7.4 Elliptic curves and the Eichler-Shimura relation

Elliptic curves are smooth, projective algebraic curves of genus one with a specified base point. They have a rich algebraic structure and play a central role in number theory and algebraic geometry, with applications ranging from cryptography to the proof of Fermat's Last Theorem.

The Eichler-Shimura relation connects elliptic curves and modular forms, establishing an isomorphism between certain Hecke modules. This deep result has profound implications for understanding the arithmetic of elliptic curves and their L-functions, forming a crucial link in the modularity theorem.

Definition of elliptic curves

• Elliptic curves are smooth, projective algebraic curves of genus one with a specified base point
• They can be defined over various fields, including the complex numbers, rational numbers, and finite fields
• Elliptic curves have a rich algebraic structure and are central objects of study in number theory and algebraic geometry

Weierstrass equations

• Elliptic curves can be described by Weierstrass equations of the form $y^2 = x^3 + ax + b$, where $a$ and $b$ are constants satisfying certain conditions
• The Weierstrass equation provides a canonical way to represent elliptic curves and study their properties
• The discriminant $\Delta = -16(4a^3 + 27b^2)$ determines the singularity of the curve; if $\Delta \neq 0$, the curve is smooth

Group law on elliptic curves

• Elliptic curves admit a natural group structure, where the group operation is defined geometrically
• The group law is given by the chord-and-tangent process: given two points $P$ and $Q$ on the curve, the sum $P + Q$ is defined as the reflection of the third intersection point of the line through $P$ and $Q$ with the curve
• The group law satisfies the associative, identity, and inverse properties, making elliptic curves an example of an abelian group

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 a finite field, denoted by $\#E(\mathbb{F}_q)$, is an important quantity in cryptography and coding theory
• Hasse's theorem bounds the number of points: $|\#E(\mathbb{F}_q) - (q+1)| \leq 2\sqrt{q}$

Complex points on elliptic curves

• Over the complex numbers, elliptic curves can be viewed as complex tori $\mathbb{C}/\Lambda$, where $\Lambda$ is a lattice in $\mathbb{C}$
• The complex points on an elliptic curve form a Lie group isomorphic to the additive group of $\mathbb{C}$ modulo the lattice $\Lambda$
• The Weierstrass $\wp$-function and its derivative provide a parametrization of complex points on an elliptic curve

Modular forms and elliptic curves

• Modular forms are holomorphic functions on the upper half-plane satisfying certain transformation properties under the action of congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$
• Elliptic curves and modular forms are intimately connected through the modularity theorem and the Eichler-Shimura relation

Modular curves and modular forms

• Modular curves are algebraic curves associated with congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$, such as the modular curve $X_0(N)$
• Modular forms of weight $k$ for a congruence subgroup $\Gamma$ are holomorphic functions $f$ on the upper half-plane satisfying $f(\gamma z) = (cz+d)^k f(z)$ for all $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma$
• The space of modular forms of weight $k$ for $\Gamma$ is a finite-dimensional vector space over $\mathbb{C}$

Hecke operators on modular forms

• Hecke operators $T_n$ are linear operators acting on the space of modular forms of a given weight for a congruence subgroup
• For a prime $p$, the Hecke operator $T_p$ is defined by $T_p f(z) = p^{k-1} f(pz) + \frac{1}{p} \sum_{i=0}^{p-1} f(\frac{z+i}{p})$
• Hecke operators are self-adjoint with respect to the Petersson inner product and commute with each other

Eigenforms and Hecke eigenvalues

• An eigenform is a modular form that is an eigenvector for all Hecke operators $T_n$
• The eigenvalues of $T_n$ acting on an eigenform $f$ are called the Hecke eigenvalues of $f$
• Normalized eigenforms are eigenforms whose Fourier coefficients are algebraic integers and the first coefficient is 1

