Near the Pacific Ocean

I arrived in Seattle last night and hung out with Lucas, who I'm staying with. He has yet to awaken but we will have some hangouts before I leave for the airport since my flight to Oakland isn't until 7:15pm.

It's been fun to hang out and it's been GREAT to have non-ridiculous phone service. I got charged some ridiculous fees for having to use my data plan in Germany...

Anyway tomorrow night it looks like I'll be staying in San Francisco with Kendra Ryan and Josh, and then heading up to Santa Rosa to see my mom and get back to Arcata!

Yay for Norcal!

But to be entirely honest, it will be nice when I finally get back to Santa Cruz...

Heidelberg, nearing the end

So I'll be leaving Heidelberg tomorrow morning. I got an e-mail from the consulate only yesterday saying they had received my passport, and I could make an appointment to pick it up; predictably there were no appointment slots left but I was able to request an emergency appointment and the request was granted this afternoon, so I should be able to stop by quickly and pick it up no problem. Then I'll be straight back to the USA, which will be a big relief for me I think.

I'm looking forward to seeing everyone, and I'm looking forward to getting home and getting things back to normal. I'm tired of all this traveling around.

Heidelberg

So I managed to make it to Heidelberg at long last, on Wednesday evening. I attended a couple days of lectures and slept long nights, and only now have had the time to get on to the internet for some blogging.

The main reason I haven't been on is that there is no wireless at my hotel! It's terrible! There is internet at the conference and the university, but it's a 25 minute walk that I don't like making with my laptop in tow. But I figured it would be worth it to bring you guys an update so here ya go!

I had a bit of trouble following the lectures at the summer school, since it was building on a lot of material I'd missed in the first few days. Hopefully the talks at the conference will be better.

Heidelberg is really hot, almost like there are actually seasons in this part of the world and this one is the hot one. It's not so bad as Florence though, so I think I'll survive.

There's a big old castle in Heidelberg and I went there yesterday with Alex and Yun, a Belgian grad student we met at the summer school.

Overall it's been neat talking to people and hanging out, but I'm hoping to actually learn some real math next week, and I'm looking forward to getting home.

PAX

Penny Arcade Expo has been pretty fun! I've been competing in the PAX Pokemon League (I've beaten 9 gym leaders and 2 elite four members, and have a few challenges left for tomorrow), got a couple awesome T-shirts, had an impromptu hula hoop competition (which I lost), played Dragon Dice, an awesome game which I thought was lost to Time, and played some Four Calibur.... SOUL!

Overall it's been pretty fun. There's a lot going on, far too much to get to everything I'd like to. It's been keeping me busy and making it tough for me to take care of non-PAX things that I might like to be taking care of in the evenings. But overall it's been a great experience.

I'll be leaving for Germany Monday morning, pretty early, so I'm a little nervous about getting to the airport on time. I may just get a taxi so that I don't have to worry about it.

The Countdown

So I've been home in Santa Cruz hanging out for a while and that's been nice, but I'm about to get back on the road again on Wednesday, when I'll be driving up to Seattle for PAX. I'm excited to see my friends who are going, especially Eugene, Lucas, Lauren, Cambria, Gabe, and Puppo!

Today I made sure all my boxes were packed and moved them downstairs, as well as moving my bookshelves downstairs, and tonight I'll put my dresser downstairs and take apart my desk, because tomorrow morning I am moving all my stuff out into storage until I get back.

It's been a little bit frustrating having so much to do and not being able to do any of it because I don't have a car myself, but I'm glad that my friends are coming to help me out tomorrow and I'm confident we'll get it all done. And then it will be all fun times!

Back Home (mostly)

I'm back in the states, in San Francisco, waiting to say hi to some friends before I head down to Santa Cruz and settle back in (for all of ten days -_- ).

I'm pretty tired and ready to be at home for a while, where there is wireless all the time and plugs that fit my devices and a big bed and a substantially less public shower and a refrigerator and all that.

I'm a little weirded out by american money now, though. It's all... clothy? And dirty. And the denominations are weird. 2 euro coins are dreadfully convenient.

Travel Woes Update and excerpts from an awesome conversation

So I am now in Dusseldorf; it is 1:25am, and I am rather tired.

My flight yesterday from Rome to SFO started at 6:20 am. However the first bus to the Da Vinci airport, apparently, gets there at 6:00 am. Which is not soon enough. I probably should have taken a taxi. However, hindsight is 20/20 and it is far too late to change the past. All the flights for that day were overbooked, and the cheapest flights I could get on were over $2000. So I went on the internet at the price of .20 euro a minute to find a cheaper flight... and I found one for about half the price of the others, and snapped it up... before discovering that it left the NEXT EVENING then had a 30 hour+ layover in Dusseldorf.

