190:
772:. Wiles used this to prove Fermat's Last Theorem, and the full TaniyamaâShimuraâWeil conjecture was finally proved by Diamond, Conrad, Diamond & Taylor; and Breuil, Conrad, Diamond & Taylor; building on Wiles's work, they incrementally chipped away at the remaining cases until the full result was proved in 1999. Once fully proven, the conjecture became known as the modularity theorem.
1394:
697:, corresponding to Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves (there can also be higher-dimensional factors, so not all Hecke eigenforms correspond to rational elliptic curves). The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is
738:-series of the elliptic curve; this was the first serious evidence that the conjecture might be true. Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form. The TaniyamaâShimuraâWeil conjecture became a part of the
1582:
attached on the one hand to elliptic curves, and on the other hand to modular forms. The latter formulation has been used in the proof of the conjecture. Dealing with the level of the forms (and the connection to the conductor of the curve) is particularly delicate.
749:. He did this by attempting to show that any counterexample to Fermat's Last Theorem would imply the existence of at least one non-modular elliptic curve. This argument was completed in 1987 when Jean-Pierre Serre identified a missing link (now known as the
1127:
756:
Even after gaining serious attention, the
TaniyamaâShimuraâWeil conjecture was seen by contemporary mathematicians as extraordinarily difficult to prove or perhaps even inaccessible to proof. For example, Wiles's Ph.D. supervisor
1814:
cannot be modular. Thus, the proof of the
TaniyamaâShimuraâWeil conjecture for this family of elliptic curves (called HellegouarchâFrey curves) implies FLT. The proof of the link between these two statements, based on an idea of
1001:
483:
1523:
586:
1058:
There are several formulations of the conjecture. Showing that they are equivalent was a main challenge of number theory in the second half of the 20th century. The modularity of an elliptic curve
734:
and
Taniyama worked on improving its rigor until 1957. André Weil rediscovered the conjecture, and showed in 1967 that it would follow from the (conjectured) functional equations for some twisted
1132:
1809:
1745:
1651:
634:
1389:{\displaystyle {\begin{aligned}x(z)&=q^{-2}+2q^{-1}+5+9q+18q^{2}+29q^{3}+51q^{4}+\cdots \\y(z)&=q^{-3}+3q^{-2}+9q^{-1}+21+46q+92q^{2}+180q^{3}+\cdots \end{aligned}}}
761:
states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed was completely inaccessible".
856:
775:
Several theorems in number theory similar to Fermat's Last
Theorem follow from the modularity theorem. For example: no cube can be written as a sum of two
1657:
3045:
340:
is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the
17:
813:(a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a
2379:
2296:
2256:
714:
402:
346:), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level
3671:
722:
stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in
753:
or Ribet's theorem) in Frey's original work, followed two years later by Ken Ribet's completion of a proof of the epsilon conjecture.
3468:
332:; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level
3666:
254:
2294:
Conrad, Brian; Diamond, Fred; Taylor, Richard (1999), "Modularity of certain potentially
BarsottiâTate Galois representations",
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233:
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2250:
Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001), "On the modularity of elliptic curves over
273:
693:
for an elliptic curve. The
Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible
240:
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31:
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Freitas, Nuno; Le Hung, Bao V.; Siksek, Samir (2015), "Elliptic curves over real quadratic fields are modular",
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Discusses the
TaniyamaâShimuraâWeil conjecture 3 years before it was proven for infinitely many cases.
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The conjecture attracted considerable interest when
Gerhard Frey suggested in 1986 that it implies
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690:
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2518:
Frey, Gerhard (1986), "Links between stable elliptic curves and certain
Diophantine equations",
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8:
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2123:
Zagier, D. (1985). "Modular points, modular curves, modular surfaces and modular forms".
