Topics in modular forms and elliptic curves, 04 January-26 February 2021

Monday, Tuesday, Wednesday, 13:00-15:00 (Rio de Janeiro's time) Main teacher: Roberto Villaflor

Videos of the course in 2018


Modular and congruence groups, modular forms of a given weight, cusp forms, Eisenstein series, theta series, Weierstrass pi function, elliptic curves in Weierstrass format, elliptic curves as group, rank of elliptic curves, Mordell-Weil theorem, Hecke operators, Fourier expansions, Growth of the coefficients, L-functions of modular forms and elliptic curves, Birch Swinnerton-Dyer conjecture, functional equation of L-functions, Old forms and new forms, modular elliptic curves, Galois representations and modular forms, application to congruent numbers, Arithmetic modularity of elliptic curves and its relation with Fermat's last theorem.


See also my lecture notes Lecture notes
  1. 04/01/2021, Fibonacci numbers, Arithmetic modularity theorem in an example
  2. 05/01/2021, Elliptic functions, Weierstrass p function, lattices, torus
  3. 06/01/2021, The differential equation of Weierstrass p function, Fourier expansion of eisenstein series
  4. 11/01/2021, The algebra of modular forms
  5. 12/01/2021, Elliptic integrals
  6. 13/01/2021, Weierstrass uniformization theorem, The algebra of modular forms is generated by E_4,E_6
  7. 18/01/2021, Rudiments of algebraic geometry
  8. 19/01/2021, Elliptic curves as groups
  9. 20/01/2021, Mordell-Weil theorem, Height function
  10. 25/01/2021, Height function for elliptic curves, Neron-Tate height
  11. 26/01/2021, Torsion points, Isogeny
  12. 27/01/2021, Hecke operators for SL(2,Z)
  13. 30/01/2021, Hecke operators for SL(2,Z)
  14. 01/02/2021, Exam
  15. 02/02/2021, Riemann's zeta function
  16. 03/02/2021, Growth of coefficients of cusp forms
  17. 08/02/2021, L-functions attached to cusp forms
  18. 09/02/2021, Congruence groups,
  19. 10/02/2021,
  20. 15/02/2021, Carnaval
  21. 16/02/2021, Carnaval
  22. 17/02/2021, Carnaval
  23. 22/02/2021,
  24. 23/02/2021, Nagell-Lutz theorem, Mazur's theorem
  25. 24/02/2021, Rank records of elliptic curves