Birch and Swinnerton-Dyer conjecture

<h2 id="background">Background</h2>
Mordell (1922) proved <a href="/facts/Mordell%2527s_theorem/3ko0yKew">Mordell's theorem</a>: the group of <a href="/facts/Rational_point/j5v4cxuE">rational points</a> on an elliptic curve has a finite <a href="/facts/Generating_set_of_a_group/B9q5lnFY">basis</a>. 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 <a href="/facts/Infinite_set/9rTN0xGH">infinite</a> then some point in a finite basis must have infinite <a href="/facts/Order_(group_theory)/Cl3tKGFg">order</a>. The number of independent basis points with infinite order is called the <a href="/facts/Rank_of_an_abelian_group/zo0y8Mj4">rank</a> of the curve, and is an important <a href="/facts/Invariant_(mathematics)/9TeNN2ed">invariant</a> 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) it is unknown if these methods handle all curves.
An L-function L(E, s) can be defined for an elliptic curve E by constructing an <a href="/facts/Euler_product/Lvsv4U3M">Euler product</a> from the number of points on the curve modulo each <a href="/facts/Prime_number/L20FLkgn">prime</a> p. This L-function is analogous to the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> and the <a href="/facts/Dirichlet_L-series/my8hc6Qy">Dirichlet L-series</a> that is defined for a binary <a href="/facts/Quadratic_form/l2n1jXPe">quadratic form</a>. It is a special case of a <a href="/facts/Hasse%25E2%2580%2593Weil_L-function/1THWBhB2">Hasse–Weil L-function</a>.
The natural definition of L(E, s) only converges for values of s in the complex plane with Re(s) > 3/2. <a href="/facts/Helmut_Hasse/WFtCX2jI">Helmut Hasse</a> conjectured that L(E, s) could be extended by <a href="/facts/Analytic_continuation/aBCuEgvt">analytic continuation</a> to the whole complex plane. This conjecture was first proved by Deuring (1941) for elliptic curves with <a href="/facts/Complex_multiplication/YQNcbNXP">complex multiplication</a>. It was subsequently shown to be true for all elliptic curves over Q, as a consequence of the <a href="/facts/Modularity_theorem/aI3HRxwj">modularity theorem</a> in 2001.
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.

<h2 id="history">History</h2>
In the early 1960s <a href="/facts/Peter_Swinnerton-Dyer/D66SILlt">Peter Swinnerton-Dyer</a> used the <a href="/facts/EDSAC_2/Q2vqMPul">EDSAC-2</a> computer at the <a href="/facts/University_of_Cambridge_Computer_Laboratory/Dd9unFtw">University of Cambridge Computer Laboratory</a> to calculate the number of points modulo p (denoted by Np) for a large number of primes p on elliptic curves whose rank was known. From these numerical results Birch & Swinnerton-Dyer (1965) conjectured that Np for a curve E with rank r obeys an asymptotic law

∏
          
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            ≤
            x
          
        
        
          
            
              N
              
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    {\displaystyle \prod _{p\leq x}{\frac {N_{p}}{p}}\approx C\log(x)^{r}{\mbox{ as }}x\rightarrow \infty }

where C is a constant.
Initially, this was based on somewhat tenuous trends in graphical plots; this induced a measure of skepticism in <a href="/facts/J._W._S._Cassels/AC33eVoA">J. W. S. Cassels</a> (Birch's Ph.D. advisor).<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> Over time the numerical evidence stacked up.
This in turn led them to make a general conjecture about the behavior of a curve's L-function L(E, s) at s = 1, namely that it would have a zero of order r at this point. This was a far-sighted conjecture for the time, given that the analytic continuation of L(E, s) was only established for curves with complex multiplication, which were also the main source of numerical examples. (NB that the <a href="/facts/Reciprocal_(mathematics)/yczdYyCw">reciprocal</a> of the L-function is from some points of view a more natural object of study; on occasion, this means that one should consider poles rather than zeroes.)
The conjecture was subsequently extended to include the prediction of the precise leading <a href="/facts/Taylor_coefficient/M0aTuftH">Taylor coefficient</a> of the L-function at s = 1. It is conjecturally given by<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