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The conjecture, going back to the 1950s, was completely proven by 1999 using ideas of
996:{\displaystyle f(z)=q-2q^{2}-3q^{3}+2q^{4}-2q^{5}+6q^{6}+\cdots ,\qquad q=e^{2\pi iz}}
3488:
3435:
3306:
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Proceedings of the
International Congress of Mathematicians, Vol. 1, 2 (ZĂŒrich, 1994)
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2349:
2315:
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802:
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173:
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In 2013, Freitas, Le Hung, and Siksek proved that elliptic curves defined over real
727:
3478:
3364:
3341:
2925:
2866:
2817:
2724:
2715:
Shimura, Goro (1989), "Yutaka Taniyama and his time. Very personal recollections",
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2310:
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2949:
2882:
2855:(1967), "Ăber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen",
2841:
2742:
2704:
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2584:
2535:
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137:
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3063:
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has a solution with non-zero integers, hence a counter-example to FLT. Then as
1010:
not equal to 37, one can verify the property about the coefficients. Thus, for
758:
679:
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291:
133:
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2181:
2127:. Lecture Notes in Mathematics. Vol. 1111. Springer. pp. 225â248.
797:
The modularity theorem is a special case of more general conjectures due to
176:, extended Wiles's techniques to prove the full modularity theorem in 2001.
3555:
3529:
3519:
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165:
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145:
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82:
60:
2404:
Contains a gentle introduction to the theorem and an outline of the proof.
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308:
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A synthetic presentation (in French) of the main ideas can be found in
214: in this section. Unsourced material may be challenged and removed.
2963:(1995b), "Modular forms, elliptic curves, and Fermat's last theorem",
817:. Most cases of these extended conjectures have not yet been proved.
3286:
3014:
2231:
2032:
671:
370:
The modularity theorem implies a closely related analytic statement:
3004:
2929:
2821:
2564:
2423:
189:
2372:"A proof of the full ShimuraâTaniyamaâWeil conjecture is announced"
1900:
1586:
The most spectacular application of the conjecture is the proof of
478:{\displaystyle L(E,s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}
2470:
850:, with discriminant (and conductor) 37, is associated to the form
3598:
3583:
1980:
776:
351:
305:
287:
2044:
3578:
2796:(1995), "Ring-theoretic properties of certain Hecke algebras",
2215:"From the TaniyamaâShimura conjecture to Fermat's Last Theorem"
2020:
1055:, who proved it in 1994 for a large family of elliptic curves.
701:
to the original curve (but not, in general, isomorphic to it).
2904:(1995a), "Modular elliptic curves and Fermat's last theorem",
2673:(1987), "Sur les représentations modulaires de degré 2 de Gal(
2408:
Diamond, Fred (1996), "On deformation rings and Hecke rings",
1431:
are modular of weight 0 and level 37; in other words they are
2596:
Ribet, Kenneth A. (1990), "On modular representations of Gal(
1847:
1066:
can be expressed also by saying that there is a non-constant
723:
2088:
1992:
689:
Some modular forms of weight two, in turn, correspond to
666:-series are also thought of as the Fourier coefficients of
1819:(1985), is difficult and technical. It was established by
2249:
2038:
1887:
1518:{\displaystyle x\!\left({\frac {az+b}{cz+d}}\right)=x(z)}
2967:, Basel, Boston, Berlin: BirkhĂ€user, pp. 243â245,
2333:
581:{\displaystyle f(E,q)=\sum _{n=1}^{\infty }a_{n}q^{n}.}
160:. Later, a series of papers by Wiles's former students
2520:
Annales Universitatis Saraviensis. Series Mathematicae
768:, proved the TaniyamaâShimuraâWeil conjecture for all
670:. The function obtained in this way is, remarkably, a
1759:
1676:
1606:
1455:
1130:
859:
600:
512:
405:
2453:
2050:
1103:
For example, a modular parametrization of the curve
2293:
2026:
2008:
1970:
1968:
1863:
1804:{\displaystyle \Delta ={\frac {1}{256}}(abc)^{2p}}
1803:
1739:
1645:
1517:
1388:
1017:, there are 6 solutions of the equation modulo 3:
995:
628:
580:
477:
1917:"Some History of the Shimura-Taniyama Conjecture"
1578:Another formulation depends on the comparison of
1459:
27:Relates rational elliptic curves to modular forms
3643:
1965:
1953:
678:and is also an eigenform (an eigenvector of all
2717:The Bulletin of the London Mathematical Society
1941:
1894:Harris, Michael (2020). "Virtues of Priority".
1875:
3053:
2219:Annales de la Faculté des Sciences de Toulouse
3039:
686:, which follows from the modularity theorem.
