Editing Birch and Swinnerton-Dyer conjecture

{{Short description|Unproved conjecture in mathematics}}
{{Use dmy dates|date=April 2022}}
{{Millennium Problems}}
In [[mathematics]], the '''Birch and Swinnerton-Dyer conjecture''' (often called the ''' Birch–Swinnerton-Dyer 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. It is named after mathematicians [[Bryan John Birch]] and [[Peter Swinnerton-Dyer]], who developed the conjecture during the first half of the 1960s with the help of machine computation. Only special cases of the conjecture have been proven.

The modern formulation of the conjecture relates to arithmetic data associated with an elliptic curve ''E'' over a [[number field]] ''K'' to the behaviour of the [[Hasse–Weil L-function|Hasse–Weil ''L''-function]] ''L''(''E'',&nbsp;''s'') of ''E'' at ''s''&nbsp;=&nbsp;1. More specifically, it is conjectured that the [[Rank of an abelian group|rank]] of the [[abelian group]] ''E''(''K'') of points of ''E'' is the order of the zero of ''L''(''E'',&nbsp;''s'') at ''s'' = 1. The first non-zero coefficient in the [[Taylor expansion]] of ''L''(''E'',&nbsp;''s'') at ''s'' = 1 is given by more refined arithmetic data attached to ''E'' over ''K'' {{harv|Wiles|2006}}.

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.<ref>[http://www.claymath.org/millennium-problems/birch-and-swinnerton-dyer-conjecture Birch and Swinnerton-Dyer Conjecture] at Clay Mathematics Institute</ref>

== Background ==
{{Harvtxt|Mordell|1922}} proved [[Mordell's theorem]]: the group of [[rational point]]s on an elliptic curve has a finite [[Generating set of a group|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 set|infinite]] then some point in a finite basis must have infinite [[order (group theory)|order]]. The number of ''independent'' basis points with infinite order is called the [[rank of an abelian group|rank]] of the curve, and is an important [[Invariant (mathematics)|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) it is unknown if these methods handle all curves.

An ''L''-function '''''L''(''E'',&nbsp;''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 number|prime]] ''p''. This ''L''-function is analogous to the [[Riemann zeta function]] and the [[Dirichlet L-series]] that is defined for a binary [[quadratic form]]. It is a special case of a [[Hasse–Weil L-function]].

The natural definition of ''L''(''E'',&nbsp;''s'') only converges for values of ''s'' in the complex plane with Re(''s'') > 3/2. [[Helmut Hasse]] conjectured that ''L''(''E'',&nbsp;''s'') could be extended by [[analytic continuation]] to the whole complex plane. This conjecture was first proved by {{Harvtxt|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]] 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.

== History ==

In the early 1960s [[Peter Swinnerton-Dyer]] used the [[EDSAC 2|EDSAC-2]] computer at the [[University of Cambridge Computer Laboratory]] to calculate the number of points modulo ''p'' (denoted by ''N<sub>p</sub>'') for a large number of primes ''p'' on elliptic curves whose rank was known. From these numerical results {{harvtxt|Birch|Swinnerton-Dyer|1965}} conjectured that ''N<sub>p</sub>'' for a curve ''E'' with rank ''r'' obeys an asymptotic law

:<math>\prod_{p\leq x} \frac{N_p}{p} \approx C\log (x)^r \mbox{ as } x \rightarrow \infty </math>

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).<ref>{{citation|title=Visions of Infinity: The Great Mathematical Problems|first=Ian|last=Stewart|author-link=Ian Stewart (mathematician)|publisher=Basic Books|year=2013|isbn=9780465022403|page=253|url=https://books.google.com/books?id=dzdSy3diraUC&pg=PA253|quote=Cassels was highly skeptical at first}}.</ref> 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'',&nbsp;''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'',&nbsp;''s'') was only established for curves with complex multiplication, which were also the main source of numerical examples. (NB that the [[Reciprocal (mathematics)|reciprocal]] 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 [[Taylor coefficient]] of the ''L''-function at ''s''&nbsp;=&nbsp;1. It is conjecturally given by<ref>{{cite journal |url=https://people.maths.bris.ac.uk/~matyd/BSD2011/bsd2011-Cremona.pdf |title=Numerical evidence for the Birch and Swinnerton-Dyer Conjecture |first=John |last=Cremona |year=2011 |journal=Talk at the BSD 50th Anniversary Conference, May 2011 }}, page 50</ref>

