344:), the rank of elliptic curves should be 1/2 on average. Even stronger half of all elliptic curves should have rank 0 (meaning that the infinite part of its MordellâWeil group is trivial) and the other half should have rank 1; all remaining ranks consist of a total of 0% of all elliptic curves over
4303:
D. Goldfeld, Conjectures on elliptic curves over quadratic fields, in Number Theory, Carbondale 1979 (Proc. Southern
Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), Lecture Notes in Math. 751, Springer-Verlag, New York, 1979, pp. 108â118.
1674:
1105:
3543:
2790:. The model makes further predictions on upper bounds which are consistent with all currently known lower bounds from example families of elliptic curves in special cases (such as restrictions on the type of torsion groups).
2262:
3169:
181:
81:. 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. If the number of rational points on a curve is
3660:
2682:
1114:, one can obtain a well-defined quantity. Obtaining estimates for this quantity is therefore obtaining upper bounds for the size of the average rank of elliptic curves (provided that an average exists).
2027:
or not. This problem has a long history of opinions of experts in the field about it. Park et al. give an account. A popular article can be found in Quanta magazine. For technical reasons instead of
805:
4475:
M. Bhargava and A. Shankar, Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0, Annals of
Mathematics 181 (2015), 587â621
1875:
3589:
3036:
2788:
3219:
2469:
1760:
1466:
2914:
2189:
265:
3426:
1592:
3261:
1715:
4459:
M. Bhargava and A. Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Annals of
Mathematics 181 (2015), 191â242
1348:
unconditional proof of the boundedness of the average rank of elliptic curves is obtained by using a certain exact sequence involving the
Mordell-Weil group of an elliptic curve
971:
3112:
504:
220:
3081:
2128:
2505:
2306:
2061:
2025:
4494:
Jennifer Park, Bjorn Poonen, John Voight, Melanie
Matchett Wood, A heuristic for boundedness of ranks of elliptic curves. J. Eur. Math. Soc. 21 (2019), no. 9, pp. 2859â2903.
3333:
As of 2024 there is no consensus among the experts if the rank of an elliptic curve should be expected to be bounded uniformly only in terms of its base number field or not.
1926:
1407:
3293:
2372:
1376:
1331:
1257:
934:
838:
689:
436:
402:
3388:
4070:
4048:
2604:
633:
364:
326:
2876:
1541:
3320:
2945:
2822:
2743:. Their model was geared along the known results on distribution of elliptic curves in low ranks and their Tate-Shafarevich groups. It predicts a conjectural bound
2574:
2532:
2399:
2333:
2088:
1200:
1122:
In the past two decades there has been some progress made towards the task of finding upper bounds for the average rank. A. Brumer showed that, conditioned on the
580:
2714:
963:
904:
867:
1230:
1148:
659:
533:
1966:
1946:
1895:
1820:
1800:
1780:
1585:
1561:
1506:
1486:
1427:
1297:
1277:
1172:
600:
553:
287:
52:
3461:
1979:, using a method used by Birch and Swinnerton-Dyer in their original computation of the analytic rank of elliptic curves which led to their famous conjecture.
3038:, therefor a uniform bound for all number fields is impossible. They attribute the failure of their model in this case to the existence of elliptic curves
1259:
without assuming either the BirchâSwinnerton-Dyer conjecture or the
Generalized Riemann Hypothesis. This is achieved by computing the average size of the
2194:
328:
to be bounded or not. It has been shown that there exist curves with rank at least 29, but it is widely believed that such curves are rare. Indeed,
3350:
2797:
a general number field the same model would predict the same bound, which however cannot hold. Park et al. show the existence of number fields
4512:
4336:
N. M. Katz and P. Sarnak, Random
Matrices, Frobenius Eigenvalues, and Monodromy, Amer. Math. Soc. Colloq. Publ. 45, Amer. Math. Soc., 1999.
1123:
303:
3121:
2580:. This was the consensus among the leading experts up to the 2010s. However Mestre in 1982 proved unconditionally that for elliptic curves
117:
3545:, as the group of rational points is no longer finitely generated. In this case the rank will always be infinite. For local fields, the
439:
3594:
2609:
3171:
of all such elliptic curves that do not come from base change of a proper subfield. The model then predicts that the analog bound
3662:
one has an infinite filtration where the successive quotients are finite groups of a well classified structure. But for general
4674:
697:
3679:
2685:
1127:
1825:
3552:
3447:-rational points. This holds much more generally than only for global fields, by a result of NĂ©ron this is true for all
2728:
3330:) it is not clear which modified heuristics predicts correct values let alone which approach would prove such bounds.
2954:
4075:
In 2024, Elkies and
Klagsbrun discovered a curve with a rank of at least 29 (under the GRH, the rank is exactly 29):
1564:
2746:
306:. There is currently no consensus among the experts on whether one should expect the ranks of elliptic curves over
3174:
2542:
2404:
1726:
1432:
78:
2881:
2145:
1669:{\displaystyle 0\rightarrow E(\mathbb {Q} )/pE(\mathbb {Q} )\rightarrow \operatorname {Sel} _{p}(E)\rightarrow }
232:
28:
4669:
3393:
3047:
2724:
94:
3341:
Park et al. argue that their model (suitably modified) should not only apply to number fields, but to general
4638:
3228:
1100:{\displaystyle \lim _{X\rightarrow \infty }{\frac {\sum _{H(E(A,B))\leq X}r(E)}{\sum _{H(E(A,B))\leq X}1}}.}
3674:
A common conjecture is that there is no bound on the largest possible rank for an elliptic curve. In 2006,
2546:
1973:
Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves
1679:
1975:, Bhargava and Shankar computed the 2-Selmer rank of elliptic curves on average. They did so by counting
408:
on the set of rational elliptic curves. To define such a function, recall that a rational elliptic curve
3086:
448:
189:
3053:
2093:
2474:
2282:
2030:
1994:
4424:
M. P. Young, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (2006), 205â250.
3440:
1900:
1381:
86:
66:
4392:
D. R. Heath-Brown, The average analytic rank of elliptic curves, Duke Math. J. 122 (2004), 591â623.
3266:
2349:
1351:
1306:
1235:
909:
813:
664:
411:
377:
3360:
661:
are integers that satisfy this property and define a height function on the set of elliptic curves
4053:
4031:
2587:
605:
347:
309:
4581:
1174:, still assuming the same two conjectures. Finally, Young showed that one can obtain a bound of
223:
374:
In order to obtain a reasonable notion of 'average', one must be able to count elliptic curves
2276:
1151:
4591:
4495:
4476:
4460:
4440:
4432:
4408:
4400:
4376:
4368:
4344:
4320:
4312:
2827:
2732:
2538:
1987:
It is in general an open problem whether the rank of all elliptic curves over a fixed field
1511:
4428:
4396:
4364:
4340:
4308:
3298:
2923:
2800:
2552:
2510:
2377:
2311:
2066:
1177:
1110:
It is not known whether or not this limit exists. However, by replacing the limit with the
558:
4517:
4436:
4425:
4404:
4393:
4372:
4361:
4348:
4337:
4316:
4305:
3666:
there is no universal analog in place of the rank that is an interesting object of study.