2380:Notices of the American Mathematical Society
2297:Journal of the American Mathematical Society
2257:Journal of the American Mathematical Society
2155:
2111:
1924:Notices of the American Mathematical Society
1667:was the first to notice, the elliptic curve
2791:
2077:http://www.lmfdb.org/EllipticCurve/Q/37/a/1
764:In 1995, Andrew Wiles, with some help from
3046:
3032:
2919:
2811:
2728:
2637:
2469:
2309:
2269:
2230:
2180:
2110:. For more details see Hellegouarch
1899:
1740:{\displaystyle y^{2}=x(x-a^{p})(x+b^{p})}
274:Learn how and when to remove this message
130:modularity conjecture for elliptic curves
2773:
1869:
2785:
2774:Taniyama, Yutaka (1956), "Problem 12",
2714:
2407:
2341:Modular forms and Fermat's last theorem
2014:
14:
3644:
3469:Clifford's theorem on special divisors
2986:
2959:
2900:
2369:
2212:
2122:
2063:
2062:For the calculations, see for example
2002:
1998:
1893:
715:Wiles's proof of Fermat's Last Theorem
358:-expansion, followed if need be by an
3027:
3005:
2752:
2669:
2595:
2545:
1986:
1974:
1959:
365:
2851:
2517:
1947:
1914:
1881:
212:adding citations to reliable sources
183:
172:, culminating in a joint paper with
2549:(1991), "Number theory as gadfly",
24:
3627:Vector bundles on algebraic curves
3561:Weber's theorem (Algebraic curves)
3158:Hasse's theorem on elliptic curves
3148:Counting points on elliptic curves
2051:Freitas, Le Hung & Siksek 2015
1760:
792:
550:
443:
152:proved the modularity theorem for
25:
3698:
3672:Conjectures that have been proved
2980:
2552:The American Mathematical Monthly
2027:Conrad, Diamond & Taylor 1999
1646:{\displaystyle a^{p}+b^{p}=c^{p}}
629:{\displaystyle q=e^{2\pi i\tau }}
1590:(FLT). Suppose that for a prime
832:For example, the elliptic curve
639:we see that we have written the
188:
126:TaniyamaâShimuraâWeil conjecture
3249:Hurwitz's automorphisms theorem
2606:) arising from modular forms",
2338:; Stevens, Glenn, eds. (1997),
2242:
2205:
2149:
2116:
2093:
2081:
2069:
2056:
1833:
1092:. In particular, the points of
967:
199:needs additional citations for
3667:Theorems in algebraic geometry
3474:Gonality of an algebraic curve
3385:Differential of the first kind
1908:
1789:
1776:
1734:
1715:
1712:
1693:
1512:
1506:
1268:
1262:
1144:
1138:
869:
863:
528:
516:
421:
409:
381:we may attach a corresponding
13:
1:
3617:BirkhoffâGrothendieck theorem
3327:Nagata's conjecture on curves
3198:SchoofâElkiesâAtkin algorithm
3072:Five points determine a conic
3010:"TaniyamaâShimura Conjecture"
2989:"ShimuraâTaniyama conjecture"
2693:10.1215/S0012-7094-87-05413-5
2311:10.1090/S0894-0347-99-00287-8
2271:10.1090/S0894-0347-01-00370-8
1989:, pp. 203â205, 223, 226.
1915:Lang, Serge (November 1995).
1857:
662:, so the coefficients of the
32:Serre's modularity conjecture
3188:Supersingular elliptic curve
2162:sur les courbes elliptiques"
591:If we make the substitution
179:
156:, which was enough to imply
7:
3677:20th century in mathematics
3395:Riemann's existence theorem
3322:Hilbert's sixteenth problem
3214:Elliptic curve cryptography
3127:Fundamental pair of periods
2994:Encyclopedia of Mathematics
2156:Hellegouarch, Yves (1974).
