4.6 Elliptic curves and the ABC conjecture

Elliptic curves are smooth, projective algebraic curves of genus one with a specified base point. They form an abelian group under a defined group law and play a crucial role in number theory, cryptography, and coding theory.

The ABC conjecture connects prime factors of relatively prime integers to their sum. If proven, it would have significant consequences for various areas of mathematics, including elliptic curves and Diophantine equations.

Elliptic curves and ABC conjecture

• Elliptic curves are a fundamental object of study in number theory and algebraic geometry with deep connections to the ABC conjecture
• Understanding the properties and structure of elliptic curves is essential for grasping the significance and implications of the ABC conjecture
• The study of elliptic curves over various fields, their group law, and associated theorems provide the mathematical framework for exploring the ABC conjecture

Definition of elliptic curves

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• Elliptic curves are smooth, projective algebraic curves of genus one with a specified base point
• Can be defined over any field, including the complex numbers, rational numbers, and finite fields
• Possess a rich algebraic structure, forming an abelian group under a specified group law
• Play a crucial role in various branches of mathematics, such as number theory, cryptography, and coding theory

Weierstrass equation

• Elliptic curves can be described by a Weierstrass equation of the form $y^2 = x^3 + ax + b$, where $a$ and $b$ are constants
• The discriminant $\Delta = -16(4a^3 + 27b^2)$ determines the singularity of the curve
  • If $\Delta \neq 0$, the curve is non-singular and defines an elliptic curve
  • If $\Delta = 0$, the curve is singular and not an elliptic curve
• The Weierstrass equation provides a standard form for representing and studying elliptic curves

Group law on elliptic curves

• Elliptic curves possess a group structure under a well-defined addition operation
• The group law allows for the addition of two points on the curve to obtain another point on the curve
• The group law is defined geometrically using the chord-and-tangent method
  • Given two points $P$ and $Q$ on the curve, draw a line through them
  • The line intersects the curve at a third point $R$
  • The reflection of $R$ about the $x$-axis is the sum $P + Q$
• The group law satisfies the axioms of an abelian group (associativity, identity, inverses, commutativity)

Elliptic curves over finite fields

• Elliptic curves can be defined over finite fields $\mathbb{F}_q$, where $q$ is a prime power
• The number of points on an elliptic curve over $\mathbb{F}_q$, denoted by $\#E(\mathbb{F}_q)$, is finite
• Hasse's theorem provides bounds for $\#E(\mathbb{F}_q)$: $|q + 1 - \#E(\mathbb{F}_q)| \leq 2\sqrt{q}$
• Elliptic curves over finite fields have applications in cryptography and coding theory (error-correcting codes)

Elliptic curve cryptography

• Elliptic curve cryptography (ECC) is a public-key cryptosystem based on the algebraic structure of elliptic curves over finite fields
• ECC relies on the difficulty of the elliptic curve discrete logarithm problem (ECDLP)
  • Given points $P$ and $Q$ on an elliptic curve, finding an integer $k$ such that $Q = kP$ is computationally infeasible
• ECC offers similar security levels to other cryptosystems (RSA) with smaller key sizes, making it more efficient
• Used in various cryptographic protocols, such as key exchange (ECDH), digital signatures (ECDSA), and encryption

Rank of elliptic curves

• The rank of an elliptic curve $E$ over a field $K$, denoted by $\text{rank}(E(K))$, is the number of independent points of infinite order in the group $E(K)$
• The rank measures the "size" of the group of rational points on the curve
• Determining the rank of an elliptic curve is a difficult problem, and there is no general algorithm for computing it
• The Birch and Swinnerton-Dyer conjecture relates the rank to the behavior of the L-function associated with the elliptic curve

Torsion points on elliptic curves

• Torsion points on an elliptic curve are points of finite order under the group law
• The set of torsion points on an elliptic curve $E$ over a field $K$ forms a subgroup, denoted by $E(K)_{\text{tors}}$
• The structure of the torsion subgroup is well-understood for elliptic curves over $\mathbb{Q}$
  • Mazur's theorem classifies the possible torsion subgroups of elliptic curves over $\mathbb{Q}$
  • There are only 15 possible torsion subgroups, with orders ranging from 1 to 12
• Torsion points play a role in the study of rational points on elliptic curves and in the Mordell-Weil theorem

Mordell-Weil theorem

• The Mordell-Weil theorem states that for an elliptic curve $E$ over a number field $K$, the group of $K$-rational points $E(K)$ is finitely generated
• The group $E(K)$ can be decomposed as $E(K) \cong \mathbb{Z}^r \oplus E(K)_{\text{tors}}$, where $r$ is the rank and $E(K)_{\text{tors}}$ is the torsion subgroup
• The theorem implies that the rational points on an elliptic curve can be described by a finite set of generators
• The proof of the Mordell-Weil theorem is non-constructive and does not provide an algorithm for finding the generators

Elliptic curves and Diophantine equations

• Elliptic curves are closely related to Diophantine equations, which are polynomial equations with integer coefficients and solutions
• Many Diophantine equations can be transformed into the study of rational points on elliptic curves
• The Mordell-Weil theorem implies that the set of rational solutions to certain Diophantine equations is finite
• Elliptic curves provide a powerful tool for studying and solving certain types of Diophantine equations (Fermat's equation, Catalan's equation)

ABC conjecture

• The ABC conjecture is a profound and far-reaching conjecture in number theory that connects the prime factors of three relatively prime integers to their sum
• If proven true, the ABC conjecture would have significant consequences for various areas of mathematics, including elliptic curves and Diophantine equations
• The conjecture remains unproven, but substantial progress has been made towards its resolution