2948:
2690:
2340:
1341:
1205:
939:
880:
843:
405:
329:
55:
4280:
3538:{\displaystyle K\in \{\mathbb {R} ,\mathbb {C} ,\mathbb {Q} _{p},\mathbb {F} _{q}((x))\}}
1215:
1133:
638:
512:
4616:
4276:
1951:
1931:
1880:
1805:
1785:
1765:
1570:
1546:
1491:
1471:
1412:
1282:
1262:
1157:
1111:
585:
538:
272:
109:
74:
37:
32:
4540:
4360:
A. Brumer, The average rank of elliptic curves. I, Invent. Math. 109 (1992), 445â472.
4663:
2740:
965:
in hand, one can then define the "average rank" as a limit, provided that it exists:
299:
70:
3455:
3342:
1345:
1300:
1209:
337:
82:
59:
4444:
4412:
4292:
The following table contains some historical data on elliptic curve rank records.
3678:
discovered an elliptic curve with a rank of at least 28. It was shown that under
4480:
4464:
4050:
are known. In particular Elkies gave an infinite family of elliptic curves over
3854:
In 2020, Elkies and Zev
Klagsbrun discovered a curve with a rank of exactly 20:
3675:
2736:
2336:
20:
2257:{\displaystyle (B_{K}<\infty )\Leftrightarrow ({\tilde {B}}_{K}<\infty )}
2723:
In 2016 Park et al. introduced a new random model drawing on analogies to the
3435:
For the question of boundedness of ranks of elliptic curves over some field
2374:, which in particular includes elliptic curves, the existence of a constant
1968:, then one would be able to bound the Mordell-Weil rank on average as well.
333:
3221:
should hold, however Park et al. also show the existence of a number field
4455:
4453:
108:
and
Mordell's theorem can be stated as the existence of an isomorphism of
4595:
4569:
4499:
4380:
4324:
3591:
one can talk about dimensions as manifolds or algebraic varieties, for
3083:, which their model does not take into account. Instead of the family
3164:{\displaystyle {\mathcal {E}}_{K}^{\circ }\subset {\mathcal {E}}_{K}}
4513:"Without a Proof, Mathematicians Wonder How Much Evidence Is Enough"
1212:
showed that the average rank of elliptic curves is bounded above by
176:{\displaystyle E(K)\cong \mathbb {Z} ^{r}\oplus E(K)_{\text{tors}},}
4586:
3443:
over that field that guarantees finite generation for the group
1409:
the Mordell-Weil group of rational points on the elliptic curve
3655:{\displaystyle K\in \{\mathbb {Q} _{p},\mathbb {F} _{q}((x))\}}
16:
Number of independent rational basis points with infinite order
3353:
over a finite field. They also point out that function fields
2677:{\displaystyle {\text{rk}}(E(\mathbb {Q} ))\leq O(\log(N(E)))}
4541:"Chow's K/k-image and K/k-trace, and the Lang-NĂ©ron theorem"
3150:
3128:
3093:
1928:. Therefore, if one can compute or obtain an upper bound on
4028:
Many other examples of (families of) elliptic curves over
4568:
Klagsbrun, Zev; Sherman, Travis; Weigandt, James (2019).
535:. Moreover, this model is unique if for any prime number
2534:, but confers a favorable attitude towards such bounds.
810:
It can then be shown that the number of elliptic curves
800:{\displaystyle H(E)=H(E(A,B))=\max\{4|A|^{3},27B^{2}\}.}
4639:"New records for ranks of elliptic curves with torsion"
298:
The rank is related to several outstanding problems in
4056:
4034:
3597:
3555:
3464:
3396:
3363:
3301:
3269:
3231:
3177:
3124:
3089:
3056:
2957:
2926:
2884:
2830:
2803:
2749:
2693:
2612:
2590:
2555:
2513:
2477:
2407:
2380:
2352:
2314:
2285:
2197:
2148:
2096:
2069:
2033:
1997:
1954:
1934:
1903:
1883:
1828:
1808:
1788:
1768:
1729:
1682:
1595:
1573:
1549:
1514:
1494:
1474:
1435:
1415:
1384:
1354:
1309:
1285:
1265:
1238:
1218:
1180:
1160:
1136:
974:
942:
912:
883:
846:
816:
700:
667:
641:
608:
588:
561:
541:
515:
451:
414:
380:
350:
340:
conjectured that in a suitable asymptotic sense (see
312:
275:
235:
192:
120:
85:
then some point in a finite basis must have infinite
40:
2916:
such that there are infinitely many elliptic curves
3549:-rational points have other useful structures, for
2471:- such a bound does not translate directly to some
1870:{\displaystyle \#\operatorname {Sel} _{p}(E)=p^{s}}
4570:"The Elkies curve has rank 28 subject only to GRH"
4064:
4042:
3654:
3584:{\displaystyle K\in \{\mathbb {R} ,\mathbb {C} \}}
3583:
3537:
3420:
3382:
3314:
3287:
3255:
3213:
3163:
3106:
3075:
3030:
2939:
2908:
2870:
2816:
2782:
2708:
2676:
2598:
2568:
2526:
2499:
2463:
2393:
2366:
2327:
2300:
2256:
2183:
2122:
2082:
2055:
2019:
1960:
1940:
1920:
1889:
1869:
1814:
1794:
1774:
1754:
1709:
1668:
1579:
1555:
1535:
1500:
1480:
1460:
1421:
1401:
1370:
1325:
1291:
1271:
1251:
1224:
1194:
1166:
1142:
1099:
957:
928:
898:
861:
832:
799:
683:
653:
627:
594:
574:
547:
527:
498:
430:
396:
358:
320:
281:
259:
214:
175:
46:
3031:{\displaystyle {\text{rk}}(E(K_{n}))\geq 2^{n}=}
2279:in 1950 held the existence of an absolute bound
976:
746:
4550:. Department of Mathematics Stanford University
2138:that occurs for infinitely many different such
1877:, is an upper bound for the Mordell-Weil rank
1154:showed that one can obtain an upper bound of
3263:While as of 2024 it cannot be ruled out that
2783:{\displaystyle B_{\mathbb {Q} }\in \{20,21\}}
404:somehow. This requires the introduction of a
229:, for which comparatively much is known, and
8:
3649:
3604:
3578:
3562:
3532:
3471:
3208:
3196:
2777:
2765:
1587:. Then we have the following exact sequence
906:the MordellâWeil rank of the elliptic curve
791:
749:
3214:{\displaystyle B_{K}^{\circ }\in \{20,21\}}