:<math>\frac{L^{(r)}(E,1)}{r!} = \frac{\#\mathrm{Sha}(E)\Omega_E R_E \prod_{p|N}c_p}{(\#E_{\mathrm{tor}})^2}</math>

where the quantities on the right-hand side are invariants of the curve, studied by Cassels, [[John Tate (mathematician)|Tate]], [[Igor Shafarevich|Shafarevich]] and others {{harv|Wiles|2006}}:

<math>\#E_{\mathrm{tor}}</math> is the order of the [[torsion group]],

<math>\#\mathrm{Sha}(E)=</math>{{math|#Ш(E)}} is the order of the [[Tate–Shafarevich group]],

<math>\Omega_E</math> is the real period of ''E'' multiplied by the number of connected components of ''E'',

<math>R_E</math> is the [[Néron–Tate height|regulator]] of ''E'' which is defined via the [[canonical height]]s of a basis of rational points,

<math>c_p</math> is the [[Tamagawa number]] of ''E'' at a prime ''p'' dividing the conductor ''N'' of ''E''. It can be found by [[Tate's algorithm]].

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. [[John Tate (mathematician)|John Tate]] expressed this in 1974 in a famous quote.<ref>{{cite journal |url=https://eudml.org/doc/142261 |title=The arithmetic of elliptic curves. |first=John T. |last=Tate |year=1974 |journal= Invent Math |volume= 23 |issue=3–4 |pages=179–206 |doi= 10.1007/BF01389745 |bibcode=1974InMat..23..179T }}, page 198</ref>{{rp|198}}
<blockquote>
This remarkable conjecture relates the behavior of a function <math>L</math> at a point where it is not at present known to be defined to the order of a group {{math|Ш}} which is not known to be finite!
</blockquote>
By the [[modularity theorem]] proved in 2001 for elliptic curves over <math>\mathbb{Q}</math> the left side is now known to be well-defined and the finiteness of {{math|Ш(E)}} is known when additionally the analytic rank is at most 1, i.e., if <math>L(E,s)</math> vanishes at most to order 1 at <math>s=1</math>. Both parts remain open.

== Current status ==

[[File:BSD data plot for elliptic curve 800h1.svg|350px|right|thumb|A plot, in blue, of <math>\prod_{p\leq X} \frac{N_p}{p}</math> for the curve ''y''<sup>2</sup>&nbsp;=&nbsp;''x''<sup>3</sup>&nbsp;−&nbsp;5''x'' as ''X'' varies over the first 100000 primes. The ''X''-axis is in log(log) scale -''X'' is drawn at distance proportional to <math>\log(\log(X))</math> from 0- and the ''Y''-axis is in a logarithmic scale, so the conjecture predicts that the data should tend to a line of slope equal to the rank of the curve, which is 1 in this case -that is, the quotient <math>\frac{\log\left(\prod_{p\leq X} \frac{N_p}{p}\right)}{\log C+r\log(\log X))}\rightarrow 1</math> as <math>X\rightarrow\infty</math>, with ''C'', ''r'' as in the text. For comparison, a line of slope 1 in (log(log),log)-scale -that is, with equation <math>\log y=a+\log(\log x)</math>- is drawn in red in the plot.]]

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

# {{harvtxt|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 (number theory)|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 {{harvtxt|Arthaud|1978}}.
# {{Harvtxt|Gross|Zagier|1986}} showed that if a [[modular elliptic curve]] has a first-order zero at ''s'' = 1 then it has a rational point of infinite order; see [[Gross–Zagier theorem]].
#{{harvtxt|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.
# {{harvtxt|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.
# {{harvtxt|Breuil|Conrad|Diamond|Taylor|2001}}, extending work of {{harvtxt|Wiles|1995}}, proved that [[Modularity theorem|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 ''L''-functions of all elliptic curves over '''Q''' are defined at ''s'' = 1.
# {{harvtxt|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 {{harvtxt|Nekovář|2009}} and {{harvtxt|Dokchitser|Dokchitser|2010}} and with the proof of the [[main conjecture of Iwasawa theory]] for GL(2) by {{harvtxt|Skinner|Urban|2014}}, they conclude that a positive proportion of elliptic curves over '''Q''' have analytic rank zero, and hence, by {{harvtxt|Kolyvagin|1989}}, satisfy the Birch and Swinnerton-Dyer conjecture.

There are currently no proofs involving curves with a rank greater than 1.

