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.

Conferences Next Year

Something took hold of me and motivated me to take a quick look at what sorts of conferences would be going on next year, so I'm going to compile them here, just posting links and maybe a one or two sentence note.

http://math.postech.ac.kr/~pntag/Winterschool2012/Postechwinter2012.html

BSD at Postech in South Korea

http://math.ucsd.edu/~kedlaya/ants10/

Algebraic number theory in San Diego, easy to get to, with Bhargava and Lenstra!

http://www.ms.u-tokyo.ac.jp/~t-saito/conf/agwtodai/agwtodai.html

Mostly I just want to go to Tokyo... arithmetic geometry, p-adic hodge theory... I dunno.

http://www.birs.ca/events/2012/summer-schools/12ss131

Diophantine equations in Toronto; should have David from this conference I'm at now.

http://www.mathi.uni-heidelberg.de/~iwasawa2012/

If I enjoy Heidelberg and if I start getting into Iwasawa theory. Coates and Ralph Greenberg.

Chris Second Talk

Chris' second talk, from Tuesday morning (it is now Thursday afternoon?! I guess I'll be writing these up during my vacation rather than keeping up with them daily)

The talk started by defining a function lambda from rational numbers to C, but taking the sum of a_n/n times e^(2pi i n r) where a_n are the coefficients from the Fourier expansion of the L-function of an elliptic curve over Q.

Then it turns out (Due to Manin and Drinfield) that the image of lambda is in 1/t times a lattice for some t, where that lattice is the image of H1(E(C),Z) (the free groups of paths on points of E(C)) under the map of integration of the invariant differential along those paths, extended linearly.

A bunch of other stuff happened (mostly formulas which are used in the proofs of the following) but I'm not going to copy it down and I'm skipping a bit, to a point where we're considering a map phi from the modular curve X0(N) to our elliptic curve E, and the differential w_x which is the pullback of the invariant differential on E under phi.

then the integral of w_x along a path from a rational point r to p*r in X0(N) is 1/c times the integral of w along the image of that path in E, which will be a loop, specifically it is in 1/ct Lambda, the where Lambda with a capital L is the lattice from earlier.

A corollary of this is that the images of the rational cusps of X0(N) in E are torsion points, since the integral of ct[phi(r)] is in Lambda.

Chris followed with an unpublished proposition (!) which was that for any elliptic curve one can find an isogenous curve such that t is coprime to all odd primes of semistable reduction.

Now we define modular symbols, which are functions from Q to Q coming from an elliptic curve.

[r] := Re(lambda(r))/Omega+

recall (though I might not have typed it up) that Omega+ is the positive integral of w_x along the curve from i*infinity to a generator (the one that makes it positive) of the +1 eigenspace of the action of the Galois group on the lattice Lambda. Technically we should define this as [r]+ and do something similar for [r]- for when we're in an appropriate imaginary field but we will be working in totally real fields so we won't worry about that.

Now notice that [-r]=[r] and [r+1]=[r]. In the future we generally deal with modular symbols in the context of sums of [a/m] where a ranges through the group of units modulo m.

In the next talks things get more interesting; there's a lot of definitions here and a lot are somewhat complicated.

Barbecue on the Beach!

Today was a good day; thought I started spacing out a bit during the lessons (I didn't get as much sleep as I'd have liked last night because of the tour of Alghero) I talked with some people afterwards and felt like I really gained a lot of understanding.

This evening we went to the beach and we played what is apparently a standard Italian children's game in the water where you bounce a ball and count the bounces and when you get to seven you whack it at someone. It was quite fun, although nobody was quite clear on the rules until at least an hour into the game. We created a great metaphor for it, in which the people standing in the circle around the edges were roots of unity and the person in the middle (we needed someone in the middle to keep the ball up in the air at all, but it turned into a sort of penalty box) was the p-sylow subgroup. Then we counted up and when we got to seven, seven ramified so something happened but it varied a bit. And the Sylow subgroup contained a lot of the action, which was kind of fun but not somewhere that you want to be all the time. And we weren't really sure of the rules; people had pretty good ideas for what the rules were but they were all conjectural results and nobody could really prove anything. When someone was surprised by some decision we made as a mob, we would say "nothing changed, it was always like that we just didn't know before. That's how mathematics works!"

We had some cheap Sardinian beer then had a barbecue dinner on the beach, with some delicious steak, steak fries, bread, and fried eggplant and zucchini. There were also a few bottles of wine that went about; a mild red, a rather forgettable white wine, and then nearer the end of dinner a buttery white wine. A friend of mine had gotten a glass of white wine beforehand which was much better than what they served with dinner; it reminded me of carrot juice in a weird way and had almost no aftertaste.

