Day 2: Vlad's second talk

My notes from the second of Vlad's talks:

restated the BSD conjecture RE: taylor series of L-functions at s=1.

Fixing a finite place v, we can find a minimal model for an elliptic curve; one which has coefficients in O_k_v and ordv(disc(E)) minimal, but this is only well defined up to twelfth powers of units in O_k_v.

Turning to this minimal model and reducing modulo v and taking the nonsingular part, we get a containment

E(Kv) > E0(Kv) > E1(Kv)

where Eo are the points of nonzero reduction and E1 are the points that reduce to the identity.

Then taking E0/E1 we get the nonsingular curve over the residue field. The size of this is called the Tamigawa number, c_v.

We can characterize the Tamigawa number and ord_v of the minimal discriminant by the reduction of E at v:

if E has good reduction at v, ord_v(disc) =0, c_v = 1

if E has multiplicative reduction, ord_v(disc) is a positive integer, n; if it is split, c_v=n; if not c_v=1 or 2 depending on if n is odd or even

if E has additive reduction and v has characteristic not 2 or 3, then ord_v(disc) must be one of 2,3,4,6,8,9,10, and c_v is at most 4 and can be found via Tate's algorithm.

Over field extensions, good reduction stays good and split mult stays split; eventually non-split becomes split and additive becomes either multiplicative or good over some field extension.

if ord(j_E) >= 0, the reduction is potentially good; if not the reduction is potentially multiplicative.

If E has model y^2 = x^3 +ax +b we can characterize K(sqrt(-6b))/K:

if the extension is trivial E has split multiplicative reduction; if it is quadratic unramified the reduction is nonsplit and if it is ramified it is additive and potentially multiplicative. I'm not sure what the place v is here.

This talk was yesterday and seems a bit weak to me at the moment, but today we reviewed some pretty serious results that come from this sort of categorization by reduction so I can appreciate now that at least it was going somewhere.