So here I am in Dusseldorf. UGH.

Anyway that's depressing so here's something awesome from a conversation with my friend Cambria: (edited for clarity and conciseness)

9:13:16 PM Cambria Scalapino: what was the weirdest thing that has happened to you so far?

9:13:22 PM Mitchell Owen: Hmmm

9:13:57 PM Mitchell Owen: Candidate: being solicited by a prostitute

9:14:24 PM Mitchell Owen: Candidate: discovering a square filled with Harry potter posters

9:14:57 PM Mitchell Owen: Candidate: returning to a hostel and passing the David at sunrise

9:15:17 PM Mitchell Owen: Candidate: tossed my drink on someone

9:15:36 PM Cambria Scalapino: ...was it on purpose?

9:15:41 PM Mitchell Owen: Yes

9:15:36 PM Mitchell Owen: Candidate: had people debate whether or not I was gay

9:16:17 PM Mitchell Owen: Candidate: bought a glorious sparkly pink hat which is my best friend and we have followed each other everywhere

9:17:38 PM Cambria Scalapino: oh man HATS.

9:18:34 PM Mitchell Owen: Candidate: talked to a stranger about my Tegmark multiverse theory of self and applications

9:19:14 PM Mitchell Owen: Candidate: stayed up until all hours talking to Christians about my religious beliefs

9:19:21 PM Cambria Scalapino: XD

9:19:56 PM Mitchell Owen: Candidate: a philosophy student from Oxford got totally wasted and was hilarious

9:20:13 PM Mitchell Owen: Candidate: carried a German girl most of the way up a hill

Back in Rome

So I made my way back to Rome via Barcelona. I didn't have much time to spend in Barcelona, but I did have enough to meet two people at my hostel who were taking the same flight to Rome that I was! So hopefully I have some friends to visit the catacombs with this afternoon.

Then on Tuesday morning I'll be flying in to SFO. And then home!

Sato Tate for Picard Curves

So I gave my talk this morning and it went okay, though it was pretty short because I didn't really have all that much to say. I'm trying to do some more computations now but I have to do all kinds of software update crap so... bleh. Anyway here are my notes for my talk they are pretty thorough but I didn't follow them as closely as all that.

Sato Tate for Picard Curves

Benasque 2011

Mitchell Owen

Work conducted under Martin Weissman

As work for my undergraduate thesis, I did some computations for Sato-Tate over a certain extremely tame family of genus 3 curves known as Picard curves. A Picard curve is one isomorphic to a curve with projective model zy^3 = x^4 + ax^3z+bx^2z^2 +cxz^3 + dz^4. We can assume that the coefficient for cubes is zero by completing the quartic (replace x with x + a/4 or something) so long as our characteristic is not 2, but since I only dealt with equidistribution statements discarding a finite number of primes is fine; we will also want to discard primes dividing the discriminant which leaves us with primes of good reduction; however since this is a finite number of primes again it doesn't really matter. (i.e. we could count points over these fields and the information would be washed out.)

Picard curves have local L-factors that are sextic [insert here], and the work of Upton tells us that we should expect the image of Galois to be GU_3. I'm still not entirely sure what this means but you can see that the trace of Frobenius is equal to a_p, which allows us to make some very simple computations.

I wrote a program to use a naïve point count method to check the Sato-Tate distribution for these curves computationally. Of course as you all probably know counting points is very very slow, so this is definitely a computation worth revisiting using some methods with twists which Drew sent me an e-mail about but I haven't had a chance to read through in detail yet.

So what I did end up actually able to do was count points over F_p for p up to 30,000 for four curves, Then calculated the first couple moments and put together some fairly clunky histograms compared to the beautiful things Drew showed us last week.

So based on what I was told as an undergraduate I expected the traces to be distributed like random traces of matrices in U_3. One could use the Weyl integration formula with information about N_p^2 and N_p^3 but it's fairly messy and I don't know how.

What I could do was calculate the even moments and from Diaconis and Shashahani it is easy to see that the first two even moments are 2 and 12. My computations for the following three curves were fairly close, although a couple of the twelves looked a bit like 13s; I have been told that this is called Chebyshev's bias and that it isn't surprising.

Then I had another curve which obviously has some extra automorphisms and it's moments were totally different; 4 and 46.

And here are the histograms!

Chris' Third Talk

Consider the following elliptic curve:

C: y^2 = x^3 + 5^2

Which has additive reduction over Q5 but good reduction over Q5(5^1/3)

Now I_Q5 (the Inertia group, see previous posts) acts on VlE through C3 for all l other than 5. In our example, I_Q5 injects into VlE tensor Ql bar, and acts by a two by two diagonal matrix with entries phi and phi inverse, where phi is some character of order 3.