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    {\displaystyle {\frac {L^{(r)}(E,1)}{r!}}={\frac {\#\mathrm {Sha} (E)\Omega _{E}R_{E}\prod _{p|N}c_{p}}{(\#E_{\mathrm {tor} })^{2}}}}

where the quantities on the right-hand side are invariants of the curve, studied by Cassels, <a href="/facts/John_Tate_(mathematician)/nBntLCG0">Tate</a>, <a href="/facts/Igor_Shafarevich/n6UINBX1">Shafarevich</a> and others (Wiles 2006):

 
 
 
 #
 
 E
 
 
 t
 o
 r
 
 
 
 
 
 {\displaystyle \#E_{\mathrm {tor} }}
 
 is the order of the <a href="/facts/Torsion_group/TEvLhDtm">torsion group</a>,

 
 
 
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 h
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 E
 )
 =
 
 
 {\displaystyle \#\mathrm {Sha} (E)=}
 
#Ш(E) is the order of the <a href="/facts/Tate%25E2%2580%2593Shafarevich_group/yCDadEcM">Tate–Shafarevich group</a>,

 
 
 
 
 Ω
 
 E
 
 
 
 
 {\displaystyle \Omega _{E}}
 
 is the real period of E multiplied by the number of connected components of E,

 
 
 
 
 R
 
 E
 
 
 
 
 {\displaystyle R_{E}}
 
 is the <a href="/facts/N%25C3%25A9ron%25E2%2580%2593Tate_height/IpAkTcqU">regulator</a> of E which is defined via the <a href="/facts/Canonical_height/IpAkTcqU">canonical heights</a> of a basis of rational points,

 
 
 
 
 c
 
 p
 
 
 
 
 {\displaystyle c_{p}}
 
 is the <a href="/facts/Tamagawa_number/NRYVaI7S">Tamagawa number</a> of E at a prime p dividing the conductor N of E. It can be found by <a href="/facts/Tate%2527s_algorithm/dVePyx1L">Tate's algorithm</a>.
At the time of the inception of the conjecture little was known, not even the well-definedness of the left side (referred to as analytic) or the right side (referred to as algebraic) of this equation. <a href="/facts/John_Tate_(mathematician)/nBntLCG0">John Tate</a> expressed this in 1974 in a famous quote.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>: 198 

<blockquote>
This remarkable conjecture relates the behavior of a function 
 
 
 
 L
 
 
 {\displaystyle L}
 
 at a point where it is not at present known to be defined to the order of a group Ш which is not known to be finite!

</blockquote>
By the <a href="/facts/Modularity_theorem/aI3HRxwj">modularity theorem</a> proved in 2001 for elliptic curves over 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 the left side is now known to be well-defined and the finiteness of Ш(E) is known when additionally the analytic rank is at most 1, i.e., if 
 
 
 
 L
 (
 E
 ,
 s
 )
 
 
 {\displaystyle L(E,s)}
 
 vanishes at most to order 1 at 
 
 
 
 s
 =
 1
 
 
 {\displaystyle s=1}
 
. Both parts remain open.

<h2 id="current-status">Current status</h2>

The Birch and Swinnerton-Dyer conjecture has been proved only in special cases:

<ol><li>Coates & Wiles (1977) proved that if E is a curve over a number field F with complex multiplication by an <a href="/facts/Imaginary_quadratic_field/dUnSDN4e">imaginary quadratic field</a> K of <a href="/facts/Class_number_(number_theory)/LLn5Pwj1">class number</a> 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 <a href="/facts/Abelian_extension/vIdW4Hnv">abelian extension</a> of K by Arthaud (1978).</li>
<li>Gross & Zagier (1986) showed that if a <a href="/facts/Modular_elliptic_curve/7FVv9htb">modular elliptic curve</a> has a first-order zero at s = 1 then it has a rational point of infinite order; see <a href="/facts/Gross%25E2%2580%2593Zagier_theorem/WYRrdqLj">Gross–Zagier theorem</a>.</li>
<li>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 first-order zero at s = 1 has rank 1.</li>
<li>Rubin (1991) showed that for elliptic curves defined over an imaginary quadratic field K with complex multiplication by K, if the L-series of the elliptic curve was not zero at s = 1, then the p-part of the Tate–Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes p > 7.</li>
<li>Breuil et al. (2001), extending work of Wiles (1995), proved that <a href="/facts/Modularity_theorem/aI3HRxwj">all elliptic curves defined over the rational numbers are modular</a>, which extends results #2 and #3 to all elliptic curves over the rationals, and shows that the L-functions of all elliptic curves over Q are defined at s = 1.</li>
<li>Bhargava & Shankar (2015) proved that the average rank of the Mordell–Weil group of an elliptic curve over Q is bounded above by 7/6. Combining this with the p-parity theorem of Nekovář (2009) and Dokchitser & Dokchitser (2010) and with the proof of the <a href="/facts/Main_conjecture_of_Iwasawa_theory/WfPLvMTT">main conjecture of Iwasawa theory</a> for GL(2) by Skinner & Urban (2014), they conclude that a positive proportion of elliptic curves over Q have analytic rank zero, and hence, by Kolyvagin (1989), satisfy the Birch and Swinnerton-Dyer conjecture.</li></ol>
There are currently no proofs involving curves with a rank greater than 1.
There is extensive numerical evidence for the truth of the conjecture.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h2 id="consequences">Consequences</h2>
Much like the <a href="/facts/Riemann_hypothesis/Td8Jq5r0">Riemann hypothesis</a>, this conjecture has multiple consequences, including the following two:

<ul><li>Let n be an odd <a href="/facts/Square-free/mJhGWeM6">square-free</a> integer. Assuming the Birch and Swinnerton-Dyer conjecture, n is the area of a right triangle with rational side lengths (a <a href="/facts/Congruent_number/BGL6BjHl">congruent number</a>) if and only if the number of triplets of integers (x, y, z) satisfying 2x2 + y2 + 8z2 = n is twice the number of triplets satisfying 2x2 + y2 + 32z2 = n. This statement, due to <a href="/facts/Tunnell%2527s_theorem/veEoQNnE">Tunnell's theorem</a> (Tunnell 1983), is related to the fact that n is a congruent number if and only if the elliptic curve y2 = x3 − n2x has a rational point of infinite order (thus, under the Birch and Swinnerton-Dyer conjecture, its L-function has a zero at 1). The interest in this statement is that the condition is easily verified.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li>
<li>In a different direction, certain analytic methods allow for an estimation of the order of zero in the center of the <a href="/facts/Critical_strip/MnfRu7lY">critical strip</a> of families of L-functions. Admitting the BSD conjecture, these estimations correspond to information about the rank of families of elliptic curves in question. For example: suppose the <a href="/facts/Generalized_Riemann_hypothesis/17nLmBbu">generalized Riemann hypothesis</a> and the BSD conjecture, the average rank of curves given by y2 = x3 + ax+ b is smaller than 2.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li>
<li>Because of the existence of the functional equation of the L-function of an elliptic curve, BSD allows us to calculate the parity of the rank of an elliptic curve. This is a conjecture in its own right called the parity conjecture, and it relates the parity of the rank of an elliptic curve to its global root number. This leads to many explicit arithmetic phenomena which are yet to be proved unconditionally. For instance:
<ul><li>Every positive integer n ≡ 5, 6 or 7 (mod 8) is a <a href="/facts/Congruent_number/BGL6BjHl">congruent number</a>.</li>
<li>The elliptic curve given by y2 = x3 + ax + b where a ≡ b (mod 2) has infinitely many solutions over 
 
 
 
 
 Q
 
 (
 
 ζ
 
 8
 
 
 )
 
 
 {\displaystyle \mathbb {Q} (\zeta _{8})}
 
.</li>
<li>Every positive rational number d can be written in the form d = s2(t3 – 91t – 182) for s and t in 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
.</li>
<li>For every rational number t, the elliptic curve given by y2 = x(x2 – 49(1 + t4)2) has rank at least 1.</li>
<li>There are many more examples for elliptic curves over number fields.</li></ul></li></ul>
<h2 id="generalizations">Generalizations</h2>
There is a version of this conjecture for general <a href="/facts/Abelian_varieties/dGSDVruG">abelian varieties</a> over number fields. A version for abelian varieties over 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 is the following:<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>: 462

