4.4 Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer conjecture connects the arithmetic of elliptic curves to their L-functions. It predicts that the order of vanishing of an elliptic curve's L-function at a specific point equals the curve's rank.

This conjecture has profound implications for number theory and algebraic geometry. It links algebraic properties of elliptic curves to analytic properties of L-functions, offering insights into the structure of rational points on these curves.

Statement of the conjecture

• The Birch and Swinnerton-Dyer conjecture is a central problem in number theory that connects the arithmetic of elliptic curves to their L-functions
• It provides a deep link between the algebraic structure of elliptic curves and analytic properties of their associated L-functions
• The conjecture has far-reaching consequences and has been a driving force in the development of modern algebraic number theory

Rank of elliptic curves

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• The rank of an elliptic curve $E$ over a number field $K$, denoted $rank(E(K))$, is the rank of the Mordell-Weil group $E(K)$ as a finitely generated abelian group
• The rank measures the number of independent rational points of infinite order on the elliptic curve
• Determining the rank is a difficult problem and the conjecture predicts a deep connection between the rank and the order of vanishing of the L-function

L-functions of elliptic curves

• The L-function $L(E,s)$ of an elliptic curve $E$ over a number field $K$ is a complex analytic function that encodes arithmetic information about the curve
• It is defined as an Euler product over the primes of $K$, with local factors determined by the reduction type of $E$ at each prime
• The L-function is expected to have an analytic continuation to the entire complex plane and satisfy a functional equation

Order of vanishing

• The Birch and Swinnerton-Dyer conjecture predicts that the order of vanishing of the L-function $L(E,s)$ at $s=1$ is equal to the rank of the elliptic curve $E$
• In other words, the conjecture asserts that $ord_{s=1} L(E,s) = rank(E(K))$
• This connection between the analytic behavior of the L-function and the algebraic rank is a central feature of the conjecture

Tate-Shafarevich group

• The Tate-Shafarevich group, denoted $Ш(E/K)$, is a mysterious group associated to an elliptic curve $E$ over a number field $K$
• It measures the failure of the local-global principle for principal homogeneous spaces of $E$
• The Birch and Swinnerton-Dyer conjecture predicts that the Tate-Shafarevich group is finite and its order is related to the leading coefficient of the L-function at $s=1$

Connections to other problems

• The Birch and Swinnerton-Dyer conjecture has deep connections to several other major problems in number theory
• It provides a unifying framework that relates seemingly disparate areas of mathematics
• Proving the conjecture would have significant implications for our understanding of elliptic curves and their role in modern number theory

Congruent number problem

• The congruent number problem asks which integers can be the area of a right triangle with rational side lengths
• It can be reformulated in terms of the rank of elliptic curves of the form $y^2 = x^3 - n^2x$
• The Birch and Swinnerton-Dyer conjecture implies a criterion for determining whether a given integer is a congruent number

Fermat's Last Theorem

• Fermat's Last Theorem states that the equation $x^n + y^n = z^n$ has no non-zero integer solutions for $n > 2$
• The proof of Fermat's Last Theorem by Andrew Wiles relied on establishing the modularity of certain elliptic curves
• The Birch and Swinnerton-Dyer conjecture played a crucial role in guiding the strategy of the proof and highlighting the importance of elliptic curves

ABC conjecture

• The ABC conjecture is a deep conjecture in number theory that relates the prime factors of three integers satisfying $a+b=c$
• It has connections to the Birch and Swinnerton-Dyer conjecture through the study of elliptic curves and their L-functions
• Some special cases of the ABC conjecture have been proven using techniques inspired by the Birch and Swinnerton-Dyer conjecture

Computational evidence

• Extensive computational evidence has been gathered in support of the Birch and Swinnerton-Dyer conjecture
• Numerical calculations have verified the conjecture for a large number of elliptic curves
• However, the conjecture remains unproven in general and computational methods have limitations

Elliptic curve databases

• Large databases of elliptic curves have been compiled, such as the L-functions and Modular Forms Database (LMFDB)
• These databases contain information about the rank, L-function, and other invariants of millions of elliptic curves
• They provide a rich source of data for testing the Birch and Swinnerton-Dyer conjecture and exploring patterns and relationships

Verification for specific cases

• The Birch and Swinnerton-Dyer conjecture has been verified for many individual elliptic curves using a combination of theoretical and computational techniques
• For example, the conjecture has been proven for all elliptic curves of rank 0 and 1 over the rational numbers
• Computational methods, such as Heegner point constructions and descent calculations, have been used to confirm the conjecture in specific cases

Limitations of computational methods

• Despite the extensive computational evidence, the Birch and Swinnerton-Dyer conjecture remains unproven in general
• Computational methods are limited by the size and complexity of the elliptic curves being considered
• As the rank and the coefficients of the elliptic curves increase, the calculations become more challenging and time-consuming
• Proving the conjecture will require new theoretical insights and techniques beyond computational verification

Partial results

• While the Birch and Swinnerton-Dyer conjecture remains unproven in full generality, significant partial results have been established
• These results provide strong evidence for the conjecture and have deepened our understanding of elliptic curves and their L-functions
• The partial results have been achieved through a combination of sophisticated techniques from algebraic geometry, harmonic analysis, and representation theory

Coates-Wiles theorem

• The Coates-Wiles theorem, proven by John Coates and Andrew Wiles, establishes a special case of the Birch and Swinnerton-Dyer conjecture
• It states that if an elliptic curve $E$ over a number field $K$ has complex multiplication and rank 0, then the L-function $L(E,s)$ does not vanish at $s=1$
• This result provided the first concrete evidence for the conjecture and highlighted the importance of complex multiplication in the study of elliptic curves

