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
- Koblitz, Neal, Introduction to elliptic curves and modular
forms, Graduate Texts in Mathematics, 97. Springer-Verlag, New York,
- Silverman, Joseph H., Advanced topics in the arithmetic of
elliptic curves. Graduate Texts in Mathematics, 151. Springer-Verlag,
New York, 1994.
- Silverman, Joseph H., The arithmetic of elliptic curves. Graduate
Texts in Mathematics, 106. Springer-Verlag, New York, 1992.
- Diamond, Fred; Shurman, Jerry A first course in modular forms.
Graduate Texts in Mathematics, 228. Springer-Verlag, New York, 2005.
Dale Husemoller, Elliptic curves, volume 111, Graduate Texts in
Mathematics, Springer-Verlag, New York, second edition, 2004.
J. S. Milne, Elliptic curves, www.jmilne.org/math/index.html
ZAGIER, D., Elliptic modular forms and their applications, Universitext, Springer, 2008.
Lang, S., Introduction to modular forms, Grund. Math. Wiss. 222, springer, 1995.
- 04/01/2021, Fibonacci numbers, Arithmetic modularity theorem in an example
- 05/01/2021, Elliptic functions, Weierstrass p function, lattices, torus
- 06/01/2021, The differential equation of Weierstrass p function, Fourier expansion of eisenstein series
- 11/01/2021, The algebra of modular forms
- 12/01/2021, Elliptic integrals
- 13/01/2021, Weierstrass uniformization theorem, The algebra of modular forms is generated by E_4,E_6
- 18/01/2021, Rudiments of algebraic geometry
- 19/01/2021, Elliptic curves as groups
- 20/01/2021, Mordell-Weil theorem, Height function
- 25/01/2021, Height function for elliptic curves, Neron-Tate height
- 26/01/2021, Torsion points, Isogeny
- 27/01/2021, Hecke operators for SL(2,Z)
- 30/01/2021, Hecke operators for SL(2,Z)
- 01/02/2021, Exam
- 02/02/2021, Riemann's zeta function
- 03/02/2021, Growth of coefficients of cusp forms
- 08/02/2021, L-functions attached to cusp forms
- 09/02/2021, Congruence groups,
- 15/02/2021, Carnaval
- 16/02/2021, Carnaval
- 17/02/2021, Carnaval
- 23/02/2021, Nagell-Lutz theorem, Mazur's theorem
- 24/02/2021, Rank records of elliptic curves