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    {\displaystyle \lim _{s\to 1}{\frac {L(A/\mathbb {Q} ,s)}{(s-1)^{r}}}={\frac {\#\mathrm {Sha} (A)\Omega _{A}R_{A}\prod _{p|N}c_{p}}{\#A(\mathbb {Q} )_{\text{tors}}\cdot \#{\hat {A}}(\mathbb {Q} )_{\text{tors}}}}.}

All of the terms have the same meaning as for elliptic curves, except that the square of the order of the torsion needs to be replaced by the product 
 
 
 
 #
 A
 (
 
 Q
 
 
 )
 
 tors
 
 
 ⋅
 #
 
 
 
 A
 ^
 
 
 
 (
 
 Q
 
 
 )
 
 tors
 
 
 
 
 {\displaystyle \#A(\mathbb {Q} )_{\text{tors}}\cdot \#{\hat {A}}(\mathbb {Q} )_{\text{tors}}}
 
 involving the <a href="/facts/Dual_abelian_variety/xRSRm5KU">dual abelian variety</a> 
 
 
 
 
 
 
 A
 ^
 
 
 
 
 
 {\displaystyle {\hat {A}}}
 
. Elliptic curves as 1-dimensional abelian varieties are their own duals, i.e. 
 
 
 
 
 
 
 E
 ^
 
 
 
 =
 E
 
 
 {\displaystyle {\hat {E}}=E}
 
, which simplifies the statement of the BSD conjecture. The regulator 
 
 
 
 
 R
 
 A
 
 
 
 
 {\displaystyle R_{A}}
 
 needs to be understood for the pairing between a basis for the free parts of 
 
 
 
 A
 (
 
 Q
 
 )
 
 
 {\displaystyle A(\mathbb {Q} )}
 
 and 
 
 
 
 
 
 
 A
 ^
 
 
 
 (
 
 Q
 
 )
 
 
 {\displaystyle {\hat {A}}(\mathbb {Q} )}
 
 relative to the Poincare bundle on the product 
 
 
 
 A
 ×
 
 
 
 A
 ^
 
 
 
 
 
 {\displaystyle A\times {\hat {A}}}
 
.
The rank-one Birch-Swinnerton-Dyer conjecture for modular elliptic curves and modular abelian varieties of <a href="/facts/General_linear_group/B54gNILS">GL(2)</a>-type over <a href="/facts/Totally_real_number_field/4rulk9kd">totally real number fields</a> was proved by <a href="/facts/Shou-Wu_Zhang/mwkykgLj">Shou-Wu Zhang</a> in 2001.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>
Another generalization is given by the <a href="/facts/Bloch%25E2%2580%2593Kato_conjecture_(L-functions)/enVGIRfP">Bloch-Kato conjecture</a>.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

