Birch and SwinnertonDyer conjecture
From Wikipedia, the free encyclopedia
Millennium Prize Problems 


In mathematics, the Birch and SwinnertonDyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. The conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute, which has offered a $1,000,000 prize for the first correct proof.^{[1]} It is named after mathematicians Bryan Birch and Peter SwinnertonDyer who developed the conjecture during the first half of the 1960s with the help of machine computation. As of 2016^{[update]}, only special cases of the conjecture have been proved.
The conjecture relates arithmetic data associated to an elliptic curve E over a number field K to the behaviour of the Hasse–Weil Lfunction L(E, s) of E at s = 1. More specifically, it is conjectured that the rank of the abelian group E(K) of points of E is the order of the zero of L(E, s) at s = 1, and the first nonzero coefficient in the Taylor expansion of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K (Wiles 2006).
Contents
Background[edit]
Mordell (1922) proved Mordell's theorem: the group of rational points on an elliptic curve has a finite basis. 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 infinite then some point in a finite basis must have infinite order. The number of independent basis points with infinite order is called the rank of the curve, and is an important invariant property of an elliptic curve.
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.
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) these cannot be generalised to handle all curves.
An Lfunction L(E, s) can be defined for an elliptic curve E by constructing an Euler product from the number of points on the curve modulo each prime p. This Lfunction is analogous to the Riemann zeta function and the Dirichlet Lseries that is defined for a binary quadratic form. It is a special case of a Hasse–Weil Lfunction.
The natural definition of L(E, s) only converges for values of s in the complex plane with Re(s) > 3/2. Helmut Hasse conjectured that L(E, s) could be extended by analytic continuation to the whole complex plane. This conjecture was first proved by Deuring (1941) for elliptic curves with complex multiplication. It was subsequently shown to be true for all elliptic curves over Q, as a consequence of the modularity theorem.
Finding rational points on a general elliptic curve is a difficult problem. Finding the points on an elliptic curve modulo a given prime p is conceptually straightforward, as there are only a finite number of possibilities to check. However, for large primes it is computationally intensive.
History[edit]
In the early 1960s Peter SwinnertonDyer used the EDSAC computer at the University of Cambridge Computer Laboratory to calculate the number of points modulo p (denoted by N_{p}) for a large number of primes p on elliptic curves whose rank was known. From these numerical results Birch & SwinnertonDyer (1965) conjectured that N_{p} for a curve E with rank r obeys an asymptotic law
where C is a constant.
Initially this was based on somewhat tenuous trends in graphical plots; this induced a measure of skepticism in J. W. S. Cassels (Birch's Ph.D. advisor).^{[2]} Over time the numerical evidence stacked up.
This in turn led them to make a general conjecture about the behaviour of a curve's Lfunction L(E, s) at s = 1, namely that it would have a zero of order r at this point. This was a farsighted conjecture for the time, given that the analytic continuation of L(E, s) there was only established for curves with complex multiplication, which were also the main source of numerical examples. (NB that the reciprocal of the Lfunction is from some points of view a more natural object of study; on occasion this means that one should consider poles rather than zeroes.)
The conjecture was subsequently extended to include the prediction of the precise leading Taylor coefficient of the Lfunction at s = 1. It is conjecturally given by
where the quantities on the right hand side are invariants of the curve, studied by Cassels, Tate, Shafarevich and others: these include the order of the torsion group, the order of the Tate–Shafarevich group, and the canonical heights of a basis of rational points (Wiles 2006).
Current status[edit]
The Birch and SwinnertonDyer conjecture has been proved only in special cases:
 Coates & Wiles (1977) proved that if E is a curve over a number field F with complex multiplication by an imaginary quadratic field K of class number 1, F = K or Q, and L(E, 1) is not 0 then E(F) is a finite group. This was extended to the case where F is any finite abelian extension of K by Arthaud (1978).
 Gross & Zagier (1986) showed that if a modular elliptic curve has a firstorder zero at s = 1 then it has a rational point of infinite order; see Gross–Zagier theorem.
 Kolyvagin (1989) showed that a modular elliptic curve E for which L(E, 1) is not zero has rank 0, and a modular elliptic curve E for which L(E, 1) has a firstorder zero at s = 1 has rank 1.
 Rubin (1991) showed that for elliptic curves defined over an imaginary quadratic field K with complex multiplication by K, if the Lseries of the elliptic curve was not zero at s = 1, then the ppart of the Tate–Shafarevich group had the order predicted by the Birch and SwinnertonDyer conjecture, for all primes p > 7.
 Breuil et al. (2001), extending work of Wiles (1995), proved that all elliptic curves defined over the rational numbers are modular, which extends results 2 and 3 to all elliptic curves over the rationals, and shows that the Lfunctions of all elliptic curves over Q are defined at s = 1.
Nothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for the truth of the conjecture.^{[3]}
Consequences[edit]
Much like the Riemann hypothesis, this conjecture has multiple consequences, including the following two:
 Let n be an odd squarefree integer. Assuming the Birch and SwinnertonDyer conjecture, n is the area of a right triangle with rational side lengths (a congruent number) if and only if the number of triplets of integers (x, y, z) satisfying 2x^{2} + y^{2} + 8z^{2} = n is twice the number of triplets satisfying 2x^{2} + y^{2} + 32z^{2} = n. This statement, due to Tunnell's theorem (Tunnell 1983), is related to the fact that n is a congruent number if and only if the elliptic curve y^{2} = x^{3} − n^{2}x has a rational point of infinite order (thus, under the Birch and SwinnertonDyer conjecture, its Lfunction has a zero at 1). The interest in this statement is that the condition is easily verified.^{[4]}
 In a different direction, certain analytic methods allow for an estimation of the order of zero in the center of the critical strip of families of Lfunctions. Admitting the BSD conjecture, these estimations correspond to information about the rank of families of elliptic curves in question. For example: suppose the generalized Riemann hypothesis and the BSD conjecture, the average rank of curves given by y^{2} = x^{3} + ax+ b is smaller than 2.^{[5]}
Notes[edit]
 ^ Birch and SwinnertonDyer Conjecture at Clay Mathematics Institute
 ^ Stewart, Ian (2013), Visions of Infinity: The Great Mathematical Problems, Basic Books, p. 253, ISBN 9780465022403,
Cassels was highly skeptical at first
.  ^ Cremona, John (2011). "Numerical evidence for the Birch and SwinnertonDyer Conjecture" (PDF). Talk at the BSD 50th anniversary conference, May 2011.
 ^ Koblitz, Neal (1993). Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics. 97 (2nd ed.). SpringerVerlag. ISBN 0387979662.
 ^ HeathBrown, D. R. (2004). "The Average Analytic Rank of Elliptic Curves". Duke Mathematical Journal. 122 (3): 591–623. doi:10.1215/S0012709404122353. MR 2057019.
References[edit]
 Arthaud, Nicole (1978). "On Birch and SwinnertonDyer's conjecture for elliptic curves with complex multiplication". Compositio Mathematica. 37 (2): 209–232. MR 504632.
 Bhargava, Manjul; Shankar, Arul (2015). "Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0". Annals of Mathematics. 181 (2): 587–621. arXiv:1007.0052. doi:10.4007/annals.2015.181.2.4.
 Birch, Bryan; SwinnertonDyer, Peter (1965). "Notes on Elliptic Curves (II)". J. Reine Angew. Math. 165 (218): 79–108. doi:10.1515/crll.1965.218.79.
 Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001). "On the Modularity of Elliptic Curves over Q: Wild 3Adic Exercises". Journal of the American Mathematical Society. 14 (4): 843–939. doi:10.1090/S0894034701003708.
 Coates, J.H.; Greenberg, R.; Ribet, K.A.; Rubin, K. (1999). Arithmetic Theory of Elliptic Curves. Lecture Notes in Mathematics. 1716. SpringerVerlag. ISBN 3540665463.
 Coates, J.; Wiles, A. (1977). "On the conjecture of Birch and SwinnertonDyer". Inventiones Mathematicae. 39 (3): 223–251. doi:10.1007/BF01402975. Zbl 0359.14009.
 Deuring, Max (1941). "Die Typen der Multiplikatorenringe elliptischer Funktionenkörper". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 14 (1): 197–272. doi:10.1007/BF02940746.
 Dokchitser, Tim; Dokchitser, Vladimir (2010). "On the BirchSwinnertonDyer quotients modulo squares". Annals of Mathematics. 172 (1): 567–596. doi:10.4007/annals.2010.172.567. MR 2680426.
 Gross, Benedict H.; Zagier, Don B. (1986). "Heegner points and derivatives of Lseries". Inventiones Mathematicae. 84 (2): 225–320. doi:10.1007/BF01388809. MR 0833192.
 Kolyvagin, Victor (1989). "Finiteness of E(Q) and X(E, Q) for a class of Weil curves". Math. USSR Izv. 32: 523–541. doi:10.1070/im1989v032n03abeh000779.
 Mordell, Louis (1922). "On the rational solutions of the indeterminate equations of the third and fourth degrees". Proc. Cambridge Phil. Soc. 21: 179–192.
 Nekovář, Jan (2009). "On the parity of ranks of Selmer groups IV". Compositio Mathematica. 145 (6): 1351–1359. doi:10.1112/S0010437X09003959.
 Rubin, Karl (1991). "The 'main conjectures' of Iwasawa theory for imaginary quadratic fields". Inventiones Mathematicae. 103 (1): 25–68. doi:10.1007/BF01239508. Zbl 0737.11030.
 Skinner, Christopher; Urban, Éric (2014). "The Iwasawa main conjectures for GL_{2}". Inventiones Mathematicae. 195 (1): 1–277. doi:10.1007/s0022201304481.
 Tunnell, Jerrold B. (1983). "A classical Diophantine problem and modular forms of weight 3/2". Inventiones Mathematicae. 72 (2): 323–334. doi:10.1007/BF01389327. Zbl 0515.10013.
 Wiles, Andrew (1995). "Modular elliptic curves and Fermat's last theorem". Annals of Mathematics. Second Series. 141 (3): 443–551. ISSN 0003486X. JSTOR 2118559. MR 1333035.
 Wiles, Andrew (2006). "The Birch and SwinnertonDyer conjecture" (PDF). In Carlson, James; Jaffe, Arthur; Wiles, Andrew. The Millennium prize problems. American Mathematical Society. pp. 31–44. ISBN 9780821836798. MR 2238272.
External links[edit]
 "Birch and SwinnertonDyer Conjecture". PlanetMath.
 The Birch and SwinnertonDyer Conjecture: An Interview with Professor Henri Darmon by Agnes F. Beaudry