2549:expressed their disbelieve in such a bound
2464:{\displaystyle {\text{rk}}(A(K))\leq c_{A}}
1755:{\displaystyle \operatorname {Sel} _{p}(E)}
1461:{\displaystyle \operatorname {Sel} _{p}(E)}
69:(generalized to arbitrary number fields by
2909:{\displaystyle n\in \mathbb {Z} _{\geq 0}}
2184:{\displaystyle B_{K}\leq {\tilde {B}}_{K}}
260:{\displaystyle r\in \mathbb {Z} _{\geq 0}}
4617:"History of elliptic curves rank records"
4585:
4281:"History of elliptic curves rank records"
4271:
4269:
4267:
4265:
4058:
4057:
4055:
4036:
4035:
4033:
3628:
3624:
3623:
3613:
3609:
3608:
3596:
3574:
3573:
3566:
3565:
3554:
3511:
3507:
3506:
3496:
3492:
3491:
3483:
3482:
3475:
3474:
3463:
3421:{\displaystyle B_{K}^{\circ }<\infty }
3406:
3401:
3395:
3368:
3362:
3306:
3300:
3279:
3274:
3268:
3241:
3236:
3230:
3187:
3182:
3176:
3155:
3149:
3148:
3138:
3133:
3127:
3126:
3123:
3118:they suggest to consider only the family
3098:
3092:
3091:
3088:
3061:
3055:
3021:
3020:
3011:
2995:
2976:
2958:
2956:
2931:
2925:
2897:
2893:
2892:
2883:
2862:
2848:
2847:
2838:
2829:
2808:
2802:
2756:
2755:
2754:
2748:
2692:
2628:
2627:
2613:
2611:
2592:
2591:
2589:
2560:
2554:
2518:
2512:
2491:
2480:
2479:
2476:
2454:
2453:
2438:
2408:
2406:
2385:
2379:
2360:
2359:
2351:
2319:
2313:
2292:
2291:
2290:
2284:
2239:
2228:
2227:
2205:
2196:
2175:
2164:
2163:
2153:
2147:
2097:
2095:
2074:
2068:
2047:
2036:
2035:
2032:
2011:
2000:
1999:
1996:
1953:
1933:
1911:
1910:
1902:
1882:
1861:
1836:
1827:
1807:
1787:
1767:
1734:
1728:
1686:
1684:
1681:
1645:
1631:
1630:
1616:
1609:
1608:
1594:
1572:
1548:
1518:
1516:
1513:
1493:
1473:
1440:
1434:
1414:
1392:
1391:
1383:
1364:
1363:
1358:
1353:
1319:
1318:
1313:
1308:
1284:
1264:
1239:
1237:
1217:
1184:
1179:
1159:
1135:
1052:
998:
991:
979:
973:
941:
922:
921:
916:
911:
882:
845:
826:
825:
820:
815:
785:
769:
764:
755:
699:
677:
676:
671:
666:
640:
613:
607:
587:
566:
560:
540:
514:
475:
462:
450:
424:
423:
418:
413:
390:
389:
384:
379:
352:
351:
349:
314:
313:
311:
274:
248:
244:
243:
234:
206:
191:
164:
142:
138:
137:
119:
39:
3451:of finite type over their prime field.
93:basis points with infinite order is the
4261:
1983:Conjectures on the boundedness of ranks
4490:
4488:
3256:{\displaystyle B_{K}^{\circ }\geq 68.}
2716:which itself is unbounded for varying
1802:, defined as the non-negative integer
1130:that one can obtain an upper bound of
7:
3114:of all elliptic curves defined over
2947:(in fact those elliptic curves have
2741:random matrix theory for L-functions
2090:the (potentially infinite) bound on
1710:{\displaystyle {}_{E}\rightarrow 0.}
267:is a nonnegative integer called the
4511:Hartnett, Kevin (31 October 2018).
2268:Elliptic curves over number fields
1202:; still assuming both conjectures.
3415:
3377:
3322:are finite for every number field
3107:{\displaystyle {\mathcal {E}}_{K}}
2339:in 1960 conjectured for a general
2248:
2214:
1829:
986:
499:{\displaystyle E:y^{2}=x^{3}+Ax+B}
215:{\displaystyle E(K)_{\text{tors}}}
77:on an elliptic curve has a finite
14:
3337:Elliptic curves over other fields
3076:{\displaystyle K_{0}\subsetneq K}
2123:{\displaystyle {\text{rk}}(E(K))}
1118:Upper bounds for the average rank
100:In mathematical terms the set of
2576:in various generality regarding
2500:{\displaystyle {\tilde {B}}_{K}}
2301:{\displaystyle B_{\mathbb {Q} }}
2056:{\displaystyle {\tilde {B}}_{K}}
2020:{\displaystyle {\tilde {B}}_{K}}
1124:BirchâSwinnerton-Dyer conjecture
304:BirchâSwinnerton-Dyer conjecture
3345:, in particular including when
1921:{\displaystyle E(\mathbb {Q} )}
1402:{\displaystyle E(\mathbb {Q} )}
1337:Bhargava and Shankar's approach
3646:
3643:
3637:
3634:
3529:
3526:
3520:
3517:
3326:(Park et al. even state it is
3288:{\displaystyle B_{K}^{\circ }}
3025:
3004:
2985:
2982:
2969:
2963:
2852:
2831:
2703:
2697:
2686:conductor of an elliptic curve
2671:
2668:
2665:
2659:
2653:
2644:
2635:
2632:
2624:
2618:
2485:
2458:
2444:
2428:
2425:
2419:
2413:
2367:{\displaystyle K=\mathbb {Q} }
2251:
2233:
2223:
2220:
2217:
2198:
2169:
2117:
2114:
2108:
2102:
2041:
2005:
1915:
1907:
1851:
1845:
1749:
1743:
1701:
1698:
1692:
1663:
1660:
1654:
1638:
1635:
1627:
1613:
1605:
1599:
1530:
1524:
1455:
1449:
1396:
1388:
1371:{\displaystyle E/\mathbb {Q} }
1326:{\displaystyle E/\mathbb {Q} }
1252:{\displaystyle {\frac {7}{6}}}
1128:Generalized Riemann hypothesis
1077:
1074:
1062:
1056:
1043:
1037:
1023:
1020:
1008:
1002:
983:
952:
946:
929:{\displaystyle E/\mathbb {Q} }
893:
887:
856:
850:
833:{\displaystyle E/\mathbb {Q} }
765:
756:
740:
737:
725:
719:
710:
704:
684:{\displaystyle E/\mathbb {Q} }
431:{\displaystyle E/\mathbb {Q} }
397:{\displaystyle E/\mathbb {Q} }
203:
196:
161:
154:
130:
124:
1:
4445:10.1090/S0894-0347-05-00503-5
4413:10.1215/S0012-7094-04-12235-3
3383:{\displaystyle B_{K}=\infty }
4065:{\displaystyle \mathbb {Q} }
4043:{\displaystyle \mathbb {Q} }
2599:{\displaystyle \mathbb {Q} }
628:{\displaystyle p^{6}\nmid B}
359:{\displaystyle \mathbb {Q} }
341:
321:{\displaystyle \mathbb {Q} }
104:-rational points is denoted
4481:10.4007/annals.2015.181.2.4
4465:10.4007/annals.2015.181.1.3
3439:to make sense, one needs a
3042:over general number fields
936:. With the height function
438:can be given in terms of a
4691:
4072:each of rank at least 19.