122:TaniyamaâShimura conjecture
18:TaniyamaâShimura conjecture
10:
3703:
3525:Moduli of algebraic curves
2254:: wild 3-adic exercises",
827:
811:automorphic representation
770:semistable elliptic curves
708:
704:
154:semistable elliptic curves
29:
3662:Theorems in number theory
3607:
3569:
3538:
3502:
3451:
3444:
3418:
3350:
3267:
3231:
3206:
3140:
3109:
3100:
3062:
2680:Duke Mathematical Journal
2488:10.1007/s00222-014-0550-z
1074:, from the modular curve
691:holomorphic differentials
104:
96:
74:
66:
52:
42:
3292:CayleyâBacharach theorem
3219:Elliptic curve primality
2784:English translation in (
2609:Inventiones Mathematicae
2457:Inventiones Mathematicae
2089:https://oeis.org/A007653
1826:
674:of weight two and level
658:of the complex variable
30:Not to be confused with
3551:RiemannâHurwitz formula
3515:GromovâWitten invariant
3375:Compact Riemann surface
3163:Mazur's torsion theorem
2182:10.4064/aa-26-3-253-263
2125:Arbeitstagung Bonn 1984
1096:can be parametrized by
373:To each elliptic curve
313:classical modular curve
3168:Modular elliptic curve
2370:Darmon, Henri (1999),
1805:
1741:
1647:
1597:, the Fermat equation
1580:Galois representations
1519:
1390:
997:
630:
582:
554:
479:
447:
300:can be obtained via a
144:in a particular way.
3082:Rational normal curve
2907:Annals of Mathematics
2858:Mathematische Annalen
2799:Annals of Mathematics
2758:Fermat's Last Theorem
2730:10.1112/blms/21.2.186
2411:Annals of Mathematics
1846:was already known by
1806:
1742:
1648:
1588:Fermat's Last Theorem
1520:
1391:
998:
747:Fermat's Last Theorem
711:Fermat's Last Theorem
709:Further information:
684:HasseâWeil conjecture
631:
583:
534:
480:
427:
158:Fermat's Last Theorem
120:(formerly called the
109:Fermat's Last Theorem
3622:Stable vector bundle
3494:Weil reciprocity law
3484:RiemannâRoch theorem
3464:BrillâNoether theory
3400:RiemannâRoch theorem
3317:Genusâdegree formula
3178:MordellâWeil theorem
3153:Division polynomials
2987:Darmon, H. (2001) ,
2344:, Berlin, New York:
2336:Silverman, Joseph H.
1757:
1674:
1604:
1453:
1128:
857:
598:
510:
492:of the coefficients
403:
223:"Modularity theorem"
208:improve this article
3682:Arithmetic geometry
3445:Structure of curves
3337:Quartic plane curve
3259:Hyperelliptic curve
3239:De Franchis theorem
3183:NagellâLutz theorem
2622:1990InMat.100..431R
2480:2015InMat.201..159F
2213:Ribet, K. (1990b).
1539:, for all integers
805:seeks to attach an
490:generating function
396:, commonly written
39:
3452:Divisors on curves
3244:Faltings's theorem
3193:Schoof's algorithm
3173:Modularity theorem
3007:Weisstein, Eric W.
2871:10.1007/BF01361551
2671:Serre, Jean-Pierre
2639:10338.dmlcz/147454
2630:10.1007/BF01231195
2211:See the survey of
2133:10.1007/BFb0084592
2066:, pp. 225â248
2039:Breuil et al. 2001
1801:
1737:
1643:
1515:
1386:
1384:
1006:For prime numbers
993:
751:epsilon conjecture
626:
578:
475:
366:Related statements
136:over the field of
118:modularity theorem
38:Modularity theorem
37:
3639:
3638:
3635:
3634:
3546:HasseâWitt matrix
3489:Weierstrass point
3436:Smooth completion
3405:TeichmĂŒller space
3307:Cubic plane curve
3227:
3226:
3141:Arithmetic theory
3122:Elliptic integral
3117:Elliptic function
2910:, Second Series,
2802:, Second Series,
2792:Taylor, Richard;
2767:978-1-85702-521-7
2760:, Fourth Estate,
2414:, Second Series,
