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HunterA
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Best ResponseYou've already chosen the best response.0The Birch and SwinnertonDyer Conjecture by A. Wiles A polynomial relation f(x, y) = 0 in two variables deﬁnes a curve C0. If the coeﬃcients of the polynomial are rational numbers then one can ask for solutions of the equation f(x, y) = 0 with x, y ∈ Q,in other words for rational points on the curve. The set of all such points is denoted C0(Q). If we consider a nonsingular projective model C of the curve then topologically C is classiﬁed by its genus,and we call this the genus of C0 also. Note that C0(Q) and C(Q) are either both ﬁnite or both inﬁnite. Mordell conjectured, and in 1983 Faltings proved,the following deep result Theorem [F1]. If the genus of C0 is greater than or equal to two, then C0(Q) is ﬁnite. As yet the proof is not eﬀective so that one does not possess an algorithm for ﬁnding the rational points. (There is an eﬀective bound on the number of solutions but that does not help much with ﬁnding them.) The case of genus zero curves is much easier and was treated in detail by Hilbert and Hurwitz [HH]. They explicitly reduce to the cases of linear and quadratic equations. The former case is easy and the latter is resolved by the criterion of Legendre. In particular for a nonsingular projective model C we ﬁnd that C(Q) is nonempty if and only if C has padic points for all primes p,and this in turn is determined by a ﬁnite number of congruences. If C(Q) is nonempty then C is parametrized by rational functions and there are inﬁnitely many rational points. The most elusive case is that of genus 1. There may or may not be rational solutions and no method is known for determining which is the case for any given curve. Moreover when there are rational solutions there may or may not be inﬁnitely many. If a nonsingular projective model C has a rational point then C(Q) has a natural structure as an abeliangroup with this point as the identity element. In this case we call C an elliptic curve over Q. (For a history of the development of this idea see [S]). In 1922 Mordell ([M]) proved that this group is ﬁnitely generated,thus fulﬁlling an implicit assumption of Poincar´e. Theorem. If C is an elliptic curve over Q then C(Q) Z r ⊕ C(Q) tors for some integer r ≥ 0, where C(Q) tors is a ﬁnite abelian group. The integer r is called the rank of C. It is zero if and only if C(Q) is ﬁnite. We can ﬁnd an aﬃne model for an elliptic curve over Q in Weierstrass form C: y 2 = x 3 + ax + b with a, b ∈ Z. We let ∆ denote the discriminant of the cubic and set Np := #{solutions of y 2 ≡ x 3 + ax + b mod p} ap := p − Np. Then we can deﬁne the incomplete Lseries of C (incomplete because we omit the Euler factors for primes p2∆) by L(C, s) := p2∆ (1 − app−s + p 1−2s )−1 . We view this as a function of the complex variable s and this Euler product is then known to converge for Re(s) > 3/2. A conjecture going back to Hasse (see the commentary on 1952(d) in [We1]) predicted that L(C, s) should have a holomorphic continuation as a function of s to the whole complex plane. This has now been proved ([W],[TW],[BCDT]). We can now state the millenium prize problem: Conjecture (Birch and SwinnertonDyer). The Taylor expansion of L(C, s) at s = 1 has the form L(C, s) = c(s − 1) r + higher order terms with c = 0 and r = rank(C(Q)). In particular this conjecture asserts that L(C, 1) = 0 ⇔ C(Q) is inﬁnite. 2Remarks. 1. There is a reﬁned version of this conjecture. In this version one has to deﬁne Euler factors at primes p2∆ to obtain the completed Lseries, L ∗ (C, s). The conjecture then predicts that L ∗ (C, s) ∼ c ∗ (s − 1) r with c ∗ = XCR∞w∞ p2∆ wp/C(Q) tors  2 . Here XC is the order of the TateShafarevich group of the elliptic curve C, a group which is not known in general to be ﬁnite although it is conjectured to be so. It counts the number of equivalence classes of homogeneous spaces of C which have points in all local ﬁelds. The term R∞ is an r × r determinant whose matrix entries are given by a height pairing applied to a system of generators of C(Q)/C(Q) tors . The wp’s are elementary local factors and w∞ is a simple multiple of the real period of C. For a precise deﬁnition of these factors see [T1] or [T3]. It is hoped that a proof of the conjecture would also yield a proof of the ﬁniteness of XC. 2. The conjecture can also be stated over any number ﬁeld as well as for abelian varieties,see [T1]. Since the original conjecture was stated much more elaborate conjectures concerning special values of Lfunctions have appeared,due to Tate, Lichtenbaum,Deligne,Bloch,Beilinson and others,see [T2],[Bl] and [Be]. In particular these relate the ranks of groups of algebraic cycles to the order of vanishing (or the order of poles) of suitable Lfunctions. 