Number Theory

1. Modular Forms and Elliptic Curves
   1. Modular Forms
      1. Definitions and Key Properties
         1. Holomorphic modular forms
         2. Cusp forms
         3. Weight, level, and character
         4. Fourier coefficients
      2. Types of Modular Forms
         1. Eisenstein series
         2. Galois representations
         3. Hecke eigenforms
         4. Theta functions
      3. Modularity Theorem
         1. History and background (Taniyama-Shimura-Weil conjecture)
         2. Proof overview and key contributors
         3. Implications for Fermat's Last Theorem
         4. Relationship with elliptic curves and L-functions
      4. Mathematical Structures and Theorems
         1. Hecke operators and algebra
         2. Petersson inner product
         3. Atkin-Lehner theory
         4. Newforms and oldforms
      5. Applications
         1. Number theory and partitions
         2. Representation of numbers by quadratic forms
         3. Link with string theory and theoretical physics
   2. Elliptic Curves
      1. Definitions and Properties
         1. Weierstrass equation
         2. Projective coordinates
         3. Group law on elliptic curves
         4. Elliptic curve over various fields (Q, R, C, finite fields)
      2. Rational Points
         1. Mordell-Weil theorem
         2. Heights and descent method
         3. Torsion subgroup (Mazur's theorem)
      3. L-functions and Conjectures
         1. Definition of L-functions for elliptic curves
         2. Birch and Swinnerton-Dyer conjecture
            1. Statement of the conjecture
            2. Rank of an elliptic curve
            3. Analytic and arithmetic rank
            4. Current progress and open problems
      4. Complex Multiplication
         1. Endomorphism ring
         2. Class field theory connection
         3. Applications in cryptography
      5. Cryptographic Applications
         1. Basics of elliptic curve cryptography (ECC)
         2. ECC key exchange and digital signatures
         3. Security advantages over other cryptographic systems
      6. Algorithmic Challenges
         1. Computation of rational points
         2. Use of elliptic curves in primality testing
         3. Elliptic curve factorization
      7. Interdisciplinary Impacts
         1. Role in string theory and physics
         2. Model creation in complex analysis and geometry