L-functions of modular forms

• The L-function of a modular form $f(z) = \sum_{n=0}^{\infty} a_n e^{2\pi i n z}$ is defined as $L(f, s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ for $\operatorname{Re}(s) > k+1$
• L-functions of eigenforms have an Euler product expansion $L(f, s) = \prod_p (1 - a_p p^{-s} + p^{k-1-2s})^{-1}$
• The modularity theorem establishes a correspondence between L-functions of elliptic curves and L-functions of modular forms

Galois representations

• Galois representations are continuous homomorphisms from absolute Galois groups to matrix groups over various fields
• They provide a powerful tool to study the arithmetic properties of elliptic curves and their connection to modular forms

Galois groups and representations

• The absolute Galois group $G_K = \operatorname{Gal}(\overline{K}/K)$ of a field $K$ is the group of automorphisms of an algebraic closure $\overline{K}$ that fix $K$
• A Galois representation is a continuous homomorphism $\rho: G_K \to \operatorname{GL}_n(F)$, where $F$ is a field (e.g., $\mathbb{Q}_l$, $\mathbb{C}$)
• The study of Galois representations reveals arithmetic information about the underlying objects (e.g., elliptic curves, modular forms)

Tate modules of elliptic curves

• For an elliptic curve $E$ over a field $K$ and a prime $l$, the $l$-adic Tate module $T_l(E)$ is the inverse limit of the $l$-power torsion points $E[l^n]$
• The Tate module $T_l(E)$ is a free $\mathbb{Z}_l$-module of rank 2, where $\mathbb{Z}_l$ is the ring of $l$-adic integers
• The absolute Galois group $G_K$ acts on $T_l(E)$, giving rise to a Galois representation $\rho_l: G_K \to \operatorname{GL}_2(\mathbb{Z}_l)$

l-adic representations

• An $l$-adic representation is a continuous homomorphism $\rho: G_K \to \operatorname{GL}_n(\mathbb{Q}_l)$, where $\mathbb{Q}_l$ is the field of $l$-adic numbers
• The $l$-adic representation associated with an elliptic curve $E$ is obtained by extending the Tate module representation $\rho_l$ to $\mathbb{Q}_l$
• $l$-adic representations encode arithmetic information about elliptic curves, such as the action of Frobenius elements on torsion points

Modularity of Galois representations

• The modularity of a Galois representation $\rho: G_{\mathbb{Q}} \to \operatorname{GL}_2(\mathbb{Q}_l)$ means that $\rho$ arises from a modular form
• More precisely, $\rho$ is modular if there exists a normalized eigenform $f$ such that the characteristic polynomial of $\rho(\operatorname{Frob}_p)$ equals the Hecke polynomial $1 - a_p X + p X^2$ for almost all primes $p$
• The modularity theorem for elliptic curves states that the $l$-adic representation associated with an elliptic curve over $\mathbb{Q}$ is modular

Eichler-Shimura relation

• The Eichler-Shimura relation is a fundamental result connecting the arithmetic of elliptic curves and modular forms
• It establishes an isomorphism between certain Hecke modules arising from elliptic curves and modular curves

Statement of Eichler-Shimura relation

• Let $\Gamma = \Gamma_0(N)$ be a congruence subgroup and $X = X_0(N)$ the corresponding modular curve
• The Eichler-Shimura relation states that there is an isomorphism of Hecke modules $H^1(X, \mathbb{Q}) \cong S_2(\Gamma) \oplus \overline{S_2(\Gamma)}$, where $S_2(\Gamma)$ is the space of cusp forms of weight 2 for $\Gamma$
• This isomorphism is compatible with the action of Hecke operators on both sides

Hecke correspondences on modular curves

• Hecke correspondences are algebraic curves $T_n$ on $X \times X$ that generalize the Hecke operators on modular forms
• The Hecke correspondence $T_n$ induces a map $T_n^*$ on the cohomology group $H^1(X, \mathbb{Q})$
• The action of $T_n^*$ on $H^1(X, \mathbb{Q})$ corresponds to the action of the Hecke operator $T_n$ on $S_2(\Gamma) \oplus \overline{S_2(\Gamma)}$ under the Eichler-Shimura isomorphism