635:. We can then assume that
54:defined over the field of
3441:Mordell-Weil-type theorem
2731:of number fields and the
2275:According to Park et al.
25:rank of an elliptic curve
3682:it has exactly rank 28:
3357:are known to exist with
2725:Cohen-Lenstra heuristics
1336:
442:, that is, we can write
1991:is bounded by a number
4675:Analytic number theory
4283:. University of Zagreb
4066:
4044:
3656:
3585:
3539:
3422:
3384:
3316:
3289:
3257:
3215:
3165:
3108:
3077:
3032:
2941:
2910:
2872:
2871:{\displaystyle =2^{n}}
2818:
2784:
2710:
2678:
2600:
2570:
2528:
2501:
2465:
2395:
2368:
2329:
2302:
2258:
2185:
2124:
2084:
2057:
2021:
1962:
1942:
1922:
1891:
1871:
1816:
1796:
1776:
1756:
1711:
1670:
1581:
1565:TateâShafarevich group
1557:
1537:
1536:{\displaystyle {}_{E}}
1502:
1482:
1462:
1423:
1403:
1372:
1327:
1293:
1273:
1253:
1226:
1196:
1168:
1150:for the average rank.
1144:
1101:
959:
930:
900:
863:
834:
801:
685:
655:
629:
596:
576:
549:
529:
500:
432:
398:
360:
322:
283:
261:
216:
177:
48:
4067:
4045:
3657:
3586:
3540:
3432:cannot be ruled out.
3423:
3385:
3317:
3315:{\displaystyle B_{K}}
3290:
3258:
3216:
3166:
3109:
3078:
3050:of a proper subfield
3033:
2942:
2940:{\displaystyle K_{n}}
2911:
2873:
2824:of increasing degree
2819:
2817:{\displaystyle K_{n}}
2785:
2711:
2679:
2601:
2571:
2569:{\displaystyle B_{K}}
2529:
2527:{\displaystyle B_{K}}
2502:
2466:
2396:
2394:{\displaystyle c_{A}}
2369:
2330:
2328:{\displaystyle r_{E}}
2303:
2259:
2186:
2125:
2085:
2083:{\displaystyle B_{K}}
2058:
2022:
1963:
1943:
1923:
1892:
1872:
1817:
1797:
1777:
1757:
1712:
1671:
1582:
1558:
1538:
1503:
1483:
1463:
1424:
1404:
1373:
1328:
1294:
1274:
1254:
1227:
1197:
1195:{\displaystyle 25/14}
1169:
1145:
1102:
960:
931:
901:
864:
835:
802:
686:
656:
630:
597:
577:
575:{\displaystyle p^{4}}
550:
530:
501:
433:
399:
361:
323:
284:
262:
217:
178:
49:
4054:
4032:
3595:
3553:
3462:
3394:
3361:
3299:
3267:
3229:
3175:
3122:
3087:
3054:
2955:
2924:
2882:
2828:
2801:
2747:
2739:heuristics based on
2709:{\displaystyle N(E)}
2691:
2610:
2588:
2553:
2511:
2475:
2405:
2378:
2350:
2312:
2283:
2195:
2146:
2094:
2067:
2031:
1995:
1977:binary quartic forms
1952:
1932:
1901:
1881:
1826:
1806:
1786:
1766:
1727:
1719:This shows that the
1680:
1593:
1571:
1547:
1512:
1492:
1472:
1433:
1413:
1382:
1352:
1307:
1283:
1263:
1236:
1216:
1178:
1158:
1134:
972:
958:{\displaystyle H(E)}
940:
910:
899:{\displaystyle r(E)}
881:
862:{\displaystyle H(E)}
844:
840:with bounded height
814:
698:
665:
639:
606:
586:
559:
539:
513:
449:
412:
378:
348:
310:
273:
233:
190:
118:
73:) says the group of
58:or more generally a
38:
3670:Largest known ranks
3411:
3284:
3246:
3192:
3143:
2130:of elliptic curves
1303:of elliptic curves
1225:{\displaystyle 1.5}
1143:{\displaystyle 2.3}
654:{\displaystyle A,B}
528:{\displaystyle A,B}
302:, most notably the
4381:10.1007/BF01232033
4325:10.1007/BFb0062705
4062:
4040:
3652:
3581:
3535:
3418:
3397:
3380:
3312:
3285:
3270:
3253:
3232:
3211:
3178:
3161:
3125:
3104:
3073:
3028:
2937:
2906:
2868:
2814:
2780:
2706:
2674:
2596:
2566:
2524:
2497:
2461:
2391:
2364:
2325:
2298:
2254:
2181:
2120:
2080:
2053:
2017:
1958:
1938:
1918:
1887:
1867:
1812:
1792:
1772:
1762:, also called the
1752:
1707:
1666:
1577:
1553:
1533:
1498:
1478:
1458:
1419:
1399:
1368:
1323:
1289:
1269:
1249:
1222:
1192:
1164:
1140:
1097:
1087:
1033:
990:
955:
926:
896:
859:
830:
797:
681:
651:
625:
592:
572:
545:
525:
509:for some integers
496:
428:
394:
356:
318:
279:
257:
212:
173:
44:
4643:NMBRTHRY Archives
4615:Dujella, Andrej.