3. There is an analogous conjecture for elliptic curves over function ﬁelds. It has been proved in this case by M. Artin and J. Tate [T1] that the Lseries has a zero of order at least r,but the conjecture itself remains unproved. In the function ﬁeld case it is now known to be equivalent to the ﬁniteness of the TateShafarevich group,[T1],[Mi] X corollary 9.7. 4. A proof of the conjecture in the stronger form would give an eﬀective means of ﬁnding generators for the group of rational points. Actually one only needs the integrality of the term XC in the expression for L ∗ (C, s) above, without any interpretation as the order of the TateShafarevich group. This was shown by Manin [Ma] subject to the condition that the elliptic curves were modular,a property which is now known for all elliptic curves by [W],[TW],[BCDT]. (A modular elliptic curve is one which occurs as a factor of the Jacobian of a modular curve.) Early History Problems on curves of genus 1 feature prominently in Diophantus’ Arith 3metica. It is easy to see that a straight line meets an elliptic curve in three points (counting multiplicity) so that if two of the points are rational then so is the third. 1 In particular if a tangent is taken to a rational point then it meets the curve again in a rational point. Diophantus implicitly uses this method to obtain a second solution from a ﬁrst. However he does not iterate this process and it is Fermat who ﬁrst realizes that one can sometimes obtain inﬁnitely many solutions in this way. Fermat also introduced a method of ‘descent’ which sometimes permits one to show that the number of solutions is ﬁnite or even zero. One very old problem concerned with rational points on elliptic curves is the congruent number problem. One way of stating it is to ask which rational integers can occur as the areas of rightangled triangles with rational length sides. Such integers are called congruent numbers. For example,Fibonacci was challenged in the court of Frederic II with the problem for n = 5 and he succeeded in ﬁnding such a triangle. He claimed moreover that there was no such triangle for n = 1 but the proof was fallacious and the ﬁrst correct proof was given by Fermat. The problem dates back to Arab manuscripts of the 10 th century (for the history see [We2] chapter 1, §VII and [Di] chapter XVI). It is closely related to the problem of determining the rational points on the curve Cn: y 2 = x 3 − n 2 x. Indeed Cn(Q) is inﬁnite ⇐⇒ n is a congruent number Assuming the Birch and SwinnertonDyer conjecture (or even the weaker statement that Cn(Q) is inﬁnite ⇔ L(Cn, 1) = 0) one can show that any n ≡ 5, 6, 7 mod 8 is a congruent number and moreover Tunnell has shown,again assuming the conjecture,that for n odd and squarefree n is a congruent number ⇐⇒ #{x,y, z ∈ Z: 2x 2 + y 2 + 8z 2 = n} = 2 × #{x, y, z ∈ Z: 2x 2 + y 2 + 32z 2 = n}, with a similar criterion if n is even ([Tu]). Tunnell proved the implication left to right unconditionally with the help of the main theorem of [CW] described below. Recent History It was the 1901 paper of Poincar´e [P] which started the modern interest in the theory of rational points on curves and which ﬁrst raised questions about the minimal 1 This was apparently ﬁrst explicitly pointed out by Newton. 4number of generators of C(Q). The conjecture itself was ﬁrst stated in the form we have given in the early 1960’s (see [BS]). In the intervening years the theory of Lfunctions of elliptic curves (and other varieties) had been developed by a number of authors but the conjecture was the ﬁrst link between the Lfunction and the structure of C(Q). It was found experimentally using one of the early computers EDSAC at Cambridge. The ﬁrst general result proved was for elliptic curves with complex multiplication. (The curves with complex multiplication fall into a ﬁnite number of families including {y 2 = x 3 − Dx} and {y 2 = x 3 − k} for varying D, k = 0.) This theorem was proved in 1976 and is due to Coates and Wiles [CW]. It states that if C is a curve with complex multiplication and L(C, 1) = 0 then C(Q) is ﬁnite. In 1983 Gross and Zagier showed that if C is a modular elliptic curve and L(C, 1) = 0 but L (C, 1) = 0,then an earlier construction of Heegner actually gives a rational point of inﬁnite order. Using new ideas together with this result, Kolyvagin showed in 1990 that for modular elliptic curves,if L(C, 1) = 0 then r = 0 and if L(C, 1) = 0 but L (C, 1) = 0 then r = 1. In the former case Kolyvagin needed an analytic hypothesis which was conﬁrmed soon afterwards; see [Da] for the history of this and for further references. Finally as noted in remark 4 above it is now known that all elliptic curves over Q are modular so that we now have the following result: Theorem. If L(C, s) ∼ c(s − 1)m with c = 0 and m = 0 or 1 then the conjecture holds. In