Tim's Second Talk

Talk from Tuesday morning.

On Monday we described the ramification filtration of a Galois group of a finite extension of Qp.

Define the inertia group of K/Qp to be the set of galois elements whose induced action on Fp is trivial, and we notice that G1 < I=G0 is the p-sylow subgroup. Then we call G0/G1 tame interia; which has order prime to p and is cyclic and embeds into K^x_Qp.

Define: An arithmetic frobenius map to be Frob_p any element of G_Qp reducing to Frobenius in G_Fp. This is only well defined up to Inertia, which is usually quite large, but in the future we will take inertial invariants of things to make it well defined.

Def: a G_Qp module is unramified if Inertia acts trivially on M; this makes M a Frob_p module.

Theorem(Neron, Ogg, Shafarevich): F/Qp finite, residue field Fq, E/F elliptic curve, l not p, then E/F has good reduction at l iff TlE is unramified.

In this case, the characteristic polynomial of Frob_p on TlE is x^2-a_p x+q where a_p is the standard q+1 - #E(Fq).

Define the local polynomial Fp(T):= det(1-Frob_p^-1 *T | (VlE*) ^ I_F) for an elliptic curve E/F.

cor: if E/Qp has good reduction then Fp(T) = 1-aT+pT^2.

Among other things, this means (VlE*)^I is 2-dimensional.

Recall that any curve acquires semistable reduction over some finite field extension. Then potentially good curves E/Qp, p not 2 or 3, acquire good reduction over Qp(deltaE^(1/12)), which implies that I_Qp acts through inertia over the Galois extension containing that field, which means its action is cyclic of order dividing 12; a difficult exercise is to show that 12 cannot ever happen.

This lecture contained a great number of examples which I have not included and probably should review in more detail, but it is dinner time.