4596:10.1090/mcom/3348
2961:
2616:
2606:there is a bound
2488:
2411:
2236:
2172:
2100:
2044:
2008:
1961:{\displaystyle E}
1941:{\displaystyle p}
1890:{\displaystyle r}
1815:{\displaystyle s}
1795:{\displaystyle E}
1775:{\displaystyle p}
1580:{\displaystyle E}
1556:{\displaystyle p}
1501:{\displaystyle E}
1488:-Selmer group of
1481:{\displaystyle p}
1422:{\displaystyle E}
1292:{\displaystyle 3}
1272:{\displaystyle 2}
1247:
1167:{\displaystyle 2}
1092:
1048:
994:
975:
595:{\displaystyle A}
548:{\displaystyle p}
282:{\displaystyle E}
224:torsion group of
209:
167:
67:Mordell's theorem
47:{\displaystyle E}
4682:
4654:
4653:
4651:
4649:
4634:
4628:
4627:
4625:
4623:
4612:
4606:
4605:
4603:
4602:
4589:
4580:(316): 837â846.
4565:
4559:
4558:
4556:
4555:
4545:
4536:
4530:
4529:
4527:
4525:
4508:
4502:
4500:10.4171/JEMS/893
4492:
4483:
4473:
4467:
4457:
4448:
4422:
4416:
4390:
4384:
4358:
4352:
4334:
4328:
4301:
4295:
4294:
4289:
4288:
4273:
4250:
4249:
4246:
4243:
4240:
4237:
4234:
4231:
4228:
4225:
4222:
4219:
4216:
4213:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4174:
4171:
4168:
4165:
4162:
4153:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4101:
4098:
4095:
4071:
4069:
4068:
4063:
4061:
4049:
4047:
4046:
4041:
4039:
4025:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3961:
3958:
3955:
3952:
3949:
3946:
3937:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3850:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3762:
3761:
3758:
3755:
3752:
3749:
3746:
3743:
3740:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3661:
3659:
3658:
3653:
3633:
3632:
3627:
3618:
3617:
3612:
3590:
3588:
3587:
3582:
3577:
3569:
3544:
3542:
3541:
3536:
3516:
3515:
3510:
3501:
3500:
3495:
3486:
3478:
3427:
3425:
3424:
3419:
3410:
3405:
3389:
3387:
3386:
3381:
3373:
3372:
3321:
3319:
3318:
3313:
3311:
3310:
3294:
3292:
3291:
3286:
3283:
3278:
3262:
3260:
3259:
3254:
3245:
3240:
3220:
3218:
3217:
3212:
3191:
3186:
3170:
3168:
3167:
3162:
3160:
3159:
3154:
3153:
3142:
3137:
3132:
3131:
3113:
3111:
3110:
3105:
3103:
3102:
3097:
3096:
3082:
3080:
3079:
3074:
3066:
3065:
3046:which come from
3037:
3035:
3034:
3029:
3024:
3016:
3015:
3000:
2999:
2981:
2980:
2962:
2959:
2949:positive density
2946:
2944:
2943:
2938:
2936:
2935:
2915:
2913:
2912:
2907:
2905:
2904:
2896:
2877:
2875:
2874:
2869:
2867:
2866:
2851:
2843:
2842:
2823:
2821:
2820:
2815:
2813:
2812:
2789:
2787:
2786:
2781:
2761:
2760:
2759:
2715:
2713:
2712:
2707:
2684:in terms of the
2683:
2681:
2680:
2675:
2631:
2617:
2614:
2605:
2603:
2602:
2597:
2595:
2575:
2573:
2572:
2567:
2565:
2564:
2533:
2531:
2530:
2525:
2523:
2522:
2506:
2504:
2503:
2498:
2496:
2495:
2490:
2489:
2481:
2470:
2468:
2467:
2462:
2457:
2443:
2442:
2412:
2409:
2400:
2398:
2397:
2392:
2390:
2389:
2373:
2371:
2370:
2365:
2363:
2334:
2332:
2331:
2326:
2324:
2323:
2307:
2305:
2304:
2299:
2297:
2296:
2295:
2263:
2261:
2260:
2255:
2244:
2243:
2238:
2237:
2229:
2210:
2209:
2190:
2188:
2187:
2182:
2180:
2179:
2174:
2173:
2165:
2158:
2157:
2129:
2127:
2126:
2121:
2101:
2098:
2089:
2087:
2086:
2081:
2079:
2078:
2062:
2060:
2059:
2054:
2052:
2051:
2046:
2045:
2037:
2026:
2024:
2023:
2018:
2016:
2015:
2010:
2009:
2001:
1967:
1965:
1964:
1959:
1948:-Selmer rank of
1947:
1945:
1944:
1939:
1927:
1925:
1924:
1919:
1914:
1896:
1894:
1893:
1888:
1876:
1874:
1873:
1868:
1866:
1865:
1841:
1840:
1821:
1819:
1818:
1813:
1801:
1799:
1798:
1793:
1782:-Selmer rank of
1781:
1779:
1778:
1773:
1761:
1759:
1758:
1753:
1739:
1738:
1716:
1714:
1713:
1708:
1691:
1690:
1685:
1675:
1673:
1672:
1667:
1650:
1649:
1634:
1620:
1612:
1586:
1584:
1583:
1578:
1562:
1560:
1559:
1554:
1542:
1540:
1539:
1534:
1523:
1522:
1517:
1507:
1505:
1504:
1499:
1487:
1485:
1484:
1479:
1467:
1465:
1464:
1459:
1445:
1444:
1428:
1426:
1425:
1420:
1408:
1406:
1405:
1400:
1395:
1377:
1375:
1374:
1369:
1367:
1362:
1332:
1330:
1329:
1324:
1322:
1317:
1298:
1296:
1295:
1290:
1278:
1276:
1275:
1270:
1258:
1256:
1255:
1250:
1248:
1240:
1231:
1229:
1228:
1223:
1201:
1199:
1198:
1193:
1188:
1173:
1171:
1170:
1165:
1149:
1147:
1146:
1141:
1106:
1104:
1103:
1098:
1093:
1091:
1086:
1046:
1032:
992:
989:
964:
962:
961:
956:
935:
933:
932:
927:
925:
920:
905:
903:
902:
897:
868:
866:
865:
860:
839:
837:
836:
831:
829:
824:
806:
804:
803:
798:
790:
789:
774:
773:
768:
759:
690:
688:
687:
682:
680:
675:
660:
658:
657:
652:
634:
632:
631:
626:
618:
617:
601:
599:
598:
593:
581:
579:
578:
573:
571:
570:
554:
552:
551:
546:
534:
532:
531:
526:
505:
503:
502:
497:
480:
479:
467:
466:
440:Weierstrass form
437:
435:
434:
429:
427:
422:
403:
401:
400:
395:
393:
388:
365:
363:
362:
357:
355:
327:
325:
324:
319:
317:
288:
286:
285:
280:
266:
264:
263:
258:
256:
255:
247:
221:
219:
218:
213:
211:
210:
207:
182:
180:
179:
174:
169:
168:
165:
147:
146:
141:
89:. The number of
56:rational numbers
53:
51:
50:
45:
27:is the rational
4690:
4689:
4685:
4684:
4683:
4681:
4680:
4679:
4670:Elliptic curves
4660:
4659:
4658:
4657:
4647:
4645:
4636:
4635:
4631:
4621:
4619:
4614:
4613:
4609:
4600:
4598:
4567:
4566:
4562:
4553:
4551:
4543:
4539:Conrad, Brian.