<h2 id="notes">Notes</h2>

<ul><li>Arthaud, Nicole (1978). "On Birch and Swinnerton-Dyer's conjecture for elliptic curves with complex multiplication". <a href="/facts/Compositio_Mathematica/Igc7zupB">Compositio Mathematica</a>. 37 (2): 209–232. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0504632">0504632</a>.</li>
<li><a href="/facts/Manjul_Bhargava/DPSrSaXE">Bhargava, Manjul</a>; <a href="/facts/Arul_Shankar/TkzFrSl0">Shankar, Arul</a> (2015). "Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0". <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>. 181 (2): 587–621. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1007.0052">1007.0052</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4007%2Fannals.2015.181.2.4">10.4007/annals.2015.181.2.4</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:1456959">1456959</a>.</li>
<li><a href="/facts/Bryan_John_Birch/QVWLhz0w">Birch, Bryan</a>; <a href="/facts/Peter_Swinnerton-Dyer/D66SILlt">Swinnerton-Dyer, Peter</a> (1965). "Notes on Elliptic Curves (II)". <a href="/facts/Journal_f%25C3%25BCr_die_reine_und_angewandte_Mathematik/YCcRSaAQ">J. Reine Angew. Math.</a> 165 (218): 79–108. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2Fcrll.1965.218.79">10.1515/crll.1965.218.79</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122531425">122531425</a>.</li>
<li><a href="/facts/Christophe_Breuil/IzljeQCV">Breuil, Christophe</a>; <a href="/facts/Brian_Conrad/uR4ARcLT">Conrad, Brian</a>; <a href="/facts/Fred_Diamond/fs32yCfw">Diamond, Fred</a>; <a href="/facts/Richard_Taylor_(mathematician)/crBWfG0r">Taylor, Richard</a> (2001). <a href="https://doi.org/10.1090%2FS0894-0347-01-00370-8">"On the Modularity of Elliptic Curves over Q: Wild 3-Adic Exercises"</a>. <a href="/facts/Journal_of_the_American_Mathematical_Society/qR98drCm">Journal of the American Mathematical Society</a>. 14 (4): 843–939. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0894-0347-01-00370-8">10.1090/S0894-0347-01-00370-8</a>.</li>
<li><a href="/facts/John_Coates_(mathematician)/P7XvMrHY">Coates, J.H.</a>; Greenberg, R.; <a href="/facts/Kenneth_Alan_Ribet/6hU9bzw8">Ribet, K.A.</a>; <a href="/facts/Karl_Rubin/3xhVe6yE">Rubin, K.</a> (1999). Arithmetic Theory of Elliptic Curves. Lecture Notes in Mathematics. Vol. 1716. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-66546-3.</li>
<li><a href="/facts/John_Coates_(mathematician)/P7XvMrHY">Coates, J.</a>; <a href="/facts/Andrew_Wiles/bdc3k3pQ">Wiles, A.</a> (1977). "On the conjecture of Birch and Swinnerton-Dyer". <a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a>. 39 (3): 223–251. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1977InMat..39..223C">1977InMat..39..223C</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01402975">10.1007/BF01402975</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:189832636">189832636</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0359.14009">0359.14009</a>.</li>
<li><a href="/facts/Max_Deuring/jtUbK0Kb">Deuring, Max</a> (1941). "Die Typen der Multiplikatorenringe elliptischer Funktionenkörper". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 14 (1): 197–272. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02940746">10.1007/BF02940746</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:124821516">124821516</a>.</li>
<li>Dokchitser, Tim; Dokchitser, Vladimir (2010). "On the Birch–Swinnerton-Dyer quotients modulo squares". <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>. 172 (1): 567–596. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0610290">math/0610290</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4007%2Fannals.2010.172.567">10.4007/annals.2010.172.567</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2680426">2680426</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:9479748">9479748</a>.</li>
<li><a href="/facts/Benedict_Gross/XS6uAajW">Gross, Benedict H.</a>; <a href="/facts/Don_Zagier/8sRgRUUj">Zagier, Don B.</a> (1986). "Heegner points and derivatives of L-series". <a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a>. 84 (2): 225–320. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1986InMat..84..225G">1986InMat..84..225G</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01388809">10.1007/BF01388809</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0833192">0833192</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:125716869">125716869</a>.</li>