2387:(11): 1397â1401,
2355:978-0-387-94609-2
2158:"Points d'ordre 2
2142:978-3-540-39298-9
2108:Jean-Pierre Serre
1774:
1658:Yves Hellegouarch
1528:and likewise for
1494:
1435:, defined on the
1399:where, as above,
1098:modular functions
803:Langlands program
740:Langlands program
695:Abelian varieties
641:Fourier expansion
470:
328:for some integer
284:
283:
276:
258:
174:Christophe Breuil
114:
113:
79:Christophe Breuil
16:(Redirected from
3694:
3652:Algebraic curves
3479:Jacobian variety
3449:
3448:
3352:Riemann surfaces
3342:Real plane curve
3302:Cramer's paradox
3282:BĂ©zout's theorem
3107:
3106:
3056:algebraic curves
3048:
3041:
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3020:
3019:
3001:
2975:
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2923:
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2815:
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2749:
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2711:
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2232:10.5802/afst.698
2209:
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2169:Acta Arithmetica
2166:
2153:
2147:
2146:
2120:
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1867:
1851:
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1837:
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1808:
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1802:
1800:
1799:
1775:
1767:
1750:of discriminant
1746:
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1711:
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1666:
1652:
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1628:
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1574:
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1524:
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1521:
1516:
1499:
1495:
1493:
1479:
1465:
1445:
1437:upper half-plane
1430:
1419:
1409:. The functions
1408:
1395:
1393:
1392:
1387:
1385:
1375:
1374:
1359:
1358:
1328:
1327:
1309:
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1290:
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1215:
1185:
1184:
1166:
1165:
1120:
1095:
1091:
1087:
1073:
1065:
1061:
1047:
1046:(3) = 3 â 6 = â3
1040:
1036:
1032:
1028:
1024:
1020:
1016:
1009:
1002:
1000:
999:
994:
992:
991:
957:
956:
941:
940:
925:
924:
909:
908:
893:
892:
849:
822:quadratic fields
807:automorphic form
799:Robert Langlands
788:
781:
737:
677:
669:
665:
661:
657:
635:
633:
632:
627:
625:
624:
587:
585:
584:
579:
574:
573:
564:
563:
553:
548:
502:
484:
482:
481:
476:
471:
469:
468:
459:
458:
449:
446:
441:
394:Dirichlet series
391:
385:
380:
376:
357:
349:
339:
335:
331:
327:
298:
290:states that any
279:
272:
268:
265:
259:
257:
216:
192:
184:
140:are related to
138:rational numbers
40:
36:
21:
3702:
3701:
3697:
3696:
3695:
3693:
3692:
3691:
3687:1995 in science
3642:
3641:
3640:
3631:
3603:
3594:Delta invariant
3565:
3534:
3498:
3459:AbelâJacobi map
3440:
3414:
3410:Torelli theorem
3380:Dessin d'enfant
3360:Belyi's theorem
3346:
3332:PlĂŒcker formula
3263:
3254:Hurwitz surface
3223:
3202:
3136:
3110:Analytic theory
3102:Elliptic curves
3096:
3077:Projective line
3064:Rational curves
3058:
3052:
2983:
2978:
2930:10.2307/2118559
2921:10.1.1.169.9076
2822:10.2307/2118560
2778:(in Japanese),
2768:
2674:
2597:
2565:10.2307/2324924
2424:10.2307/2118586
2374:
2356:
2346:Springer-Verlag
2334:Cornell, Gary;
2245:
2240:
2210:
2206:
2164:
2154:
2150:
2143:
2121:
2117:
2098:
2094:
2086:
2082:
2074:
2070:
2061:
2057:
2049:
2045:
2037:
2033:
2025:
2021:
2013:
2009:
1997:
1993:
1985:
1981:
1973:
1966:
1958:
1954:
1946:
1942:
1933:
1931:
1930:(11): 1301â1307
1919:
1913:
1909:
1892:
1888:
1880:
1876:
1868:
1864:
1860:
1855:
1854:
1840:
1838:
1834:
1829:
1792:
1788:
1766:
1758:
1755:
1754:
1728:
1724:
1706:
1702:
1681:
1677:
1675:
1672:
1671:
1660:
1637:
1633:
1624:
1620:
1611:
1607:
1605:
1602:
1601:
1591:
1569:
1558:
1540:
1529:
1480:
1466:
1464:
1460:
1454:
1451:
1450:
1439:
1421:
1410:
1400:
1383:
1382:
1370:
1366:
1354:
1350:
1320:
1316:
1301:
1297:
1282:
1278:
1271:
1256:
1255:
1243:
1239:
1227:
1223:
1211:
1207:
1177:
1173:
1158:
1154:
1147:
1131:
1129:
1126:
1125:
1104:
1093:
1089:
1081:
1075:
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1063:
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1042:
1038:
1034:
1030:
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1022:
1018:
1011:
1007:
978:
974:
952:
948:
936:
932:
920:
916:
904:
900:
888:
884:
858:
855:
854:
833:
830:
795:
793:Generalizations
783:
779:
735:
720:Yutaka Taniyama
717:
707:
682:); this is the
680:Hecke operators
675:
667:
663:
659:
644:
611:
607:
599:
596:
595:
569:
565:
559:
555:
549:
538:
511:
508:
507:
501:
493:
464:
460:
454:
450:
448:
442:
431:
404:
401:
400:
389:
383:
378:
374:
368:
355:
350:, a normalized
347:
337:
333:
329:
321:
315:
296:
280:
269:
263:
260:
217:
215:
205:
193:
182:
134:elliptic curves
89:
85:
81:
59:
57:Yutaka Taniyama
35:
28:
23:
22:
15:
12:
11:
5:
3700:
3690:
3689:
3684:
3679:
3674:
3669:
3664:
3659:
3654:
3637:
3636:
3633:
3632:
3630:
3629:
3624:
3619:
3613:
3611:
3609:Vector bundles
3605:
3604:
3602:
3601:
3596:
3591:
3586:
3581:
3575:
3573:
3567:
3566:
3564:
3563:
3558:
3553:
3548:
3542:
3540:
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3527:
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3517:
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3499:
3497:
3496:
3491:
3486:
3481:
3476:
3471:
3466:
3461:
3455:
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3442:
3441:
3439:
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3428:
3422:
3420:
3416:
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3412:
3407:
3402:
3397:
3392:
3387:
3382:
3377:
3372:
3367:
3362:
3356:
3354:
3348:
3347:
3345:
3344:
3339:
3334:
3329:
3324:
3319:
3314:
3309:
3304:
3299:
3294:
3289:
3284:
3279:
3273:
3271:
3265:
3264:
3262:
3261:
3256:
3251:
3246:
3241:
3235:
3233:
3229:
3228:
3225:
3224:
3222:
3221:
3216:
3210:
3208:
3204:
3203:
3201:
3200:
3195:
3190:
3185:
3180:
3175:
3170:
3165:
3160:
3155:
3150:
3144:
3142:
3138:
3137:
3135:
3134:
3129:
3124:
3119:
3113:
3111:
3104:
3098:
3097:
3095:
3094:
3089:
3087:Riemann sphere
3084:
3079:
3074:
3068:
3066:
3060:
3059:
3051:
3050:
3043:
3036:
3028:
3022:
3021:
3002:
2982:
2981:External links
2979:
2977:
2976:
2957:
2914:(3): 443â551,
2898:
2849:
2813:10.1.1.128.531
2806:(3): 553â572,
2789:
2771:
2766:
2750:
2723:(2): 186â196,
2712:
2687:(1): 179â230,
2667:
2616:(2): 431â476,
2593:
2559:(7): 593â610,
2543:
2515:
2464:(1): 159â206,
2451:
2418:(1): 137â166,
2405:
2367:
2354:
2331:
2304:(2): 521â567,
2291:
2264:(4): 843â939,
2246:
2244:
2241:
2239:
2238:
2204:
2175:(3): 253â263.