Define Xl:G_Qp -> Zl* as the action of G_Qp on the inverse limit of l^n roots of unity; then we call Xl the l-adic cyclotomic character.

Recall that TlE wedge TlE is isomorphic to Xl by the Weil paring.

I missed a bunch in the talk at this point.

Theorem("Tate Curve"): say E/Qp has split multiplicative reduction.

Then there is some q in Zp so that v(q)=v(disc(E))=-v(j)=n>0 (not sure what this v function is)

such that

E(Qp bar) ~ Qp bar cross modulo q^Z

here q^z represents a sort of spiral lattice; think of them as integer powers of a complex number; this allows us to "rotate" the p-adic space to within a certain sort of angle/absolute value pie chunk.

so E[l]= = (Z/lZ)^2, and E[l^n]=, and TlE=Zl^2.

Now G_Qp acts on TlE as an upper diagonal two by two matrix where the upper left entry is the l-adic cyclotomic character and the lower right entry is 1.

And I_Qp acts as an upper triangular matrix with 1s on the diagonal and v(q)*phi is in the upper right.

Then we define this phi to be the tame character; i.e. it is the part of Inertia that acts in a way we can observe like this.

So here I_Qp/G1 = the product of Zl for l not p, and we are projecting onto specific Zl.

Corollary: E/Qp has split multiplicative reduction -> Fp(T)=1-T. Yay local factors!

Now the case of potentially multiplicative reduction:

say E/K: y^2=x^3+ax+b

then the quadratic twist of E by d is

Ed/K: dy^2=x^3+ax+b

which is isogenous to E over K(d^1/2)

Let phi from the Galois group of that field extension be the nontrivial character.

Then Vl(Ed)=VlE tensor phi (this was an exercise to prove. I don't entirely know what it means)

Now say our E has pot. multiplicative reduction. Now consider E twisted by -6B, which has split multiplicative reduction (seeing why was maybe an exercise?)

now phi:Gal -> +-1 nontrivially means in the non-split multiplicative case, inertia acts trivially so Frobp |-> -1, in the additive inertia can act as -1 but this isn't too big a deal I guess? in the first case we can see that Fp(T)=1+T; in the additive case Fp(T)=1.

What this finally allows us to do is characterize Fp(T) for all cases. Yay!

Oh god oh god oh god

So I finally made it to an actual conference event, a talk by Victor Rotger followed by Drew Sutherland, two of the three people I'd heard of before the conference, and I followed the talk pretty well for the most part so that was neat. I ran out of lined paper; not so neat.

Anyway after the talk Joan-Carlos Lario, one of the organizers, asked me if I wanted to do a talk on my work on Picard curves next week.

Sweet goat mother with a thousand young. Luacs the spider queen whose eyes are always watching.

I am a bit apprehensive.

Basically my research can be summed up in about a sentence, and I'm very wary about what I should do to make that one sentence of information last longer than five minutes.

Anyway it's exciting though! And it'll be a good thing to put on my CV! So yay.

Benasque!

So my alarm decided not to wake me up when I told it to this morning, combined with getting lost at the train station trying to find the bus station I ended up missing two buses, but I did manage to finally catch one and got into Benasque around 7:30pm, checked into my hotel which was literally right next to the bus stop, and then ate a big salad and an entire meat pizza by myself because I was that hungry. I think the waitstaff was mildly surprised.

I have my own room with my own bathroom and shower and plug and desk and there is wifi and the room is honestly pretty small it is not the nicest of hotel rooms but compared to all the hostel dorms I've been staying at it is like heaven.

I'm going to go out and find the Scientific center so that I know where to go tomorrow to actually get to take part in the conference, then I'm probably going to dick around a bit on the internet and go to bed early. The conference seems to be pretty small; only 14 people are attending in total! So I'm a bit intimidated that all of them will know/not know who I am I guess? I don't really know what I should be expecting. Anyway I'm out to pursue the maths!

Barcelona Duex

Hi all,

I'm still in Barcelona, about to hit bed and get up early to take the bus to Benasque. I'm still in Barcelona because the bus from Barcelona to Benasque for the conference was cancelled, since only 4 people wanted to take it. Then the conference organizers arranged for me to share a ride with someone, but it was on a different day than the bus originally was which through me off (I thought it would be today, but it was yesterday!) so now I need to catch a public bus tomorrow. It's been a bit of a hectic day and I've spent a lot of time walking around town carrying my bags (UGH) but it's almost done now and I'll be up in the mountains doing mathematics. Be prepared for my next few posts to be summaries of notes from lectures or descriptions of whatever papers are being presented!