<li><a href="/facts/Victor_Kolyvagin/7H6wd82H">Kolyvagin, Victor</a> (1989). "Finiteness of E(Q) and X(E, Q) for a class of Weil curves". Math. USSR Izv. 32 (3): 523–541. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1989IzMat..32..523K">1989IzMat..32..523K</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1070%2Fim1989v032n03abeh000779">10.1070/im1989v032n03abeh000779</a>.</li>
<li><a href="/facts/Louis_Mordell/L2Qpjnnz">Mordell, L. J.</a> (1922). <a href="https://archive.org/details/proceedingscambr21camb/page/178/mode/2up">"On the rational solutions of the indeterminate equations of the third and fourth degrees"</a>. <a href="/facts/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society/KUUzcS7S">Mathematical Proceedings of the Cambridge Philosophical Society</a>. 21: 179–192.</li>
<li><a href="/facts/Jan_Nekov%25C3%25A1%25C5%2599/3j4XY7xz">Nekovář, Jan</a> (2009). <a href="https://doi.org/10.1112%2FS0010437X09003959">"On the parity of ranks of Selmer groups IV"</a>. <a href="/facts/Compositio_Mathematica/Igc7zupB">Compositio Mathematica</a>. 145 (6): 1351–1359. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2FS0010437X09003959">10.1112/S0010437X09003959</a>.</li>
<li><a href="/facts/Karl_Rubin/3xhVe6yE">Rubin, Karl</a> (1991). "The 'main conjectures' of Iwasawa theory for imaginary quadratic fields". <a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a>. 103 (1): 25–68. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1991InMat.103...25R">1991InMat.103...25R</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01239508">10.1007/BF01239508</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120179735">120179735</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0737.11030">0737.11030</a>.</li>
<li><a href="/facts/Christopher_Skinner/CFBcb0jn">Skinner, Christopher</a>; <a href="/facts/%25C3%2589ric_Urban/F2Wzynxx">Urban, Éric</a> (2014). "The Iwasawa main conjectures for GL2". <a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a>. 195 (1): 1–277. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2014InMat.195....1S">2014InMat.195....1S</a>. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.363.2008">10.1.1.363.2008</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs00222-013-0448-1">10.1007/s00222-013-0448-1</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120848645">120848645</a>.</li>
<li><a href="/facts/Jerrold_B._Tunnell/UmNU8rE5">Tunnell, Jerrold B.</a> (1983). <a href="http://dml.cz/bitstream/handle/10338.dmlcz/137483/ActaOstrav_14-2006-1_8.pdf">"A classical Diophantine problem and modular forms of weight 3/2"</a> (PDF). <a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a>. 72 (2): 323–334. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1983InMat..72..323T">1983InMat..72..323T</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01389327">10.1007/BF01389327</a>. <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/10338.dmlcz%2F137483">10338.dmlcz/137483</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121099824">121099824</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0515.10013">0515.10013</a>.</li>
<li><a href="/facts/Andrew_Wiles/bdc3k3pQ">Wiles, Andrew</a> (1995). "Modular elliptic curves and Fermat's last theorem". <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>. Second Series. 141 (3): 443–551. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2118559">10.2307/2118559</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2118559">2118559</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1333035">1333035</a>.</li>
<li><a href="/facts/Andrew_Wiles/bdc3k3pQ">Wiles, Andrew</a> (2006). <a href="https://web.archive.org/web/20180329033023/http://www.claymath.org/sites/default/files/birchswin.pdf">"The Birch and Swinnerton-Dyer conjecture"</a> (PDF). In Carlson, James; <a href="/facts/Arthur_Jaffe/0pF6APjR">Jaffe, Arthur</a>; <a href="/facts/Andrew_Wiles/bdc3k3pQ">Wiles, Andrew</a> (eds.). The Millennium prize problems. American Mathematical Society. pp. 31–44. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-3679-8. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2238272">2238272</a>. Archived from <a href="http://www.claymath.org/sites/default/files/birchswin.pdf">the original</a> (PDF) on 29 March 2018. Retrieved 16 December 2013.</li></ul>