2148:
2141:
2115:
2092:
2080:
2068:
2055:
2043:
2031:
2019:
2007:
1991:
1979:
1964:
1952:
1940:
1907:
1886:
1874:
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1852:
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1148:
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1143:
1140:
1137:
1134:
1133:
1079:
1004:
1003:
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987:
984:
981:
977:
973:
970:
966:
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951:
947:
944:
939:
935:
931:
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923:
919:
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903:
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891:
887:
883:
880:
877:
874:
871:
868:
865:
862:
829:
826:
794:
791:
766:Richard Taylor
706:
703:
643:of a function
637:
636:
623:
620:
617:
614:
610:
606:
603:
589:
588:
577:
572:
568:
562:
558:
552:
547:
544:
541:
537:
533:
530:
527:
524:
521:
518:
515:
497:
486:
485:
474:
467:
463:
457:
453:
445:
440:
437:
434:
430:
426:
423:
420:
417:
414:
411:
408:
367:
364:
319:
292:elliptic curve
282:
281:
196:
194:
187:
181:
178:
170:Richard Taylor
150:Richard Taylor
132:) states that
112:
111:
106:
102:
101:
98:
97:First proof in
94:
93:
91:Richard Taylor
76:
75:First proof by
72:
71:
68:
67:Conjectured in
64:
63:
54:
53:Conjectured by
50:
49:
44:
26:
9:
6:
4:
3:
2:
3699:
3688:
3685:
3683:
3680:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3657:Modular forms
3655:
3653:
3650:
3649:
3647:
3628:
3625:
3623:
3620:
3618:
3615:
3614:
3612:
3610:
3606:
3600:
3597:
3595:
3592:
3590:
3587:
3585:
3582:
3580:
3577:
3576:
3574:
3572:
3571:Singularities
3568:
3562:
3559:
3557:
3554:
3552:
3549:
3547:
3544:
3543:
3541:
3537:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3507:
3505:
3501:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3456:
3454:
3450:
3447:
3443:
3437:
3434:
3432:
3429:
3427:
3424:
3423:
3421:
3419:Constructions
3417:
3411:
3408:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3390:Klein quartic
3388:
3386:
3383:
3381:
3378:
3376:
3373:
3371:
3370:Bolza surface
3368:
3366:
3365:Bring's curve
3363:
3361:
3358:
3357:
3355:
3353:
3349:
3343:
3340:
3338:
3335:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3298:
3297:Conic section
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3277:AF+BG theorem
3275:
3274:
3272:
3270:
3266:
3260:
3257:
3255:
3252:
3250:
3247:
3245:
3242:
3240:
3237:
3236:
3234:
3230:
3220:
3217:
3215:
3212:
3211:
3209:
3205:
3199:
3196:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3145:
3143:
3139:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3114:
3112:
3108:
3105:
3103:
3099:
3093:
3092:Twisted cubic
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3073:
3070:
3069:
3067:
3065:
3061:
3057:
3049:
3044:
3042:
3037:
3035:
3030:
3029:
3026:
3017:
3016:
3011:
3008:
3003:
3000:
2996:
2995:
2990:
2985:
2984:
2974:
2970:
2966:
2962:
2961:Wiles, Andrew
2958:
2955:
2951:
2947:
2943:
2939:
2935:
2931:
2927:
2922:
2917:
2913:
2909:
2908:
2903:
2902:Wiles, Andrew
2899:
2896:
2892:
2888:
2884:
2880:
2876:
2872:
2868:
2864:
2860:
2859:
2854:
2850:
2847:
2843:
2839:
2835:
2831:
2827:
2823:
2819:
2814:
2809:
2805:
2801:
2800:
2795:
2794:Wiles, Andrew
2790:
2787:
2781:
2777:
2772:
2769:
2763:
2759:
2755:
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2748:
2744:
2740:
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2623:
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2600:
2594:
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2023:
2016:
2011:
2004:
2000:
1995:
1988:
1983:
1976:
1971:
1969:
1961:
1956:
1949:
1944:
1929:
1925:
1918:
1911:
1902:
1897:
1890:
1883:
1878:
1871:
1870:Taniyama 1956
1866:
1862:
1849:
1843:
1836:
1832:
1824:
1822:
1821:Kenneth Ribet
1818:
1796:
1793:
1785:
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1167:
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1135:
1124:
1123:
1122:
1119:
1115:
1111:
1107:
1101:
1099:
1085:
1078:
1070:defined over
1069:
1062:of conductor
1056:
1054:
1049:
1045:
1014:
988:
985:
982:
979:
975:
971:
968:
964:
961:
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953:
949:
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933:
929:
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917:
913:
910:
905:
901:
897:
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889:
885:
881:
878:
875:
872:
866:
860:
853:
852:
851:
848:
844:
840:
836:
825:
824:are modular.
