Galois representations modulo $p$ that do not lift modulo $p^2$
Résumé
For every finite group H H and every finite H H -module A A , we determine the subgroup of negligible classes in H 2 ( H , A ) H^2(H,A) , in the sense of Serre, over fields with enough roots of unity. As a consequence, we show that for every odd prime p p , every integer n ≥ 3 n\geq 3 , and every field F F containing a primitive p p -th root of unity, there exists a continuous n n -dimensional mod p p representation of the absolute Galois group of F ( x 1 , … , x p ) F(x_1,\dots ,x_p) which does not lift modulo p 2 p^2 . This answers a question of Khare and Serre, and disproves a conjecture of Florence.