
1807:
1799:
1771:Kolyvagin (1989)
1747:p-parity theorem
1649:Kolyvagin (1989)
1552:
1550:
1549:
1544:
1488:
1486:
1485:
1480:
1462:
1460:
1459:
1454:
1446:
1444:
1403:
1402:
1398:
1397:
1392:
1391:
1382:
1379:
1352:
1340:from 0- and the
1339:
1337:
1336:
1331:
1271:
1269:
1268:
1263:
1261:
1256:
1255:
1246:
1243:
1209:
1207:
1206:
1201:
1183:
1181:
1180:
1175:
1148:
1144:
1142:
1141:
1136:
1134:
1114:
1110:
1108:
1107:
1102:
1078:Tate's algorithm
1055:
1053:
1052:
1047:
1045:
1044:
1014:
1012:
1011:
1006:
1004:
1003:
977:
975:
974:
969:
967:
966:
944:
941:
939:
938:
933:
919:
892:
890:
889:
884:
882:
881:
880:
837:
835:
834:
829:
827:
825:
824:
823:
814:
813:
812:
789:
788:
787:
777:
773:
760:
759:
750:
749:
731:
716:
711:
709:
701:
685:
684:
668:
601:J. W. S. Cassels
588:
586:
585:
580:
569:
565:
562:
561:
534:
529:
528:
519:
516:
454:computer at the
295:. The number of
220:Taylor expansion
136:Bryan John Birch
104:
97:
90:
50:Hodge conjecture
32:
21:
4049:
4048:
4044:
4043:
4042:
4040:
4039:
4038:
3994:
3993:
3992:
3987:
3952:
3916:
3880:
3854:
3811:
3764:
3752:
3718:Manjul Bhargava
3687:
3669:
3668:
3665:
3660:
3651:
3649:
3645:
3631:
3608:
3600:
3570:10.2307/2118559
3550:
3493:
3485:
3462:10.1.1.363.2008
3436:
3425:
3375:
3343:
3326:
3274:
3220:
3166:Dokchitser, Tim
3164:
3132:
3078:
3072:
3062:Springer-Verlag
3044:
3014:Taylor, Richard
3000:
2980:(218): 79â108.
2963:
2915:
2892:
2888:
2883:
2882:
2834:
2833:
2829:
2819:
2817:
2809:
2804:
2803:
2799:
2770:
2769:
2765:
2758:
2735:
2734:
2730:
2679:
2678:
2674:
2667:
2651:
2650:
2646:
2634:
2629:
2628:
2624:
2597:
2596:
2592:
2579:
2574:
2573:
2569:
2558:
2540:
2539:
2535:
2527:
2523:
2518:
2457:
2456:
2417:
2416:
2386:
2385:
2364:
2359:
2358:
2324:
2323:
2295:
2294:
2270:
2234:
2215:
2214:
2184:
2148:
2133:
2122:
2094:
2084:
2060:
2042:
2029:
1999:
1976:
1975:
1951:
1950:
1943:
1941:Generalizations
1933:
1915:
1906:of families of
1896:
1892:
1875:
1844:
1825:
1821:
1817:
1813:
1805:
1797:
1786:
1731:are defined at
1679:= 1 has rank 1.
1564:proved that if
1499:
1498:
1465:
1464:
1404:
1383:
1364:
1360:
1353:
1346:
1345:
1298:
1297:
1247:
1224:
1223:
1216:
1186:
1185:
1151:
1150:
1146:
1125:
1124:
1112:
1093:
1092:
1058:Tamagawa number
1036:
1031:
1030:
995:
990:
989:
958:
953:
952:
942:
901:
900:
865:
857:
856:
815:
797:
790:
779:
751:
741:
717:
702:
670:
669:
662:
661:
596:is a constant.
553:
520:
497:
496:
482:
467:
444:
278:rational points
276:: the group of
267:
198:) of points of
108:
28:
23:
22:
15:
12:
11:
5:
4047:
4045:
4037:
4036:
4031:
4026:
4021:
4016:
4011:
4006:
3996:
3995:
3989:
3988:
3986:
3985:
3980:
3975:
3969:
3967:
3954:
3953:
3951:
3950:
3945:
3940:
3935:
3930:
3924:
3922:
3918:
3917:
3915:
3914:
3909:
3904:
3899:
3894:
3888:
3886:
3882:
3881:
3879:
3878:
3873:
3868:
3862:
3860:
3856:
3855:
3853:
3852:
3843:
3834:
3825:
3819:
3817:
3813:
3812:
3810:
3809:
3804:
3795:
3787:
3778:
3772:
3770:
3766:
3765:
3753:
3751:
3750:
3743:
3736:
3728:
3722:
3721:
3709:
3699:
3685:
3664:
3663:External links
3661:
3659:
3658:
3629:
3598:
3564:(3): 443â551.