823:
818:
816:
812:
808:
804:
800:
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733:
729:
725:
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692:
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685:
681:
673:
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621:
618:
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612:
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593:
592:
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566:
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531:
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491:
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465:
461:
455:
451:
438:
435:
432:
428:
424:
418:
415:
412:
406:
399:
398:
397:
395:
392:-series is a
387:
371:
363:
361:
354:with integer
353:
345:
344:
325:
318:
314:
310:
307:
303:
299:
293:
289:
278:
275:
267:
256:
253:
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246:
242:
239:
235:
232:
228:
225: â
224:
220:
219:Find sources:
213:
209:
203:
202:
197:This section
195:
191:
186:
185:
177:
175:
171:
167:
163:
159:
155:
151:
147:
143:
142:modular forms
139:
135:
131:
127:
123:
119:
110:
107:
103:
99:
95:
92:
88:
84:
80:
77:
73:
69:
65:
62:
58:
55:
51:
48:
47:Number theory
45:
41:
33:
19:
3556:Prym variety
3530:Stable curve
3520:Hodge bundle
3510:ELSV formula
3312:Fermat curve
3269:Plane curves
3232:Higher genus
3207:Applications
3172:
3132:Modular form
3013:
2992:
2964:
2911:
2905:
2862:
2856:
2803:
2797:
2786:Shimura 1989
2779:
2775:
2757:
2754:Singh, Simon
2720:
2716:
2684:
2678:
2613:
2607:
2603:
2598:
2556:
2550:
2547:Mazur, Barry
2526:(1): iv+40,
2523:
2519:
2461:
2455:
2415:
2409:
2384:
2378:
2340:
2301:
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2261:
2255:
2251:
2243:Bibliography
2222:
2218:
2207:
2172:
2168:
2159:
2151:
2124:
2118:
2095:
2083:
2071:
2058:
2046:
2034:
2022:
2015:Diamond 1996
2010:
1994:
1982:
1955:
1943:
1932:. Retrieved
1927:
1923:
1910:
1889:
1877:
1865:
1841:
1835:
1817:Gerhard Frey
1813:
1749:
1655:
1592:
1585:
1577:
1571:
1563:
1559:
1553:
1549:
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1530:
1527:
1446:and satisfy
1441:
1426:
1422:
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1405:
1401:
1398:
1121:is given by
1117:
1113:
1109:
1105:
1102:
1083:
1076:
1068:rational map
1057:
1053:Andrew Wiles
1050:
1043:
1012:
1005:
846:
842:
838:
834:
831:
819:
815:number field
796:
784:
774:
763:
755:
744:
732:Goro Shimura
718:
688:
683:
653:
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645:
638:
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494:
487:
372:
369:
341:
323:
316:
309:coefficients
302:rational map
285:
270:
261:
251:
244:
237:
230:
218:
206:Please help
201:verification
198:
166:Fred Diamond
162:Brian Conrad
146:Andrew Wiles
129:
125:
121:
117:
115:
105:Consequences
87:Fred Diamond
83:Brian Conrad
61:Goro Shimura
3431:Polar curve
2865:: 149â156,
2853:Weil, André
2225:: 116â139.
2106:article of
2064:Zagier 1985
2003:Wiles 1995b
1999:Wiles 1995a
1661: [
1433:meromorphic
782:th powers,
759:John Coates
3646:Categories
3426:Dual curve
3054:Topics in
1987:Singh 1997
1975:Ribet 1990
1960:Serre 1987
1934:2022-11-08
1901:2003.08242
1858:References
264:March 2021
234:newspapers
3539:Morphisms
3287:Bitangent
3015:MathWorld
2999:EMS Press
2938:0003-486X
2916:CiteSeerX
2895:120553723
2879:0025-5831
2830:0003-486X
2808:CiteSeerX
2788:, p. 194)
2739:0024-6093
2701:0012-7094
2664:120614740
2648:0020-9910
2573:0002-9890
2532:0933-8268
2512:119132800
2496:0020-9910
2471:1310.7088
2432:0003-486X
2393:0002-9920
2320:0894-0347
2280:0894-0347
2191:0065-1036
1948:Frey 1986
1882:Weil 1967
1839:The case
1823:in 1987.
1761:Δ
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1380:⋯
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1179:−
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879:−
699:isogenous
672:cusp form
622:τ
616:π
551:∞
536:∑
444:∞
429:∑
343:conductor
311:from the
180:Statement
2756:(1997),
2104:Bourbaki
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503:is then
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1041:; thus
828:Example
777:coprime
705:History
386:-series
360:isogeny
352:newform
306:integer
288:theorem
248:scholar
3579:Acnode
3503:Moduli
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388:. The
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2466:arXiv
2436:JSTOR
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1896:arXiv
1848:Euler
1827:Notes
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1570:37 |
1557:with
728:NikkĆ
724:Tokyo
377:over
304:with
294:over
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