Gross-Zagier theorem

• The Gross-Zagier theorem, proven by Benedict Gross and Don Zagier, relates the height of Heegner points on an elliptic curve to the derivative of its L-function at $s=1$
• It provides a formula for the canonical height of Heegner points in terms of the derivative of the L-function
• The theorem has important implications for the Birch and Swinnerton-Dyer conjecture and has been used to prove cases of the conjecture for elliptic curves of rank 1

Kolyvagin's work

• Victor Kolyvagin made significant progress on the Birch and Swinnerton-Dyer conjecture using techniques from Iwasawa theory and Euler systems
• He introduced the concept of Euler systems, which are collections of cohomology classes that satisfy certain compatibility relations
• Kolyvagin's work led to the proof of the weak Birch and Swinnerton-Dyer conjecture for elliptic curves of rank 0 and 1 over the rational numbers

Modularity theorem

• The modularity theorem, originally known as the Taniyama-Shimura conjecture and proven by Andrew Wiles and others, establishes a deep connection between elliptic curves and modular forms
• It states that every elliptic curve over the rational numbers is modular, meaning it corresponds to a modular form of a specific level and weight
• The modularity theorem has important consequences for the Birch and Swinnerton-Dyer conjecture, as it allows techniques from the theory of modular forms to be applied to the study of elliptic curves

Generalizations and variants

• The Birch and Swinnerton-Dyer conjecture has been generalized and extended in various directions
• These generalizations and variants aim to capture more intricate arithmetic properties of elliptic curves and their L-functions
• They provide a broader framework for understanding the deep connections between algebra, geometry, and analysis in the study of elliptic curves

Birch and Swinnerton-Dyer conjecture over number fields

• The original Birch and Swinnerton-Dyer conjecture was formulated for elliptic curves over the rational numbers
• It has been generalized to elliptic curves over arbitrary number fields, taking into account the more complex arithmetic and algebraic structure
• The conjecture over number fields involves the rank of the Mordell-Weil group, the order of vanishing of the L-function, and additional arithmetic invariants such as the Tate-Shafarevich group and the regulator

Equivariant Birch and Swinnerton-Dyer conjecture

• The equivariant Birch and Swinnerton-Dyer conjecture is a refinement that takes into account the action of the absolute Galois group on the elliptic curve and its invariants
• It formulates the conjecture in terms of equivariant L-functions and equivariant Selmer groups
• The equivariant version provides a more precise description of the arithmetic properties of elliptic curves and has connections to non-commutative Iwasawa theory

p-adic Birch and Swinnerton-Dyer conjecture

• The p-adic Birch and Swinnerton-Dyer conjecture is an analogue of the classical conjecture that considers p-adic L-functions instead of complex L-functions
• It relates the rank of an elliptic curve to the order of vanishing of its p-adic L-function at certain points
• The p-adic version has important applications in Iwasawa theory and the study of Selmer groups and Galois cohomology

Birch and Swinnerton-Dyer conjecture for abelian varieties

• The Birch and Swinnerton-Dyer conjecture has been generalized to abelian varieties, which are higher-dimensional analogues of elliptic curves
• The conjecture for abelian varieties involves the rank of the Mordell-Weil group, the order of vanishing of the L-function, and additional arithmetic invariants such as the Tate-Shafarevich group and the Néron-Tate height pairing
• The study of the Birch and Swinnerton-Dyer conjecture for abelian varieties has led to important developments in the theory of motives and algebraic cycles

Current status and future directions

• Despite significant progress, the Birch and Swinnerton-Dyer conjecture remains one of the most challenging open problems in mathematics
• The conjecture has far-reaching consequences and its resolution would have a profound impact on our understanding of elliptic curves and number theory
• Current research focuses on developing new techniques and insights to tackle the remaining obstacles and make further progress towards a complete proof

Consequences of the conjecture

• A proof of the Birch and Swinnerton-Dyer conjecture would have numerous consequences and applications
• It would provide a powerful tool for determining the rank of elliptic curves and understanding their arithmetic properties
• The conjecture has implications for other areas of mathematics, such as the study of Diophantine equations, Galois representations, and algebraic cycles
• Its resolution would also have practical applications in cryptography and coding theory, where elliptic curves play a crucial role

Obstacles to a complete proof

• Proving the Birch and Swinnerton-Dyer conjecture in full generality faces significant obstacles and challenges
• One major difficulty lies in understanding the structure and finiteness of the Tate-Shafarevich group
• Another challenge is relating the algebraic and analytic ranks of elliptic curves and controlling the error terms in the asymptotic formulas
• The conjecture is also closely tied to deep conjectures in Iwasawa theory and the equivariant Tamagawa number conjecture, which remain unproven

Recent progress and breakthroughs

• In recent years, there have been significant breakthroughs and progress towards the Birch and Swinnerton-Dyer conjecture
• New techniques, such as the use of Euler systems, p-adic methods, and automorphic representations, have led to important advances
• Notable results include the proof of the conjecture for elliptic curves over function fields, the establishment of the p-part of the conjecture in certain cases, and the development of new methods for bounding Selmer groups and Tate-Shafarevich groups
• The modularity theorem and its generalizations have also opened up new avenues for studying elliptic curves and their L-functions

Open questions and research directions

• Many open questions and research directions remain in the study of the Birch and Swinnerton-Dyer conjecture
• One important area of investigation is the development of new Euler systems and their applications to the conjecture
• Another active area of research is the study of p-adic methods and their potential for proving cases of the conjecture
• Generalizations and refinements of the conjecture, such as the equivariant and p-adic versions, provide fertile ground for further exploration
• The connections between the Birch and Swinnerton-Dyer conjecture and other areas of mathematics, such as arithmetic geometry, Iwasawa theory, and automorphic forms, continue to drive new discoveries and insights