4538:
4537:
4533:
4523:
4521:
4518:Quanta Magazine
4510:
4509:
4505:
4493:
4486:
4474:
4470:
4458:
4451:
4423:
4419:
4391:
4387:
4359:
4355:
4335:
4331:
4302:
4298:
4286:
4284:
4277:Dujella, Andrej
4275:
4274:
4263:
4258:
4247:
4244:
4241:
4238:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4211:
4208:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4163:
4160:
4158:
4150:
4147:
4144:
4141:
4138:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4108:
4105:
4102:
4099:
4096:
4093:
4091:
4052:
4051:
4030:
4029:
4022:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
3998:
3995:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3942:
3934:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3881:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3767:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3706:
3672:
3622:
3607:
3593:
3592:
3551:
3550:
3505:
3490:
3460:
3459:
3454:This fails for
3392:
3391:
3364:
3359:
3358:
3339:
3302:
3297:
3296:
3265:
3264:
3227:
3226:
3173:
3172:
3147:
3120:
3119:
3090:
3085:
3084:
3057:
3052:
3051:
3007:
2991:
2972:
2953:
2952:
2927:
2922:
2921:
2891:
2880:
2879:
2858:
2834:
2826:
2825:
2804:
2799:
2798:
2750:
2745:
2744:
2689:
2688:
2608:
2607:
2586:
2585:
2556:
2551:
2550:
2514:
2509:
2508:
2478:
2473:
2472:
2434:
2403:
2402:
2381:
2376:
2375:
2348:
2347:
2341:abelian variety
2315:
2310:
2309:
2286:
2281:
2280:
2273:
2226:
2201:
2193:
2192:
2162:
2149:
2144:
2143:
2092:
2091:
2070:
2065:
2064:
2034:
2029:
2028:
1998:
1993:
1992:
1985:
1950:
1949:
1930:
1929:
1899:
1898:
1879:
1878:
1857:
1832:
1824:
1823:
1804:
1803:
1784:
1783:
1764:
1763:
1730:
1725:
1724:
1683:
1678:
1677:
1641:
1591:
1590:
1569:
1568:
1545:
1544:
1515:
1510:
1509:
1490:
1489:
1470:
1469:
1436:
1431:
1430:
1411:
1410:
1380:
1379:
1350:
1349:
1339:
1305:
1304:
1281:
1280:
1261:
1260:
1234:
1233:
1214:
1213:
1176:
1175:
1156:
1155:
1132:
1131:
1120:
1047:
993:
970:
969:
938:
937:
908:
907:
879:
878:
875:
842:
841:
812:
811:
781:
763:
696:
695:
663:
662:
637:
636:
609:
604:
603:
584:
583:
562:
557:
556:
537:
536:
511:
510:
471:
458:
447:
446:
410:
409:
406:height function
376:
375:
372:
346:
345:
308:
307:
271:
270:
242:
231:
230:
202:
188:
187:
160:
136:
116:
115:
97:of the curve.
75:rational points
36:
35:
17:
12:
11:
5:
4688:
4686:
4678:
4677:
4672:
4662:
4661:
4656:
4655:
4637:Elkies, Noam.
4629:
4607:
4560:
4531:
4503:
4484:
4468:
4449:
4417:
4385:
4353:
4329:
4296:
4260:
4259:
4257:
4254:
4253:
4252:
4060:
4038:
3879:
3878:
3852:
3851:
3671:
3668:
3651:
3648:
3645:
3642:
3639:
3636:
3631:
3626:
3621:
3616:
3611:
3606:
3603:
3600:
3580:
3576:
3572:
3568:
3564:
3561:
3558:
3534:
3531:
3528:
3525:
3522:
3519:
3514:
3509:
3504:
3499:
3494:
3489:
3485:
3481:
3477:
3473:
3470:
3467:
3417:
3414:
3409:
3404:
3400:
3379:
3376:
3371:
3367:
3351:function field
3338:
3335:
3309:
3305:
3282:
3277:
3273:
3252:
3249:
3244:
3239:
3235:
3210:
3207:
3204:
3201:
3198:
3195:
3190:
3185:
3181:
3158:
3152:
3146:
3141:
3136:
3130:
3101:
3095:
3072:
3069:
3064:
3060:
3027:
3023:
3019:
3014:
3010:
3006:
3003:
2998:
2994:
2990:
2987:
2984:
2979:
2975:
2971:
2968:
2965:
2934:
2930:
2903:
2900:
2895:
2890:
2887:
2865:
2861:
2857:
2854:
2850:
2846:
2841:
2837:
2833:
2811:
2807:
2779:
2776:
2773:
2770:
2767:
2764:
2758:
2753:
2705:
2702:
2699:
2696:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2630:
2626:
2623:
2620:
2594:
2563:
2559:
2521:
2517:
2494:
2487:
2484:
2460:
2456:
2452:
2449:
2446:
2441:
2437:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2388:
2384:
2362:
2358:
2355:
2322:
2318:
2294:
2289:
2272:
2266:
2253:
2250:
2247:
2242:
2235:
2232:
2225:
2222:
2219:
2216:
2213:
2208:
2204:
2200:
2178:
2171:
2168:
2161:
2156:
2152:
2119:
2116:
2113:
2110:
2107:
2104:
2077:
2073:
2063:one considers
2050:
2043:
2040:
2014:
2007:
2004:
1984:
1981:
1957:
1937:
1917:
1913:
1909:
1906:
1886:
1864:
1860:
1856:
1853:
1850:
1847:
1844:
1839:
1835:
1831:
1811:
1791:
1771:
1751:
1748:
1745:
1742:
1737:
1733:
1706:
1703:
1700:
1697:
1694:
1689:
1665:
1662:
1659:
1656:
1653:
1648:
1644:
1640:
1637:
1633:
1629:
1626:
1623:
1619:
1615:
1611:
1607:
1604:
1601:
1598:
1576:
1552:
1532:
1529:
1526:
1521:
1497:
1477:
1457:
1454:
1451:
1448:
1443:
1439:
1418:
1398:
1394:
1390:
1387:
1366:
1361:
1357:
1338:
1335:
1333:respectively.