We had some great conversation trying out accents, telling old jokes, talking about the social dynamics of tutoring for attractive young women.

Overall it was a great night. I'm quite tired and going to go to bed soon.

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.

Non Math Comments

Coming out of class today I noticed that my head was wonderfully empty. All this math is tiring me out enough to make me stop thinking. ME! Surprising, right?

Anyway other than that brief period of nicety there are a few things to be said.

I did a head count in the problem session a few minutes ago, and there were 27 male participants and 6 female ones.

They have not been providing nearly enough nice drinking water, and the tap water here tastes like terrible chlorine.

I went swimming at the beach yesterday, and the water was really beautifully clear and calm; it was warmer than I'm used to (NorCal) but not hot, and still a bit cool to stay in for a long time.

The food was not terribly good at first (cold rice and sandwiches?!) but it has been getting better--today for lunch there was a pasta in alfredo and chicken (among other possibilities: bow ties in marinara, kebobs), and the chicken was covered in chicken broth that dripped onto the plate and was AMAZING for dipping the bread into.

There hasn't been a lot of variety of vegetation, I've mostly been eating salads because that's all that there is, except that the vegetarians are getting more variety there.

Tonight we're going on a historic guided tour of Alghero, and after that is "Alghero by night" which I believe means "going out on the town wooo" which hopefully will be fun?

Last night we had "Sardinian Suits testing" which confused a lot of people until they specified that it was actually meant to be "sweets testing" which was a bit better. There was a champagne-like beverage which tasted mostly like a light Chardonnay and some cheese filled things and a bunch of fruit cookies and flavored macaroons but overall it was not as much chocolate as I would have liked.

I'm going to head off toward the beach to get some sun and pick up some post cards.

Tim's first talk

The goal for this talk was to examine elliptic curves over Q, then their l-adic representations (which emerge from the Tate module), then to construct the L-function.

We can define the L-function as before, and the completed L-function by adding in various factors of the gamma function to make it satisfy a nice functional equation.

This L-function is known to have analytic continuation to C for curves over Q, and holomorphic continuation to C for totally real fields.

Recall that E[n] denotes the group of n-torsion points on E over the algebraic closure of our field, and for E/Q, E[n] = C_n^2.

Identifying E[2] and E[3] is easy enough, but identifying E[4] involves taking the roots of a sextic.

However noting that the absolute Galois group of our field acts on E[n] we get a map, the mod n representation, from the absolute Galois group to GL2(Z/nZ). The image of this map will be the Galois group of Q(E[n])/Q, and since E[n] is the product of E[p^e_i] the primes dividing n, we can simply study prime power torsion.

To start, we define the (l-adic) Tate module, TlE, the inverse limit over n of E[l^n], which is isomorphic as a group to Z_l^2. Then we define VlE the l-adic representation to be TlE tensor Ql. (sometimes TlE or VlE* are also called this)

Now we can also take the inverse limit of our mod l^n representations to get a map from the absolute Galois group to Aut(TlE)=GL2(Z_l)

Theorem (Serre): if E has no CM, then the above representation is surjective for almost all l. This is false if E has CM, however.

It is conjectured that the above theorem is true for l>163, but it is unproven for any universal bound.

So we now want to study E/Qp and describe the action of the absolute Galois group there on E[l^n], TlE, and VlE.

If we take a finite extension K/Qp with Galois group G, define Gi = {g in G| g(x) is equivalent to x mod pi^i+1 for all x in O_k}, where pi is the prime in K sitting under p. This gives us the ramification filtration, G > G0 > G1 >... >{1} with each containment normal.

We call G0 Inertia I and G1 Wild Inertia.

This also seemed to skip around a fair amount. Hopefully when I write up the lectures from today things will get pulled together a bit.

BSD Summer School Talk 2

I've split things up by talk because I want the posts to each be a manageable size and not to have to write too much at once.

Chris' first talk: Modular Symbols

The aim here is to find out about BSD over number fields and their relationship to BSD over Q.

In general we will fix E/Q with a global minimal model and the Neron differential (which is the standard invariant differential).

We will consider H1(E(C),Z) the group of loops in E(C) based at 0 modulo homotopy, which since E(C) is a torus is isomorphic to Z^2. There is a map called the period map from here to C, which sends a loop to the integral of the invariant differential modulo about that loop. Its image is a lattice in C called the Neron lattice, and E(C) is isomorphic to C mod that lattice.

X0(N) is the modular curve of level N, which over C is the upper half plane adjoin the point at infinite and Q, modulo \lambda_0(N), the group of matrices in SL2(Z) which are upper triangular mod N.

