Serre's invariants for icosahedral mod 2 Galois representations

Bryden Cais

This program was written to verify and expand some of the examples of mod 2 Galois representations from section 5.2 of Serre's paper "Sur les representations modulaires de degre 2 de Gal_Q." Serre refers to Mestre's paper "La methode de graphes: Exemples et applications" for the details, but Mestre's exposition is incomplete and contains some errors. We give a complete exposition and correct these errors, and we provide several new examples in our notes on Serre's conjectures.

The main function of this program is "SerreInvariants" which takes as input a quintic, monic polynomial g with integer coefficients, whose Galois group is A_5, and a positive integer N. Since A_5 = SL_2(F_4) = GL_2(F_4), we get a mod 2 Galois representation (which is automatically odd) that turns out (for group theoretical reasons) to be independent of the identification of the Galois group of G with GL_2(F_4). Our program returns the predicted level, weight and character (=1) of a modular form giving rise to this representation, and lists the traces of Frob_l for primes l < N up to Gal(F_4/F_2)-conjugacy (if l is ramified, the value given by our program should be ignored). One can then consult William Stein's tables to try and find a corresponding eigenform of the right level and weight (at least if the weight is bigger than 1).

We also include the program "MakeList" which takes an integer B as input and returns a list of those quintic polynomials of the form

x^5 + ix^3 + jx^2 + kx + l

with -B < i,j,k,l < B that have Galois group A_5. This uses an exhaustive algorithm.

The code is documented, and the ideas underlying the algorithms we used are explained thoroughly in the aforementioned notes above.