Statement of ABC conjecture

• For any $\varepsilon > 0$, there exist only finitely many triples of relatively prime positive integers $(a, b, c)$ satisfying $a + b = c$ and $\text{rad}(abc) < c^{1-\varepsilon}$, where $\text{rad}(n)$ is the radical of $n$
• The conjecture asserts that if the sum of two relatively prime integers $a$ and $b$ is equal to $c$, then the product of their distinct prime factors (radical) is usually not much smaller than $c$
• The conjecture is stated in terms of the quality $q(a,b,c) = \frac{\log c}{\log \text{rad}(abc)}$, with the conjecture implying that the quality is bounded for all but finitely many triples

Radical of an integer

• The radical of a positive integer $n$, denoted by $\text{rad}(n)$, is the product of the distinct prime factors of $n$
• For example, $\text{rad}(12) = \text{rad}(2^2 \cdot 3) = 2 \cdot 3 = 6$
• The radical measures the "smoothness" of an integer, with smaller radicals indicating a higher degree of smoothness
• The radical plays a central role in the formulation of the ABC conjecture

Quality vs size in ABC conjecture

• The ABC conjecture relates the quality $q(a,b,c) = \frac{\log c}{\log \text{rad}(abc)}$ to the size of the integers $a$, $b$, and $c$
• The conjecture suggests that triples $(a,b,c)$ with high quality (i.e., $c$ is large compared to $\text{rad}(abc)$) are rare
• Triples with quality greater than 1 are of particular interest, as they represent instances where the radical is small relative to the size of $c$
• The ABC conjecture effectively limits the number of high-quality triples, with only finitely many exceeding any given quality threshold

Consequences of ABC conjecture

• The ABC conjecture has numerous consequences in various areas of mathematics, particularly in number theory and Diophantine equations
• Some notable consequences include:
  • Fermat's Last Theorem for sufficiently large exponents
  • The Mordell conjecture on the finite generation of rational points on curves of genus $\geq 2$
  • The Erdős-Woods conjecture on the existence of perfect powers with small gaps
  • The Wieferich prime conjecture on the non-existence of certain prime numbers
• The ABC conjecture provides a unifying framework for understanding and proving various long-standing conjectures in number theory

ABC conjecture and Fermat's last theorem

• The ABC conjecture implies Fermat's Last Theorem for sufficiently large exponents
• Fermat's Last Theorem states that the equation $x^n + y^n = z^n$ has no non-zero integer solutions for $n > 2$
• If the ABC conjecture holds, then there exists a constant $N$ such that Fermat's Last Theorem is true for all exponents $n > N$
• The proof of Fermat's Last Theorem by Andrew Wiles does not rely on the ABC conjecture, but the conjecture provides an alternative approach to the problem

ABC conjecture and Mordell conjecture

• The ABC conjecture implies the Mordell conjecture, which states that a curve of genus $\geq 2$ over a number field has only finitely many rational points
• The Mordell conjecture was proven by Gerd Faltings in 1983, but the ABC conjecture provides a more elementary proof
• The ABC conjecture can be used to bound the height of rational points on curves of genus $\geq 2$, which leads to the finite generation of rational points
• The connection between the ABC conjecture and the Mordell conjecture highlights the deep interplay between Diophantine equations and the structure of rational points on curves

Masser-Oesterlé conjecture

• The Masser-Oesterlé conjecture, also known as the $abc$ conjecture, is an equivalent formulation of the ABC conjecture
• The conjecture states that for every $\varepsilon > 0$, there exists a constant $C_\varepsilon$ such that for any relatively prime integers $a$, $b$, and $c$ satisfying $a + b = c$, the inequality $\max(|a|, |b|, |c|) \leq C_\varepsilon \cdot \text{rad}(abc)^{1+\varepsilon}$ holds
• The Masser-Oesterlé conjecture provides a more explicit bound on the size of the integers $a$, $b$, and $c$ in terms of the radical of their product
• The conjecture is named after David Masser and Joseph Oesterlé, who independently formulated it in the 1980s

Progress towards proving ABC conjecture

• Despite significant efforts, the ABC conjecture remains unproven
• Several partial results and special cases of the conjecture have been established
  • The conjecture has been proven for certain classes of triples $(a,b,c)$, such as those where $a$, $b$, and $c$ have specific divisibility properties
  • Lower bounds on the quality of triples have been obtained, providing evidence in favor of the conjecture
• In 2012, Shinichi Mochizuki proposed a proof of the ABC conjecture using his theory of Inter-universal Teichmüller Theory (IUT)
  • Mochizuki's proof is highly complex and relies on novel mathematical techniques that are not yet fully understood by the mathematical community
  • The validity of Mochizuki's proof remains a topic of ongoing discussion and verification
• The ABC conjecture continues to be a central problem in number theory, with active research aimed at its resolution

Connections between ABC conjecture and elliptic curves

• The ABC conjecture has deep connections to the theory of elliptic curves
• Elliptic curves can be used to construct high-quality ABC triples
  • For example, the Frey curve $y^2 = x(x-a)(x+b)$ associated with an ABC triple $(a,b,c)$ has good reduction properties related to the radical of $abc$
  • The study of elliptic curves and their reduction types can provide insights into the structure of ABC triples
• The ABC conjecture has implications for the rank and torsion subgroups of elliptic curves
  • If the ABC conjecture is true, it would imply that the rank of elliptic curves over $\mathbb{Q}$ is uniformly bounded
  • The conjecture also has consequences for the possible torsion subgroups of elliptic curves over number fields
• The interplay between the ABC conjecture and elliptic curves highlights the deep connections between different areas of number theory and algebraic geometry