There is extensive numerical evidence for the truth of the conjecture.<ref>{{cite journal |url=https://people.maths.bris.ac.uk/~matyd/BSD2011/bsd2011-Cremona.pdf |title=Numerical evidence for the Birch and Swinnerton-Dyer Conjecture |first=John |last=Cremona |year=2011 |journal=Talk at the BSD 50th Anniversary Conference, May 2011 }}</ref>


== Topological of Birch and Swinnerton–Dyer Conjectures ==

In 2025, Maisara Shoeib proposed a four-dimensional topological embedding that seeks to give a geometric interpretation of the conjecture.<ref name="Shoeib2025">{{Cite arxiv |title=A Topological Perspective on the Birch and Swinnerton–Dyer Conjecture |last=Shoeib |first=Maisara |date=26 May 2025 |eprint=2505.19796 }}</ref><ref>{{Cite journal |title=A Topological Perspective on the Birch and Swinnerton–Dyer Conjecture – supporting data |last=Shoeib |first=Maisara |year=2025 |journal=Zenodo |doi=10.5281/zenodo.15515153 }}</ref>

The embedding
:<math>\Phi\colon E \;\longrightarrow\; \mathbf R^{4}</math>
maps a point <math>P=(x,y)\in E(\mathbf K)</math> to a 4-tuple that combines normalised algebraic coordinates with two holomorphic-integral components taken modulo one.  
* Torsion points yield finite orbits in <math>\mathbf R^{4}</math>.  
* Points of infinite order trace non-contractible loops whose homology classes generate <math>H_{1}(\Phi(E),\mathbb Z)</math>.

Empirically, the number of topologically independent infinite loops equals the Mordell–Weil rank <math>r</math>.  Shoeib also introduces  
:<math>F_{\text{new}}(E,N)=\frac1N\sum_{p\le N}\frac{a_p\log p}{\sqrt p},\qquad a_p=p+1-\#E(\mathbf F_p),</math>  
and observes that <math>F_{\text{new}}(E,N)\sim C(\log N)^{\,r}</math>.  Tests on curves of verified rank 0 – 8 support the alignment

:<math>\text{rank}(E)=\#\{\text{independent loops}\}= \operatorname{ord}_{s=1}L(E,s).</math>

While still conjectural, this framework offers a possible geometric bridge between algebraic rank, analytic rank and the topology of elliptic curves.<ref name="Shoeib2025" />

== Consequences ==
Much like the [[Riemann hypothesis]], this conjecture has multiple consequences, including the following two:
* Let {{mvar|n}} be an odd [[square-free]] integer. Assuming the Birch and Swinnerton-Dyer conjecture, {{mvar|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 ({{mvar|x}}, {{mvar|y}}, {{mvar|z}}) satisfying {{math|2''x''<sup>2</sup> + ''y''<sup>2</sup> + 8''z''<sup>2</sup> {{=}} ''n''}} is twice the number of triplets satisfying {{math|2''x''<sup>2</sup> + ''y''<sup>2</sup> + 32''z''<sup>2</sup> {{=}} ''n''}}. This statement, due to [[Tunnell's theorem]] {{harv|Tunnell|1983}}, is related to the fact that ''n'' is a congruent number if and only if the elliptic curve {{math|''y''<sup>2</sup> {{=}} ''x''<sup>3</sup> − ''n''<sup>2</sup>''x''}} has a rational point of infinite order (thus, under the Birch and Swinnerton-Dyer conjecture, its {{mvar|L}}-function has a zero at {{math|1}}). The interest in this statement is that the condition is easily verified.<ref>{{Cite book
 | last = Koblitz |first=Neal | author-link = Neal Koblitz
 | year = 1993 | edition=2nd
 | title = Introduction to Elliptic Curves and Modular Forms
 | series = Graduate Texts in Mathematics
 | volume=97
 | publisher = Springer-Verlag
 | isbn=0-387-97966-2
 }}</ref>
*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 ''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 [[generalized Riemann hypothesis]] and the BSD conjecture, the average rank of curves given by {{math|''y''<sup>2</sup> {{=}} ''x''<sup>3</sup> + ''ax''+ ''b''}} is smaller than {{math|2}}.<ref>{{cite journal |first=D. R. |last=Heath-Brown | author-link = Roger Heath-Brown |title=The Average Analytic Rank of Elliptic Curves |journal=Duke Mathematical Journal |volume=122 |issue=3 |pages=591–623 |year=2004 |doi=10.1215/S0012-7094-04-12235-3 | mr=2057019|arxiv=math/0305114 |s2cid=15216987 }}</ref>
*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:
**Every positive integer {{math|''n'' ≡ 5, 6 or 7 (mod 8)}} is a [[congruent number]]. 
**The elliptic curve given by {{math|''y''<sup>2</sup> {{=}} ''x''<sup>3</sup> + ''ax'' + ''b''}} where {{math|''a'' ≡ ''b'' (mod 2)}} has infinitely many solutions over <math>\mathbb{Q}(\zeta_8)</math>.
**Every positive rational number {{mvar|d}} can be written in the form {{math|''d'' {{=}} ''s''<sup>2</sup>(''t''<sup>3</sup> – 91''t'' – 182)}} for {{mvar|s}} and {{mvar|t}} in <math>\mathbb{Q}</math>. 
**For every rational number {{mvar|t}}, the elliptic curve given by {{math|''y''<sup>2</sup> {{=}} ''x''(''x''<sup>2</sup> – 49(1 + ''t''<sup>4</sup>)<sup>2</sup>)}} has rank at least {{math|1}}.
**There are many more examples for elliptic curves over number fields. 