3548:
3506:(2): 323â334.
3483:
3434:
3423:
3373:
3341:
3328:Mordell, Louis
3324:
3306:(3): 523â541.
3300:Math. USSR Izv
3272:
3238:(2): 225â320.
3226:Zagier, Don B.
3218:
3182:(1): 567â596.
3162:
3144:(1): 197â272.
3130:
3096:(3): 223â251.
3076:
3070:
3042:
3030:(4): 843â939.
2998:
2961:
2933:(2): 587â621.
2913:
2903:(2): 209â232.
2889:
2887:
2884:
2881:
2880:
2851:(1): 179â198.
2827:
2797:
2763:
2756:
2728:
2691:(3): 591â623.
2672:
2665:
2644:
2622:
2590:
2567:
2556:
2533:
2520:
2519:
2517:
2514:
2501:was proved by
2476:
2473:
2467:
2464:
2444:
2440:
2436:
2430:
2427:
2404:
2400:
2396:
2393:
2371:
2367:
2346:
2343:
2337:
2334:
2308:
2305:
2290:involving the
2273:
2268:
2264:
2258:
2255:
2249:
2246:
2237:
2232:
2228:
2225:
2222:
2211:
2210:
2199:
2187:
2182:
2178:
2172:
2169:
2163:
2160:
2151:
2146:
2142:
2139:
2136:
2129:
2125:
2119:
2115:
2111:
2107:
2101:
2097:
2091:
2087:
2083:
2080:
2077:
2073:
2070:
2067:
2063:
2057:
2049:
2045:
2041:
2038:
2035:
2032:
2027:
2024:
2021:
2017:
2012:
2008:
2005:
2002:
1994:
1991:
1988:
1984:
1959:
1942:
1939:
1938:
1937:
1904:critical strip
1900:
1788:Much like the
1785:
1782:
1775:
1774:
1751:NekovĂĄĆ (2009)
1736:
1719:, proved that
1710:
1701:= 1, then the
1680:
1646:
1628:
1625:Arthaud (1978)
1615:is any finite
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1478:
1475:
1472:
1452:
1449:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1401:
1395:
1390:
1386:
1378:
1375:
1372:
1368:
1363:
1359:
1356:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1280: â 5
1272:for the curve
1259:
1254:
1250:
1242:
1239:
1236:
1232:
1215:
1214:Current status
1212:
1199:
1196:
1193:
1173:
1170:
1167:
1164:
1161:
1158:
1133:
1100:
1043:
1039:
1002:
998:
965:
961:
931:
928:
925:
922:
918:
915:
912:
908:
879:
876:
873:
868:
864:
839:
838:
822:
818:
811:
808:
805:
800:
796:
793:
786:
782:
776:
772:
768:
764:
758:
754:
748:
744:
740:
737:
734:
730:
727:
724:
720:
714:
708:
705:
700:
697:
694:
691:
688:
683:
680:
677:
673:
590:
589:
578:
575:
572:
566: as
560:
556:
552:
549:
546:
543:
540:
537:
532:
527:
523:
515:
512:
509:
505:
480:
465:
443:
440:
417:Deuring (1941)
366:quadratic form
270:Mordell (1922)
266:
263:
257:listed by the
128:elliptic curve
110:
109:
107:
106:
99:
92:
84:
81:
80:
79:
78:
73:
68:
62:
57:
52:
47:
39:
38:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4046:
4035:
4032:
4030:
4027:
4025:
4022:
4020:
4019:Number theory
4017:
4015:
4012:
4010:
4007:
4005:
4002:
4001:
3999:
3984:
3981:
3979:
3976:
3974:
3971:
3970:
3968:
3966:
3964:
3960:
3955:
3949:
3946:
3944:
3941:
3939:
3936:
3934:
3931:
3929:
3926:
3925:
3923:
3919:
3913:
3910:
3908:
3905:
3903:
3900:
3898:
3895:
3893:
3890:
3889:
3887:
3883:
3877:
3874:
3872:
3869:
3867:
3864:
3863:
3861:
3857:
3851:
3849:
3844:
3842:
3840:
3835:
3833:
3831:
3826:
3824:
3821:
3820:
3818:
3814:
3808:
3807:Selberg class
3805:
3803:
3801:
3796:
3794:
3792:
3788:
3786:
3784:
3779:
3777:
3774:
3773:
3771:
3767:
3763:
3762:number theory
3759:
3757:
3749:
3744:
3742:
3737:
3735:
3730:
3729:
3726:
3719:
3715:
3714:
3710:
3707:
3703:
3700:
3696:
3695:
3690:
3686:
3681:
3680:
3675:
3672:
3667:
3666:
3662:
3644:
3640:
3636:
3632:
3626:
3622:
3618:
3617:Wiles, Andrew
3614:
3613:Jaffe, Arthur
3607:
3603:
3602:Wiles, Andrew
3599:
3595:
3591:
3587:
3583:
3579:
3575:
3571:
3567:
3563:
3559:
3558:
3553:
3552:Wiles, Andrew
3549:
3545:
3541:
3537:
3533:
3529:
3525:
3521:
3517:
3513:
3509:
3505:
3501:
3500:
3492:
3488:
3484:
3480:
3476:
3472:
3468:
3463:
3458:
3454:
3450:
3446:
3442:
3441:
3432:
3428:
3424:
3420:
3416:
3412:
3408:
3404:
3400:
3396:
3392:
3388:
3384:
3383:
3378:
3374:
3369:
3364:
3360:
3356:
3355:
3350:
3346:
3342:
3338:
3335:
3334:
3329:
3325:
3321:
3317:
3313:
3309:
3305:
3301:
3297:
3293:
3289:
3285:
3281:
3277:
3273:
3269:
3265:
3261:
3257:
3253:
3249:
3245:
3241:
3237:
3233:
3232:
3227:
3223:
3219:
3215:
3211:
3207:
3203:
3199:
3195:
3190:
3185:
3181:
3177:
3176:
3171:
3167:
3163:
3159:
3155:
3151:
3147:
3143:
3139:
3135:
3131:
3127:
3123:
3119:
3115:
3111:
3107:
3103:
3099:
3095:
3091:
3090:
3085:
3081:
3077:
3073:
3071:3-540-66546-3
3067:
3063:
3059:
3055:
3051:
3047:
3043:
3038:
3033:
3029:
3025:
3024:
3019:
3015:
3011:
3010:Diamond, Fred
3007:
3006:Conrad, Brian
3003:
2999:
2995:
2991:
2987:
2983:
2979:
2976:
2975:
2970:
2966:
2962:
2958:
2954:
2950:
2946:
2941:
2936:
2932:
2928:
2927:
2922:
2921:Shankar, Arul
2918:
2914:
2910:
2906:
2902:
2898:
2897:
2891:
2890:
2885:
2876:
2872:
2868:
2864:
2859:
2854:
2850:
2846:
2842:
2840:
2831:
2828:
2815:
2808:
2801:
2798:
2791:
2786:
2782:
2778:
2774:
2767:
2764:
2759:
2753:
2749:
2745:
2741:
2740:
2732:
2729:
2724:
2720:
2716:
2712:
2708:
2704:
2699:
2694:
2690:
2686:
2682:
2676:
2673:
2668:
2666:0-387-97966-2
2662:
2658:
2654:
2653:Koblitz, Neal
2648:
2645:
2640:
2633:
2626:
2623:
2617:
2613:
2609:
2605:
2601:
2594:
2591:
2585:
2578:
2571:
2568:
2563:
2559:
2557:9780465022403
2553:
2549:
2548:
2543:
2537:
2534:
2530:
2525:
2522:
2515:
2513:
2511:
2506:
2504:
2503:Shou-Wu Zhang
2500:
2496:
2491:
2471:
2465:
2462:
2425:
2391:
2369:
2365:
2344:
2341:
2332:
2303:
2293:
2253:
2244:
2223:
2197:
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2010:
2006:
2000:
1992:
1986:
1974:
1973:
1972:
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1918:
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1811:
1803:
1795:
1794:
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1783:
1781:
1778:
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1764:
1761:for GL(2) by
1760:
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652:
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343:Euler product
340:
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316:
312:
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302:
298:
294:
290:
285:
283:
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186:
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132:number theory
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3978:Selmer group
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3799:
3798:Automorphic
3790:
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3755:
3712:
3706:Henri Darmon
3692:
3677:
3650:. Retrieved
3643:the original
3620:
3561:
3555:
3503:
3497:
3447:(1): 1â277.