1321:
1316:
1312:
1288:
1268:
1246:
1243:
1221:
1191:
1187:
1183:
1163:
1139:
1119:
1116:
1112:limit superior
1108:
1107:
1096:
1090:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1051:
1045:
1042:
1039:
1036:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
997:
988:
985:
982:
978:
954:
951:
948:
945:
924:
919:
915:
895:
892:
889:
886:
874:
871:
858:
855:
852:
849:
828:
823:
819:
808:
807:
796:
793:
788:
784:
780:
777:
772:
767:
762:
758:
754:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
679:
674:
670:
650:
647:
644:
624:
621:
616:
612:
591:
569:
565:
544:
524:
521:
518:
507:
506:
495:
492:
489:
486:
483:
478:
474:
470:
465:
461:
457:
454:
426:
421:
417:
392:
387:
383:
371:
368:
354:
316:
278:
254:
251:
246:
241:
238:
205:
201:
198:
195:
184:
183:
172:
163:
159:
156:
153:
150:
145:
140:
135:
132:
129:
126:
123:
110:abelian groups
43:
33:elliptic curve
15:
13:
10:
9:
6:
4:
3:
2:
4687:
4676:
4673:
4671:
4668:
4667:
4665:
4644:
4640:
4633:
4630:
4618:
4611:
4608:
4597:
4593:
4588:
4583:
4579:
4575:
4571:
4564:
4561:
4549:
4542:
4535:
4532:
4520:
4519:
4514:
4507:
4504:
4501:
4497:
4491:
4489:
4485:
4482:
4478:
4472:
4469:
4466:
4462:
4456:
4454:
4450:
4446:
4442:
4438:
4434:
4430:
4427:
4421:
4418:
4414:
4410:
4406:
4402:
4398:
4395:
4389:
4386:
4382:
4378:
4374:
4370:
4366:
4363:
4357:
4354:
4350:
4346:
4342:
4339:
4333:
4330:
4326:
4322:
4318:
4314:
4310:
4307:
4300:
4297:
4293:
4282:
4278:
4272:
4270:
4268:
4266:
4262:
4255:
4156:
4089:
4085:
4081:
4078:
4077:
4076:
4073:
4026:
3940:
3876:
3872:
3868:
3864:
3860:
3857:
3856:
3855:
3765:
3704:
3700:
3696:
3692:
3688:
3685:
3684:
3683:
3681:
3677:
3669:
3667:
3665:
3640:
3629:
3619:
3614:
3601:
3598:
3570:
3559:
3556:
3548:
3523:
3512:
3502:
3497:
3487:
3479:
3468:
3465:
3457:
3452:
3450:
3446:
3442:
3438:
3433:
3431:
3428:for all such
3412:
3407:
3402:
3398:
3374:
3369:
3365:
3356:
3352:
3348:
3344:
3343:global fields
3336:
3334:
3331:
3329:
3325:
3307:
3303:
3280:
3275:
3271:
3250:
3247:
3242:
3237:
3233:
3224:
3205:
3202:
3199:
3193:
3188:
3183:
3179:
3156:
3144:
3139:
3134:
3117:
3099:
3070:
3067:
3062:
3058:
3049:
3045:
3041:
3017:
3012:
3008:
3001:
2996:
2992:
2988:
2977:
2973:
2966:
2950:
2932:
2928:
2920:defined over
2919:
2901:
2898:
2888:
2885:
2863:
2859:
2855:
2844:
2839:
2835:
2809:
2805:
2796:
2791:
2774:
2771:
2768:
2762:
2751:
2742:
2738:
2734:
2730:
2726:
2721:
2719:
2700:
2694:
2687:
2662:
2656:
2650:
2647:
2641:
2638:
2621:
2583:
2579:
2561:
2557:
2548:
2544:
2540:
2535:
2519:
2515:
2492:
2482:
2450:
2447:
2439:
2435:
2431:
2422:
2416:
2386:
2382:
2356:
2353:
2346:defined over
2345:
2342:
2338:
2320:
2316:
2308:for the rank
2287:
2278:
2271:
2267:
2265:
2245:
2240:
2230:
2211:
2206:
2202:
2176:
2166:
2159:
2154:
2150:
2141:
2137:
2134:defined over
2133:
2111:
2105:
2075:
2071:
2048:
2038:
2012:
2002:
1990:
1982:
1980:
1978:
1974:
1969:
1955:
1935:
1904:
1884:
1862:
1858:
1854:
1848:
1842:
1837:
1833:
1809:
1789:
1769:
1746:
1740:
1735:
1731:
1722:
1717:
1704:
1695:
1687:
1657:
1651:
1646:
1642:
1624:
1621:
1617:
1602:
1596:
1588:
1574:
1566:
1563:-part of the
1550:
1527:
1519:
1495:
1475:
1452:
1446:
1441:
1437:
1416:
1385:
1359:
1355:
1347:
1343:
1334:
1314:
1310:
1302:
1301:Selmer groups
1286:
1266:
1244:
1241:
1219:
1211:
1207:
1203:
1189:
1185:
1181:
1161:
1153:
1137:
1129:
1125:
1117:
1115:
1113:
1094:
1088:
1083:
1080:
1071:
1068:
1065:
1059:
1053:
1049:
1040:
1034:
1029:
1026:
1017:
1014:
1011:
1005:
999:
995:
980:
968:
967:
966:
949:
943:
917:
913:
890:
884:
877:We denote by
872:
870:
853:
847:
821:
817:
794:
786:
782:
778:
775:
770:
760:
752:
743:
734:
731:
728:
722:
716:
713:
707:
701:
694:
693:
692:
672:
668:
648:
645:
642:
622:
619:
614:
610:
589:
567:
563:
542:
522:
519:
516:
493:
490:
487:
484:
481:
476:
472:
468:
463:
459:
455:
452:
445:
444:
443:
441:
419:
415:
407:
385:
381:
369:
367:
343:
339:
335:
331:
305:
301:
300:number theory
296:
294:
292:
276:
252:
249:
239:
236:
228:
227:
199:
193:
170:
157:
151:
148:
143:
133:
127:
121:
114:
113:
112:
111:
107:
103:
98:
96:
92:
88:
84:
80:
76:
72:
68:
64:
61:
57:
41:
34:
30:
26:
22:
4646:. Retrieved
4642:
4632:
4620:. Retrieved
4610:
4599:. Retrieved
4577:
4573:
4563:
4552:. Retrieved
4548:Brian Conrad
4547:
4534:
4522:. Retrieved
4516:
4506:
4471:
4420:
4388:
4356:
4332:
4299:
4291:
4285:. Retrieved
4154:
4087:
4083:
4079:
4074:
4027:
3938:
3880:
3874:
3870:
3866:
3862:
3858:
3853:
3763:
3702:
3698:
3694:
3690:
3686:
3673:
3663:
3546:
3456:local fields
3453:
3448:
3444:
3436:
3434:
3429:
3354:
3346:
3340:
3332:
3327:
3323:
3222:
3115:
3043:
3039:
2917:
2794:
2792:
2729:class groups
2722:
2717:
2581:
2577:
2536:
2343:
2274:
2269:
2139:
2135:
2131:
1988:
1986:
1976:
1972:
1970:
1720:
1718:
1589:
1508:, and let Đš
1378:. Denote by
1340:
1279:-Selmer and
1204:
1121:
1109:
876:
873:Average rank
809:
508:
373:
297:
290:
268:
225:
185:
105:
101:
99:
90:
62:
60:number field
29:MordellâWeil
24:
18:
3676:Noam Elkies
3390:, but that
3048:base change
1543:denote the
1152:Heath-Brown
869:is finite.