== Generalizations ==

There is a version of this conjecture for general [[abelian varieties]] over number fields. A version for abelian varieties over <math>\mathbb{Q}</math> is the following:<ref>{{cite book |last1=Hindry |first1=Marc |last2=Silverman |first2=Joseph H. |date=2000 |title=Diophantine Geometry: An Introduction |url=https://link.springer.com/book/10.1007/978-1-4612-1210-2 |location=New York, NY |publisher=Springer |series=Graduate Texts in Mathematics |volume=201 |page=462 |isbn=978-0-387-98975-4 |doi=10.1007/978-1-4612-1210-2}}</ref>{{rp|462}}

:<math> \lim_{s\to1} \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}}}.
</math>

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 <math>\#A(\mathbb Q)_{\text{tors}}\cdot\#\hat A(\mathbb Q)_{\text{tors}}</math> involving the [[dual abelian variety]] <math>\hat A</math>. Elliptic curves as 1-dimensional abelian varieties are their own duals, i.e. <math>\hat E = E</math>, which simplifies the statement of the BSD conjecture. The regulator <math>R_A</math> needs to be understood for the pairing between a basis for the free parts of <math>A(\mathbb Q)</math> and <math>\hat A(\mathbb Q)</math> relative to the Poincare bundle on the product <math>A\times\hat A</math>.

The rank-one Birch-Swinnerton-Dyer conjecture for modular elliptic curves and modular abelian varieties of [[General linear group|GL(2)]]-type over [[totally real number field]]s was proved by [[Shou-Wu Zhang]] in 2001.<ref>{{cite journal|last=Zhang |first=Wei |title=The Birch–Swinnerton-Dyer conjecture and Heegner points: a survey |journal=Current Developments in Mathematics |volume=2013 |pages=169–203|doi=10.4310/CDM.2013.v2013.n1.a3 |year=2013 |doi-access=free }}.</ref><ref>{{cite magazine|last=Leong |first=Y. K. |date=July–December 2018 |url=https://ims.nus.edu.sg/wp-content/uploads/2020/05/imprints-32-2018.pdf |title=Shou-Wu Zhang: Number Theory and Arithmetic Algebraic Geometry |issue=32 |magazine=Imprints |pages=32–36 |publisher=The Institute for Mathematical Sciences, National University of Singapore |access-date=5 May 2019}}</ref>

Another generalization is given by the [[Bloch–Kato conjecture (L-functions)|Bloch-Kato conjecture]].<ref>{{cite journal| last1=Kings | first1=Guido | title=The Bloch–Kato conjecture on special values of ''L''-functions. A survey of known results | url=http://jtnb.cedram.org/item?id=JTNB_2003__15_1_179_0 |mr=2019010 | year=2003 | journal=Journal de théorie des nombres de Bordeaux | issn=1246-7405 | volume=15 | issue=1 | pages=179–198 | doi= 10.5802/jtnb.396| doi-access=free }}</ref>