Theorem due to Wiles, Taylor, et. al.:

There is a non-constant morphism from the modular curve of level N to an elliptic curve of conductor N sending the point at infinity to the zero of E

There exists an integer, the Manin constant, such that the pullback of the invariant differential of E is a scalar multiple of gamma_x, given by an explicit formula with coefficients closely related to the Taylor expansion of the L function of E/Q.

The scalar multiple above can be made to be 1 by moving to an isogenous curve.

Our next purpose is to show that the special values of L-functions are rational multiples of the positive generator of the subgroup of eigenvectors of the period map with eigenvalue 1.

Calling the images of Q and infinity in X0(N)(C) cusps, we take the free group on paths between cusps modulo homotopy (when tensored with Qp this gives us the dual of etale cohomology of the modular curve over Qp)

We now take the map lambda to be the integral over these paths on X0(N)(C) of the pullback of the invariant differential. Because of our explicit formula for that differential, we can integrate; though this gives us a series that converges very slowly and not obviously.

Hm. Rereading this, I think I'm understanding more of what is going on, but it doesn't feel very well structured; goals were outlined but not really achieved.

Beginnings of the BSD Summer School

Here is my plan, to induce myself to study as well as to provide updates about my vacation, I'll be typing up some notes on the notes that I take on the conferences I attend.

There are three lecturers, Vladimir Dokchitser, Christian Wuthrich, and Tim Dokchitser. Their talks are respectively about Birch-Swinnerton-Dyer and Parity conjectures, modular symbols and BSD over abelian fields, and L-functions and root numbers. Here are some comments on the first two days of notes that I've taken.

Vlad's first talk:

Reviewed the definition and group law for elliptic curves, and the Mordell-Weil Theorem that the group of an elliptic curve is finitely generated as well as abelian; call the rank of the free part of that group the arithmetic rank. Questions you might ask are: what ranks occur for curves over Q? Over number fields? It is conjectured that there are curves of unbounded rank, however it is known (conjectured?) that half of them are rank 0, and half are rank 1. It is also known that every elliptic curve over a number field has a quadratic twist of rank 0 or 1.

The L function of an elliptic curve is the product over all primes of the inverses of polynomials:

(1-a_p*p^-s+p*p^-2s) where a_p is the number of points on the curve modulo p; with some adjustments for primes of bad reduction (this works with the norms of primes over generic number fields). For all p, the local factor at s=1 is the norm of p divided by the number of points on E mod p. L(E/k,s) has an analytic continuation to C and satisfies a functional equation. This is known over Q due to Wiles, Taylor, et. al.

We define the analytic rank of E/k to be the order of vanishing of the L function at s=1. Then the Birch and Swinnerton-Dyer conjecture (as reformulated with Tate) states:

i) the arithmetic rank is equal to the analytic rank

ii) the terms of the Taylor series are given by explicit formulas (too long to include here)

The formula involves the regulator or E/K, the terrifying Tate-Shafarevich group, Tamigawa numbers, an integral of the invariant differential for each infinite place, the discriminant of K, and the size of the torsion subgroup of E(k)

A useful fact for bounding the rank of an elliptic curve: if the n-torsion subgroup of an elliptic curve is in K, then we can look at points which are 1/n parts of points on our curve, which must lie in a field with Galois group inside (C_n)^2 and satisfy further conditions. We get a map from our free points to this Galois group which is at worst two-to-one, and this allows us to use Galois theory to bound the rank of the curve.

Using this alone, one can prove that if E/Q has good reduction at 2 or 3, the torsion group of E(Q) cannot be bigger than order 12. To do this we use the Hasse-Weil bound to note that E cannot have more than 5 points mod 2 or 7 points mod 3, and that we have an injective map from 2-torsion points to E(F_3) which must have size 2^n less than 7, so it must be 1,2, or 4. Similarly we must have an injection from 3-torsion to E(F_2) which has size power of 3 less than 5, so it must be 1 or 3. This allows us groups up to order 12. If there is any other prime torsion injecting in, the prime must be no bigger than 5, but five can inject in. However, if it does, the size of the torsion group cannot be divisible by 3 or 2, or we would have 15<5, 10<7, which is fail. So groups of order 1,2,3,4,5,6,12 are the only possibilities.

Third Day in Italy

It is my third day in Italy, the second day of the international BSD Summer School at the Porto Conte Research center in Alghero, Sardinia.

I've put a few pictures on facebook; yesterday we went to the beach and tonight there is a historic guided tour of Alghero; presumably more pictures will emerge from that.

The lessons have been a bit advanced for me, but I'd like to put up summaries of them every day if I can make the time; hopefully that will help me absorb them a bit better.

It's very warm hear; I'm fairly certain that I'll sweat through all of my socks before the week is up.

Anyway this is going to be my place for updates and blogging, so here I go!