Introduction to modular forms and elliptic curves, 03 January-25 February 2022
Wednesday, Thursday, Friday 8:00-10:00 (Rio de Janeiro's time)
Classroom Class code ym4unxk
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.
- 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.
Movasati, H., A differential Introduction to modular forms and elliptic curves,
- 05/01/2022, Fibonacci numbers, Arithmetic modularity theorem in an example
- 06/01/2022, Elliptic functions, Weierstrass p function, lattices, torus
- 07/01/2021, 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