Barry Mazur's paper "Modular Curves and the Eisenstein Ideal" explores the classification of elliptic curves over number fields, focusing on the modular curves \(X_1(N)\) and \(X_0(N)\). The main goal is to determine the rational points on these curves, which are related to the torsion subgroups of the Jacobians of the curves. Mazur uses advanced techniques from algebraic geometry and number theory, including modular forms and Hecke algebras, to achieve this. Key results include: 1. **Theorem (1)**: For a prime \(N \geq 5\), the torsion subgroup of the Mordell-Weil group of the Jacobian \(J\) of \(X_1(N)\) is cyclic of order \(n = \text{numerator}(\frac{N-1}{12})\), generated by the linear equivalence class of the difference of the two cusps \((0) - (1)\). 2. **Theorem (2)**: The maximal \(\mu\)-type group of \(J\) is the Shimura subgroup, which is cyclic of order \(n\). 3. **Theorem (3)**: The Mordell-Weil group of \(J^+\) (the Jacobian of the modular curve \(X_0(N)\) with the canonical involution) is a free abelian group of positive rank, provided the genus of \(X_0(N)\) is at least 73 or \(N = 37, 43, 53, 61, 67\). 4. **Theorem (4)**: The natural map from the Jacobian \(J\) to its Eisenstein quotient \(J'\) induces an isomorphism between the cyclic group of order \(n\) generated by the linear equivalence class of \((0) - (1)\) and the Mordell-Weil group of \(J'\). 5. **Theorem (5)**: There exist absolutely simple abelian varieties of arbitrarily high dimension over \(\mathbf{Q}\) with finite Mordell-Weil groups. 6. **Theorem (6)**: For prime \(N\) such that the genus of \(X_0(N)\) is greater than zero, \(X_0(N)\) has only finitely many rational points over \(\mathbf{Q}\). 7. **Theorem (7)**: For \(m\) an integer such that the genus of \(X_0(m)\) is greater than zero, the only rational points of \(X_0(m)\) are the rational cusps. 8. **Theorem (8)**: The possible torsion subgroups of the Mordell-Weil groups of elliptic curves over \(\mathbf{Q}\) are limited to 15 specificBarry Mazur's paper "Modular Curves and the Eisenstein Ideal" explores the classification of elliptic curves over number fields, focusing on the modular curves \(X_1(N)\) and \(X_0(N)\). The main goal is to determine the rational points on these curves, which are related to the torsion subgroups of the Jacobians of the curves. Mazur uses advanced techniques from algebraic geometry and number theory, including modular forms and Hecke algebras, to achieve this. Key results include: 1. **Theorem (1)**: For a prime \(N \geq 5\), the torsion subgroup of the Mordell-Weil group of the Jacobian \(J\) of \(X_1(N)\) is cyclic of order \(n = \text{numerator}(\frac{N-1}{12})\), generated by the linear equivalence class of the difference of the two cusps \((0) - (1)\). 2. **Theorem (2)**: The maximal \(\mu\)-type group of \(J\) is the Shimura subgroup, which is cyclic of order \(n\). 3. **Theorem (3)**: The Mordell-Weil group of \(J^+\) (the Jacobian of the modular curve \(X_0(N)\) with the canonical involution) is a free abelian group of positive rank, provided the genus of \(X_0(N)\) is at least 73 or \(N = 37, 43, 53, 61, 67\). 4. **Theorem (4)**: The natural map from the Jacobian \(J\) to its Eisenstein quotient \(J'\) induces an isomorphism between the cyclic group of order \(n\) generated by the linear equivalence class of \((0) - (1)\) and the Mordell-Weil group of \(J'\). 5. **Theorem (5)**: There exist absolutely simple abelian varieties of arbitrarily high dimension over \(\mathbf{Q}\) with finite Mordell-Weil groups. 6. **Theorem (6)**: For prime \(N\) such that the genus of \(X_0(N)\) is greater than zero, \(X_0(N)\) has only finitely many rational points over \(\mathbf{Q}\). 7. **Theorem (7)**: For \(m\) an integer such that the genus of \(X_0(m)\) is greater than zero, the only rational points of \(X_0(m)\) are the rational cusps. 8. **Theorem (8)**: The possible torsion subgroups of the Mordell-Weil groups of elliptic curves over \(\mathbf{Q}\) are limited to 15 specific