the cases where m = 0 or 1 some more precise results on c (which of course depends on the curve) are known by work of Rubin and Kolyvagin. Rational Points on Higher Dimensional Varieties We began by discussing the diophantine properties of curves,and we have seen that the problem of giving a criterion for whether C(Q) is ﬁnite or not is an issue only for curves of genus 1. Moreover according to the conjecture above, in the case of genus 1, C(Q) is ﬁnite if and only if L(C, 1) = 0. In higher dimensions if V is an algebraic variety,it is conjectured (see [L]) that if we remove from V (the closure of) all subvarieties which are images of P 1 or of abelian varieties then the remaining open variety W should have the property that W(Q) is ﬁnite. This has been proved in the case where V is itself a subvariety of an abelian variety by Faltings 5[F2]. This suggests that to ﬁnd inﬁnitely many points on V one should look for rational curves or abelian varieties in V . In the latter case we can hope to use methods related to the Birch and SwinnertonDyer conjecture to ﬁnd rational points on the abelian variety. As an example of this consider the conjecture of Euler from 1769 that x 4 + y 4 + z 4 = t 4 has no nontrivial solutions. By ﬁnding a curve of genus 1 on the surface and a point of inﬁnite order on this curve,Elkies [E] found the solution, 2682440 4 + 15365639 4 + 18796760 4 = 20615673 4 His argument shows that there are inﬁnitely many solutions to Euler’s equation. In conclusion,although there has been some success in the last ﬁfty years in limiting the number of rational points on varieties,there are still almost no methods for ﬁnding such points. It is to be hoped that a proof of the Birch and SwinnertonDyer conjecture will give some insight concerning this general problem. References [BCDT] Breuil,C.,Conrad,B.,Diamond,F.,Taylor,R., On the modularity of elliptic curves over Q: wild 3adic exercises,preprint. [Be] Beilinson,A.,Notes on absolute Hodge cohomology, Applications of algebraic Ktheory to algebraic geometry and number theory,Contemp. Math. 55 (1986), 35–68. [Bl] Bloch,S., Height pairings for algebraic cycles,J. Pure Appl. Algebra 34 (1984) 119–145. [BS] Birch,B.,SwinnertonDyer,H., Notes on elliptic curves II,Journ. reine u. angewandte Math. 218 (1965),79–108. [CW] Coates,J.,Wiles,A., On the conjecture of Birch and SwinnertonDyer,Invent. Math. 39,223–251 (1977). [Da] Darmon,H.,Wiles’ theorem and the arithmetic of elliptic curves,in Modular forms and Fermat’s Last Theorem pp. 549–569,Springer (1997). [Di] wingspanson,L., History of the theory of numbers vol. II. 6[E] Elkies,N., On A4 + B4 + C 4 = D4 ,Math. Comput. 51,No. 184 (1988) pp. 825–835. [F1] Faltings,G., Endlichkeits¨atze f¨ur abelsche Variet¨aten ¨uber Zahlk¨orpern,Invent. Math. 73,No. 3 (1983) pp. 549–576. [F2] Faltings,G.,The general case of S. Lang’s conjecture, Perspec. Math.,vol. 15, Academic Press,Boston (1994). [GZ] Gross,B.,Zagier,D., Heegner Points and Derivatives of Lseries,Invent. Math. 84 (1986) pp. 225–320. [HH] Hilbert,D.,Hurwitz,A., Uber die diophantischen Gleichungen von Geschlect Null; Acta Mathematica 14 (1890),pp. 217224. [K] Kolyvagin,V., Finiteness of E(Q) and X(E, Q) for a class of Weil curves,Math. USSR,Izv. 32 (1989) pp. 523–541. [L] Lang,S.,Number Theory III, Encyclopædia of Mathematical Sciences,vol. 60, SpringerVerlag,Heidelberg (1991). [M] Mordell, On the rational solutions of the indeterminate equations of the third and fourth degrees,Proc. Cambridge Phil. Soc. 21 (192223),179–192. [Ma] Manin,Y., Cyclotomic Fields and Modular Curves,Russian Mathematical Surveys vol. 26,no. 6,pp. 7–78. (1971). [Mi] Milne,J., Arithmetic Duality Theorems,Academic Press,Inc. (1986). [P] Poincar´e,H., Sur les Propri´et´es Arithm´etiques des Courbes Alg´ebriques,Jour. Math. Pures Appl. 7,Ser. 5 (1901). [S] Schappacher,N., D´eveloppement de la loi de groupe sur une cubique; Seminaire de Th´eorie des Nombres,Paris 1988/89,Progress in Mathematics 91 (1991),pp. 159184. [T1] Tate,J., On the conjectures of Birch and SwinnertonDyer and a geometric analog,Seminaire Bourbaki 1965/66,no. 306. [T2] Tate,J.,Algebraic Cycles and Poles of Zeta Functions,in Arithmetical Algebraic Geometry,Proceedings of a conference at Purdue University (1965). [T3] Tate,J., The Arithmetic of Elliptic Curves,Inv. Math. 23,pp. 179–206 (1974). [Tu] Tunnell,J., A classical diophantine problem and modular forms of weight 3/2, Invent. Math. 72 (1983) pp. 323–334. [TW] Taylor,R.,Wiles,A., Ringtheoretic properties of certain Hecke algebras,Ann. of Math. vol. 141,no 3 (1995) 553–572. 7[W] Wiles,A., Modular Elliptic Curves and Fermat’s Last Theorem,Ann. Math. 141 (1995) pp. 443–551. [We1] Weil,A., Collected Papers,Vol. II. [We2] Weil,A., Basic Number Theory,Birkha¨user,Boston (1984). 8

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