Jacobians of modular curves

• The Jacobian $J_0(N)$ of the modular curve $X_0(N)$ is an abelian variety that parametrizes degree zero divisors on $X_0(N)$
• The Hecke correspondences $T_n$ induce endomorphisms of $J_0(N)$, which we also denote by $T_n$
• The Eichler-Shimura relation implies that the $\mathbb{Q}$-vector space $\operatorname{Hom}(J_0(N), \mathbb{C})$ is isomorphic to $S_2(\Gamma_0(N)) \oplus \overline{S_2(\Gamma_0(N))}$ as a Hecke module

Isomorphism of Hecke modules

• The Eichler-Shimura isomorphism $H^1(X_0(N), \mathbb{Q}) \cong S_2(\Gamma_0(N)) \oplus \overline{S_2(\Gamma_0(N))}$ is an isomorphism of Hecke modules
• This means that the action of Hecke operators on the cohomology group $H^1(X_0(N), \mathbb{Q})$ corresponds to the action of Hecke operators on the space of cusp forms $S_2(\Gamma_0(N))$ and its complex conjugate
• The isomorphism is given by the period map, which associates to a cusp form $f$ the cohomology class of the differential form $f(z) dz$ on $X_0(N)$

Consequences for L-functions

• The Eichler-Shimura relation has important consequences for the L-functions of elliptic curves and modular forms
• It implies that the L-function of an elliptic curve $E$ of conductor $N$ is equal to the L-function of a normalized eigenform $f \in S_2(\Gamma_0(N))$
• This connection between L-functions is a key ingredient in the proof of the modularity theorem for elliptic curves over $\mathbb{Q}$

Applications and examples

• The theory of elliptic curves and modular forms has numerous applications in number theory and beyond
• Some notable examples include the proof of Fermat's last theorem, the study of congruent numbers, and the Birch and Swinnerton-Dyer conjecture

Elliptic curves over Q

• Elliptic curves over the rational numbers $\mathbb{Q}$ are of particular interest in number theory
• The Mordell-Weil theorem states that the group of rational points $E(\mathbb{Q})$ is finitely generated
• The rank of an elliptic curve, which is the rank of the free part of $E(\mathbb{Q})$, is a crucial invariant that measures the size of the rational point group

Modularity theorem for elliptic curves

• The modularity theorem, proved by Wiles, Taylor-Wiles, and others, states that every elliptic curve over $\mathbb{Q}$ is modular
• This means that for an elliptic curve $E$ of conductor $N$, there exists a normalized eigenform $f \in S_2(\Gamma_0(N))$ such that the L-function of $E$ equals the L-function of $f$
• The modularity theorem is a powerful result that connects the arithmetic of elliptic curves to the theory of modular forms

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, completed by Wiles in 1995, relies crucially on the modularity theorem for semistable elliptic curves
• The key idea is to associate an elliptic curve to a hypothetical solution of Fermat's equation and derive a contradiction using the modularity theorem and Ribet's level-lowering result

Congruent number problem

• A positive integer $n$ is called a congruent number if it is the area of a right triangle with rational side lengths
• The congruent number problem asks for a characterization of congruent numbers
• Tunnell's theorem provides a criterion for congruent numbers in terms of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves

Birch and Swinnerton-Dyer conjecture

• The Birch and Swinnerton-Dyer (BSD) conjecture is a central open problem in the arithmetic of elliptic curves
• It relates the rank of an elliptic curve $E$ over $\mathbb{Q}$ to the order of vanishing of its L-function $L(E, s)$ at $s = 1$
• The BSD conjecture also predicts a precise formula for the leading coefficient of the Taylor expansion of $L(E, s)$ at $s = 1$ in terms of arithmetic invariants of $E$
• The conjecture has been verified for specific classes of elliptic curves, but remains open in general