3444:
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3389:(1): 25â68.
3386:
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3352:
3345:NekovĂĄĆ, Jan
3336:
3331:
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3299:
3295:
3291:
3287:
3283:
3279:
3235:
3229:
3189:math/0610290
3179:
3173:
3141:
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3134:Deuring, Max
3093:
3087:
3057:
3046:Coates, J.H.
3027:
3021:
2977:
2972:
2965:Birch, Bryan
2930:
2924:
2900:
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2844:
2838:
2830:
2818:. Retrieved
2813:
2800:
2780:
2776:
2766:
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2698:math/0305114
2688:
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1868:Tunnell 1983
1858:
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1831:
1827:
1787:
1784:Consequences
1779:
1776:
1766:
1742:
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1728:
1724:
1717:Wiles (1995)
1706:
1702:
1698:
1694:
1690:
1686:
1683:Rubin (1991)
1676:
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478:
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395:) > 3/2.
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151:number field
146:
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4004:Conjectures
3837:HasseâWeil
3716:lecture by
3652:16 December
3431:Urban, Ăric
3377:Rubin, Karl
3050:Ribet, K.A.
2783:: 169â203.
2610:: 179â206.
2604:Invent Math
2497:-type over
1802:square-free
1064:at a prime
847:Shafarevich
297:independent
158:HasseâWeil
116:mathematics
3998:Categories
3965:-functions
3850:-functions
3841:-functions
3832:-functions
3802:-functions
3785:-functions
3781:Dirichlet
3758:-functions
3694:PlanetMath
3544:0515.10013
3419:0737.11030
3339:: 179â192.
3126:0359.14009
3080:Coates, J.
2886:References
2620:, page 198
1800:be an odd
1667:for which
1655:for which
851:Wiles 2006
640:reciprocal
488:with rank
322:-function
265:Background
248:Wiles 2006
3679:MathWorld
3578:0003-486X
3536:121099824
3479:120848645
3457:CiteSeerX
3411:120179735
3268:125716869
3158:124821516
3118:189832636
3084:Wiles, A.
3054:Rubin, K.
2994:122531425
2940:1007.0052
2867:1246-7405
2588:, page 50
2505:in 2001.
2475:^
2466:×
2429:^
2336:^
2307:^
2257:^
2248:#
2245:⋅
2221:#
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960:Ω
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574:→
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536:≈
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504:∏
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431:in 2001.
305:invariant
162:-function
3859:Theorems
3846:Motivic
3619:(eds.).
3604:(2006).
3489:(1983).
3347:(2009).
3056:(1999).
3016:(2001).
2814:Imprints
2723:15216987
2655:(1993).
2544:(2013),
360:and the
289:infinite
67:(solved)
3639:2238272
3594:1333035
3586:2118559
3508:Bibcode
3449:Bibcode
3391:Bibcode
3308:Bibcode
3294:,
3260:0833192
3240:Bibcode
3214:9479748
3206:2680426
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2957:1456959
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442:History
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3646:(PDF)
3609:(PDF)
3582:JSTOR
3532:S2CID
3494:(PDF)
3475:S2CID
3407:S2CID
3264:S2CID
3210:S2CID
3184:arXiv
3154:S2CID
3114:S2CID
2990:S2CID
2953:S2CID
2935:arXiv
2820:5 May
2810:(PDF)
2719:S2CID
2693:arXiv
2635:(PDF)
2580:(PDF)
2516:Notes
2495:GL(2)
943:#Đš(E)
618:) at
347:prime
293:order
282:basis
242:over
234:) at
214:) at
175:) of
3654:2013
3625:ISBN
3574:ISSN
3066:ISBN
2863:ISSN
2822:2019
2781:2013
2752:ISBN
2661:ISBN
2552:ISBN
2415:and
2276:tors
2240:tors
2190:tors
2154:tors
1853:+ 32
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1735:= 1.
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843:Tate
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