== Notes ==
{{Reflist}}

== References ==
{{Refbegin}}
*{{Cite journal |last=Arthaud |first=Nicole |title=On Birch and Swinnerton-Dyer's conjecture for elliptic curves with complex multiplication |journal=[[Compositio Mathematica]] |volume=37 |issue=2 |year=1978 |pages=209–232 |mr=504632 }}
*{{Cite journal |last1=Bhargava |first1=Manjul |author-link=Manjul Bhargava |last2=Shankar |first2=Arul |author-link2=Arul Shankar |title=Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0 |year=2015 |journal=[[Annals of Mathematics]] |volume=181 |issue=2 |pages=587–621 |doi=10.4007/annals.2015.181.2.4 |arxiv=1007.0052|s2cid=1456959 }}
*{{cite journal |last1=Birch |first1=Bryan |author-link=Bryan John Birch |last2=Swinnerton-Dyer |first2=Peter |author-link2=Peter Swinnerton-Dyer |year=1965 |title=Notes on Elliptic Curves (II) |journal=[[Journal für die reine und angewandte Mathematik|J. Reine Angew. Math.]] |volume=165 |issue=218 |pages=79–108 |doi=10.1515/crll.1965.218.79 |s2cid=122531425 }}
*{{cite journal |last1=Breuil |first1=Christophe |author-link=Christophe Breuil |last2=Conrad |first2=Brian |author-link2=Brian Conrad |last3=Diamond |first3=Fred |author-link3=Fred Diamond |last4=Taylor |first4=Richard |author-link4=Richard Taylor (mathematician) |year=2001 |title=On the Modularity of Elliptic Curves over Q: Wild 3-Adic Exercises |journal=[[Journal of the American Mathematical Society]] |volume=14 |issue=4 |pages=843–939 |doi=10.1090/S0894-0347-01-00370-8 |doi-access=free }}
* {{cite book | first1=J.H. | last1=Coates | author-link1=John Coates (mathematician) | first2=R. | last2=Greenberg | first3=K.A. | last3=Ribet | author-link3=Kenneth Alan Ribet | first4=K. | last4=Rubin | author-link4=Karl Rubin | title=Arithmetic Theory of Elliptic Curves | series=Lecture Notes in Mathematics | volume=1716 | publisher=[[Springer-Verlag]] | year=1999 | isbn=3-540-66546-3 }}
*{{Cite journal |last1=Coates |first1=J. |author-link=John Coates (mathematician) |last2=Wiles |first2=A. | author-link2=Andrew Wiles |title=On the conjecture of Birch and Swinnerton-Dyer |journal=[[Inventiones Mathematicae]] |volume=39 |year=1977 |issue=3 |pages=223–251 |doi=10.1007/BF01402975 | zbl=0359.14009 |bibcode=1977InMat..39..223C |s2cid=189832636 }}
*{{cite journal |last=Deuring |first=Max |author-link=Max Deuring |year=1941 |title=Die Typen der Multiplikatorenringe elliptischer Funktionenkörper |journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg |volume=14 |issue=1 |pages=197–272 |doi=10.1007/BF02940746 |s2cid=124821516 }}
*{{cite journal |last1=Dokchitser |first1=Tim |author-link=Tim Dokchitser |last2=Dokchitser |first2=Vladimir |author-link2=Vladimir Dokchitser |doi=10.4007/annals.2010.172.567 |mr=2680426 |title=On the Birch–Swinnerton-Dyer quotients modulo squares|journal=[[Annals of Mathematics]] |volume=172 |year=2010 |issue=1 |pages=567–596 |arxiv=math/0610290 |s2cid=9479748 }}
*{{cite journal |last1=Gross |first1=Benedict H. |author-link=Benedict Gross |last2=Zagier |first2=Don B. |author-link2=Don Zagier |doi=10.1007/BF01388809 |mr=0833192 |title=Heegner points and derivatives of L-series |journal=[[Inventiones Mathematicae]] |volume=84 |year=1986 |issue=2 |pages=225–320 |bibcode=1986InMat..84..225G |s2cid=125716869 }}
*{{cite journal |last=Kolyvagin |first=Victor |author-link=Victor Kolyvagin |year=1989 |title=Finiteness of ''E''(''Q'') and ''X''(''E'',&nbsp;''Q'') for a class of Weil curves |journal=Math. USSR Izv. |volume=32 |issue= 3|pages=523–541 |doi=10.1070/im1989v032n03abeh000779|bibcode=1989IzMat..32..523K }}
*{{cite journal
| last1=Mordell |first1=L. J. |authorlink1=Louis Mordell
| title=On the rational solutions of the indeterminate equations of the third and fourth degrees
|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]
| volume=21
| year=1922
| pages=179–192
| url=https://archive.org/details/proceedingscambr21camb/page/178/mode/2up}}
*{{cite journal |last=Nekovář |first=Jan |author-link=Jan Nekovář |title=On the parity of ranks of Selmer groups IV |journal=[[Compositio Mathematica]] |volume=145 |issue=6 |year=2009 |pages=1351–1359 |doi=10.1112/S0010437X09003959 |doi-access=free }}
*{{cite journal |last=Rubin |first=Karl |author-link=Karl Rubin |year=1991 |title=The 'main conjectures' of Iwasawa theory for imaginary quadratic fields |journal=[[Inventiones Mathematicae]] |volume=103 |issue=1 |pages=25–68 |doi=10.1007/BF01239508 | zbl=0737.11030 |bibcode=1991InMat.103...25R |s2cid=120179735 }}
*{{Cite journal |last1=Skinner |first1=Christopher |author-link=Christopher Skinner |last2=Urban |first2=Éric |author-link2=Éric Urban |title=The Iwasawa main conjectures for GL<sub>2</sub> |journal=[[Inventiones Mathematicae]] |volume=195 |issue=1 |pages=1–277 |year=2014 |doi=10.1007/s00222-013-0448-1 |bibcode=2014InMat.195....1S |s2cid=120848645 |citeseerx=10.1.1.363.2008 }}
*{{cite journal | last = Tunnell | first = Jerrold B. | author-link = Jerrold B. Tunnell | title = A classical Diophantine problem and modular forms of weight 3/2 | journal = [[Inventiones Mathematicae]] | volume = 72 | issue = 2 | pages = 323–334 | year = 1983 | zbl = 0515.10013 | doi = 10.1007/BF01389327 | bibcode = 1983InMat..72..323T | hdl = 10338.dmlcz/137483 | s2cid = 121099824 | url = http://dml.cz/bitstream/handle/10338.dmlcz/137483/ActaOstrav_14-2006-1_8.pdf }}
*{{Cite journal | last1=Wiles | first1=Andrew | author1-link=Andrew Wiles | title=Modular elliptic curves and Fermat's last theorem | jstor=2118559 | mr=1333035 | year=1995 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=141 | issue=3 | pages=443–551| doi=10.2307/2118559}}
*{{Cite encyclopedia
| last=Wiles
| first=Andrew
| author-link=Andrew Wiles
| chapter=The Birch and Swinnerton-Dyer conjecture
| editor1-last=Carlson
| editor1-first=James
| editor2-last=Jaffe
| editor2-first=Arthur
| editor2-link=Arthur Jaffe
| editor3-last=Wiles
| editor3-first=Andrew
| editor3-link=Andrew Wiles
| title=The Millennium prize problems
| publisher=American Mathematical Society
| year=2006
| isbn=978-0-8218-3679-8
| chapter-url=http://www.claymath.org/sites/default/files/birchswin.pdf
| pages=31–44
| mr=2238272
| access-date=16 December 2013
| archive-date=29 March 2018
| archive-url=https://web.archive.org/web/20180329033023/http://www.claymath.org/sites/default/files/birchswin.pdf
| url-status=dead
}}
{{Refend}}
*{{Cite arxiv |last=Shoeib |first=Maisara |title=A Topological Perspective on the Birch and Swinnerton–Dyer Conjecture |eprint=2505.19796 |date=26 May 2025}}