91:independent
31:rank of an
21:mathematics
4664:Categories
4601:2024-05-04
4587:1606.07178
4574:Math. Comp
4554:2024-05-04
4437:1086.11032
4405:1063.11013
4373:0783.14019
4349:0958.11004
4317:0417.14031
4287:2024-05-04
4256:References
3225:such that
2878:for every
2401:such that
2335:probable.
2142:. We have
1822:such that
602:, we have
555:such that
332:and later
71:André Weil
3602:∈
3560:∈
3469:∈
3416:∞
3408:∘
3378:∞
3328:plausible
3295:and even
3281:∘
3248:≥
3243:∘
3194:∈
3189:∘
3145:⊂
3140:∘
3068:⊊
2989:≥
2899:≥
2889:∈
2763:∈
2651:
2639:≤
2545:and 1982
2486:~
2432:≤
2249:∞
2234:~
2221:⇔
2215:∞
2170:~
2160:≤
2042:~
2006:~
1843:
1830:#
1741:
1702:→
1664:→
1652:
1639:→
1600:→
1447:
1346:Shankar's
1081:≤
1050:∑
1027:≤
996:∑
987:∞
984:→
620:∤
250:≥
240:∈
149:⊕
134:≅
4648:30 March
4622:30 March
3458:such as
2537:In 1966
1342:Bhargava
1206:Bhargava
1126:and the
582:divides
330:Goldfeld
269:rank of
83:infinite
4524:18 July
4429:2169047
4397:2057019
4365:1176198
4341:1659828
4309:0564926
2951:) with
2733:Keating
2541:, 1974
2539:Cassels
1210:Shankar
370:Heights
222:is the
4435:
4403:
4371:
4347:
4315:
2737:Snaith
2584:over
2547:Mestre
338:Sarnak
289:(over
186:where
23:, the
4582:arXiv
4544:(PDF)
3349:is a
2337:Honda
2277:NĂ©ron
342:below
87:order
79:basis
4650:2020
4624:2020
4526:2019
3413:<
2793:For
2727:for
2543:Tate
2246:<
2212:<
2191:and
1971:In
1721:rank
1468:the
1344:and
1232:and
1208:and
334:Katz
208:tors
166:tors
106:E(K)
95:rank
4592:doi
4496:doi
4477:doi
4461:doi
4441:doi
4433:Zbl
4409:doi
4401:Zbl
4377:doi
4369:Zbl
4345:Zbl
4321:doi
4313:Zbl
4248:497
4245:721
4242:636
4239:425
4236:173
4233:716
4230:277
4227:413
4224:149
4221:779
4218:404
4215:372
4212:608
4209:079
4206:535
4203:032
4200:067
4197:521
4194:821
4191:191
4188:118
4185:591
4182:699
4179:736
4176:475
4173:376
4170:342
4167:551
4164:058
4161:258
4151:385
4148:737
4145:954
4142:357
4139:836
4136:621
4133:054
4130:768
4127:784
4124:453
4121:297
4118:145
4115:652
4112:434
4109:218
4106:922
4103:630
4100:241
4097:183
4094:006
4023:931
4020:499
4017:121
4014:151
4011:438
4008:898
4005:989
4002:830
3999:957
3996:238
3993:434
3990:964
3987:028
3984:682
3981:270
3978:743
3975:817
3972:806
3969:258
3966:979
3963:222
3960:546
3957:034
3954:183
3951:053
3948:182
3945:710
3943:961
3935:707
3932:853
3929:859
3926:821
3923:573
3920:757
3917:270
3914:769
3911:961
3908:168
3905:487
3902:803
3899:463
3896:601
3893:319
3890:336
3887:673
3884:537
3882:244
3848:429
3845:243
3842:732
3839:448
3836:939
3833:291
3830:296
3827:008
3824:266
3821:361
3818:180
3815:319
3812:359
3809:944
3806:855
3803:374
3800:720
3797:390
3794:690
3791:985
3788:032
3785:467
3782:556
3779:030
3776:795
3773:611
3770:481
3760:502
3757:956
3754:178
3751:312
3748:230
3745:930
3742:750
3739:542
3736:338
3733:209
3730:208
3727:033
3724:585
3721:526
3718:575
3715:415
3712:762
3709:067
3680:GRH
3251:68.
2648:log
2507:or
1897:of
1834:Sel
1732:Sel
1723:of
1643:Sel
1567:of
1438:Sel
1220:1.5
1138:2.3
977:lim
747:max
691:by
19:In
4666::
4641:.
4590:.
4578:88
4576:.
4572:.
4546:.
4515:.
4487:^
4452:^
4439:.
4431:.
4426:MR
4407:.
4399:.
4394:MR
4375:.
4367:.
4362:MR
4343:.
4338:MR
4319:.
4311:.
4306:MR
4290:.
4279:.
4264:^
4159:55
4157:+
4092:27
4090:â
4086:=
4084:xy
4082:+
3941:+
3873:â
3869:=
3865:+
3863:xy
3861:+
3768:34
3766:+
3707:20
3705:â
3701:â
3697:=
3693:+
3691:xy
3689:+
3206:21
3200:20
2960:rk
2775:21
2769:20
2720:.
2615:rk
2410:rk
2264:.
2099:rk
1705:0.
1676:Đš
1429:,
1190:14
1182:25
779:27
366:.
295:.
65:.
4652:.
4626:.
4604:.
4594::
4584::
4557:.
4528:.
4498::
4479::
4463::
4447:.
4443::
4415:.
4411::
4383:.
4379::
4351:.
4327:.
4323::
4251:.
4155:x
4088:x
4080:y
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