<ul><li>Shoeib, Maisara (26 May 2025). "A Topological Perspective on the Birch and Swinnerton–Dyer Conjecture". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/2505.19796">2505.19796</a>.</li></ul>
<h2 id="external-links">External links</h2>

Wikiquote has quotations related to Birch and Swinnerton-Dyer conjecture.

<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Swinnerton-DyerConjecture.html">"Swinnerton-Dyer Conjecture"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li><a href="https://planetmath.org/BirchAndSwinnertonDyerConjecture">"Birch and Swinnerton-Dyer Conjecture"</a>. <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li>
<li><a href="https://issuu.com/thedeltaepsilon/docs/de1">The Birch and Swinnerton-Dyer Conjecture</a>: An Interview with Professor <a href="/facts/Henri_Darmon/45q7g0NL">Henri Darmon</a> by Agnes F. Beaudry</li>
<li><a href="https://www.youtube.com/watch?v=2gbQWIzb6Dg&t=3s">What is the Birch and Swinnerton-Dyer Conjecture?</a> lecture by <a href="/facts/Manjul_Bhargava/DPSrSaXE">Manjul Bhargava</a> (September 2016) given during the Clay Research Conference held at the University of Oxford</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Birch and Swinnerton-Dyer Conjecture at Clay Mathematics Institute <a href="http://www.claymath.org/millennium-problems/birch-and-swinnerton-dyer-conjecture" target="_blank">http://www.claymath.org/millennium-problems/birch-and-swinnerton-dyer-conjecture</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Stewart, Ian (2013), Visions of Infinity: The Great Mathematical Problems, Basic Books, p. 253, ISBN 9780465022403, Cassels was highly skeptical at first. <a href="9780465022403" target="_blank">9780465022403</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Cremona, John (2011). "Numerical evidence for the Birch and Swinnerton-Dyer Conjecture" (PDF). Talk at the BSD 50th Anniversary Conference, May 2011., page 50 <a href="https://people.maths.bris.ac.uk/~matyd/BSD2011/bsd2011-Cremona.pdf" target="_blank">https://people.maths.bris.ac.uk/~matyd/BSD2011/bsd2011-Cremona.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Tate, John T. (1974). "The arithmetic of elliptic curves". Invent Math. 23 (3–4): 179–206. Bibcode:1974InMat..23..179T. doi:10.1007/BF01389745., page 198 <a href="https://eudml.org/doc/142261" target="_blank">https://eudml.org/doc/142261</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Cremona, John (2011). "Numerical evidence for the Birch and Swinnerton-Dyer Conjecture" (PDF). Talk at the BSD 50th Anniversary Conference, May 2011. <a href="https://people.maths.bris.ac.uk/~matyd/BSD2011/bsd2011-Cremona.pdf" target="_blank">https://people.maths.bris.ac.uk/~matyd/BSD2011/bsd2011-Cremona.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Koblitz, Neal (1993). Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics. Vol. 97 (2nd ed.). Springer-Verlag. ISBN 0-387-97966-2. <a href="0-387-97966-2" target="_blank">0-387-97966-2</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Heath-Brown, D. R. (2004). "The Average Analytic Rank of Elliptic Curves". Duke Mathematical Journal. 122 (3): 591–623. arXiv:math/0305114. doi:10.1215/S0012-7094-04-12235-3. MR 2057019. S2CID 15216987. <a href="/wiki/Roger_Heath-Brown" target="_blank">/wiki/Roger_Heath-Brown</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Hindry, Marc; Silverman, Joseph H. (2000). Diophantine Geometry: An Introduction. Graduate Texts in Mathematics. Vol. 201. New York, NY: Springer. p. 462. doi:10.1007/978-1-4612-1210-2. ISBN 978-0-387-98975-4. <a href="978-0-387-98975-4" target="_blank">978-0-387-98975-4</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Zhang, Wei (2013). "The Birch–Swinnerton-Dyer conjecture and Heegner points: a survey". Current Developments in Mathematics. 2013: 169–203. doi:10.4310/CDM.2013.v2013.n1.a3.. <a href="https://doi.org/10.4310%2FCDM.2013.v2013.n1.a3" target="_blank">https://doi.org/10.4310%2FCDM.2013.v2013.n1.a3</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Leong, Y. K. (July–December 2018). "Shou-Wu Zhang: Number Theory and Arithmetic Algebraic Geometry" (PDF). Imprints. No. 32. The Institute for Mathematical Sciences, National University of Singapore. pp. 32–36. Retrieved 5 May 2019. <a href="https://ims.nus.edu.sg/wp-content/uploads/2020/05/imprints-32-2018.pdf" target="_blank">https://ims.nus.edu.sg/wp-content/uploads/2020/05/imprints-32-2018.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Kings, Guido (2003). "The Bloch–Kato conjecture on special values of L-functions. A survey of known results". Journal de théorie des nombres de Bordeaux. 15 (1): 179–198. doi:10.5802/jtnb.396. ISSN 1246-7405. MR 2019010. <a href="http://jtnb.cedram.org/item?id=JTNB_2003__15_1_179_0" target="_blank">http://jtnb.cedram.org/item?id=JTNB_2003__15_1_179_0</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
</ol>

Birch and Swinnerton-Dyer conjecture open-in-new

Birch and Swinnerton-Dyer conjecture