== External links ==
{{Sister project links| wikt=no | commons=no | b=no | n=no | q=Birch and Swinnerton-Dyer conjecture | s=no | v=no | voy=no | species=no | d=no}}

*{{MathWorld|urlname = Swinnerton-DyerConjecture |title = Swinnerton-Dyer Conjecture}}
*{{planetmath reference|urlname=BirchAndSwinnertonDyerConjecture|title =  Birch and Swinnerton-Dyer Conjecture}}
* [https://issuu.com/thedeltaepsilon/docs/de1 The Birch and Swinnerton-Dyer Conjecture]: An Interview with Professor [[Henri Darmon]] by Agnes F. Beaudry
* [https://www.youtube.com/watch?v=2gbQWIzb6Dg&t=3s ''What is the Birch and Swinnerton-Dyer Conjecture?''] lecture by [[Manjul Bhargava]] (September 2016) given during the Clay Research Conference held at the University of Oxford

{{L-functions-footer}}
{{Authority control}}

{{DEFAULTSORT:Birch And Swinnerton-Dyer Conjecture}}
[[Category:Conjectures]]
[[Category:Diophantine geometry]]
[[Category:Millennium Prize Problems]]
[[Category:Number theory]]
[[Category:University of Cambridge Computer Laboratory]]
[[Category:Zeta and L-functions]]
[[Category:Unsolved problems in number theory]]