A CIMPA research school on

Introduction to Galois representations and modular forms
and their computational aspects


University of the Philippines Diliman, Manila
January 9th - January 20th, 2023





     

 

Courses
     
The computational courses will be done using PARI/GP, please do install it on your computer. Instruction on installation, by Jared Asuncion, can be found here
Number Theoretical background
Ila Varma and Richell Celeste		 Introduction to modular forms
Laura Geatti and Valentijn Karemaker
Program


Introduction to the course and motivation. The action of the modular group SL2(Z) on the upper half plane in C: its fundamental domain, cusps. Lattices, modular functions, modular forms and cusp forms. The Eisenstein series as examples of modular forms: transformation law and convergence. The k/12 -formula for modular functions. The graded algebra of modular forms: dimension and generators of the spaces of forms of weight k. The Fourier expansion of the modular forms Ek, especially E4 and E6. Arithmetic identities derived from the Fourier coefficients. The discriminant function Δ and the Ramanujan τ function. Hecke operators and application to the Ramanujan conjecture (time permitting).

PROBLEMS for the first training session.

PROBLEMS for the first training session.

References:
J.P. Serre, A course in arithmetic, GTM 7, Springer
D. Zagier, Elliptic modular forms and their applications, in 123 of modular forms, Springer, pdf

Links to the recordings of the lectures on Modular Forms delivered in Yogjakarta (Indonesia), in February 2020.


Prerequisites: An introductory course in one complex variable. The slides of Lecture 1 contain the main results which are needed.
Elliptic curves with complex multiplication
Jerome T. Dimabayao and Michel Waldschmidt 		Galois representations of elliptic curves
René Schoof
Explicit class field theory
Peter Stevenhagen

Exercises		 Computational number theory
Francesco Campagna and Valerio Talamanca
Exercise session
Useful links
Solution 1, Solution 2,
Computational aspects of Modular forms
Victor Manuel Aricheta
Computations with class field theory
Jared Asuncion and Francesco Campagna
Tutorials