Galois representations attached to modular forms
Let f = q +
+∞
X
n=2
anqn
∈ Sk Γ1(N), ε


, k > 2, be a newform
with coefficient field Kf = Q(an, n > 2).
Pick a prime l of Kf above some ` ∈ N.
Theorem (Deligne, Serre)
There exists a Galois representation
ρf ,l : Gal(Q/Q) −→ GL2(Fl),
which is unramified outside `N, and such that the image of
any Frobenius element at p - `N has characteristic polynomial
x2
− apx + ε(p)pk−1
∈ Fl[x].
Nicolas Mascot p-adic computation of Galois representations