I did have a nice dinner today at an Irish pub (there are far more of these in Barcelona than I would have expected!) which was a stew with beef, carrots, peas, onions, etc.; it has been hard to efficiently eat vegetables and so my body experienced a wave of intense relief at all of the vitamins that it has been missing out on.

Barcelona

Hola! Finally, I am in Barcelona, my last destination before the conference on Sato Tate in higher dimension and then my return to California. I'm honestly looking forward to getting home; I've been gone a long time and things are just a bit too interesting here I think :P

But Barcelona is nice; the weather is warmer than France and Amsterdam, but not ridiculously hot like Italy. At night, it's a bit too warm to wear a jacket.

It's also nice to speak Spanish so that when I'm going around town I can actually communicate with people much of the time. Although Catalan and Spain-Spanish are a bit different from Mexico-Spanish which is really what I know. Regardless it seems nice enough and I've got a few days to actually experience the city before I head out.

Paris 2: Electric Bugaloo.

So In my two days in Paris, overall, I have spent about 50 euros on food. But due to poor planning and execution, most of that food was quick pickups on my way somewhere.

But tonight, oh tonight. I spent 3o of those euros on one meal.

It started with chevre wrapped in ham on a bed of lettuce with tomatoes, a basket of soft baguette slices and a glass of bourdeaux, which lasted me until just before dessert.

Next came a small steak with a plate of fried scalloped potatoes and a cheese sauce.

Then, after I was already full, a chocolate cake. Small, about cupcake sized, filled with a gooey chocolate inside, accompanied by cream and a fruity sauce. The chocolate was rich enough for me, which is saying something if you know me and my taste in chocolate.

It was the only real meal I got to eat in Paris, so I had to make it count. I think that went well.

OH NO!

Oh no everyone! I forgot something terribly important! While I was busy getting extremely sunburnt and ripped off by Italians, my amazing father turned 68 and I totally forgot about it!

If more than 2 people actually read this blog then I might try to come up with some neat internet thing to do to wish him a belated happy birthday, but as is I think I will just have to say:

Dad,

I love you, I'm sorry I forgot about your birthday; I know you know I've been busy but that's no excuse. I promise I'll call you tomorrow and wax rhapsodic about Paris.

But for now, good night.

Paris, You've Got a Lot of Nerve

Being so charming even though I:

Made reservations for a hotel on the wrong day

Had to make my way across town to find another hotel

Got lost trying to find the hotel

Have not eaten dinner as of 8:15 pm

Had no 3G access to find myself on google maps on my phone.

The trip to Paris was rather expensive and like Amsterdam the hotels are expensive. The food will also probably be expensive. But there are fountains and artwork every couple of blocks and the climate is very nice (at least today) and hopefully the food will be ridiculously delicious and everyone seems pretty nice and I've been able to do my laundry (I hope it gets dry in the next 15 minutes before I have to go get it before the laundromat closes.

I am quite hungry, but otherwise in good health and I now have internet which is great!

More updates to come when I start finding my way around, having fun, eating food, seeing the night life, doing mathematics, etc.

Amsterdam

The other day we (Richard and Brennen, the Aussies, and I) arrived in Amsterdam from Venice. The weather was beautiful: cold and rainy. After getting burnt and burnt and burnt by the Italian sun it was great to be cold. At least, after I stopped getting lost around the city and found a place to stay and put on two shirts which were dry, it was nice.

So far Amsterdam has been a bit expensive (making me feel a bit silly about not going to Istanbul which probably would have been cheaper to stay at, if more difficult to get to) and hasn't been as exciting as I'd hoped. Well, maybe unexciting is wrong, as Brennen got his bag stolen (!!!!) yesterday and had to file a police report.

Then we hung around and had some cakes and went to see the last Harry Potter movie in 3D which had the fortunate side effect of leaving me with 3D hipster glasses.

Today some of the brits from Rome are coming to Amsterdam and we'll be meeting them today which should be fun. After that we hit Paris, then move on to Barcelona, after which we'll be parting ways so that I can MATH MATH MATH MATH MATH and they can see Madrid and Morocco.

I wanted to send an update to everyone who's interested (<3 moms :D ) because there is free internet where I am at the moment, but I've got to head out and meet my friends and eat some food so I'll post more later!

Pisa and Venice

On our way out of Far-too-hot florence, we stopped at Pisa to see the tower and the surrounding... Pisa. The tower was neat. Everything was expensive. And the tickets to see the tower were only for a specific time nearly two hours after we bought them... by which time we'd lost the tickets.

So we went to Venice, which was probably a good idea because it was (a) really expensive to get there and (b) we arrived just in time to get totally lost and ask for directions to the appropriate bus stop to get in to the hostel at 12:30am.

It's much nicer and not nearly so hot as it was in Florence, which is great, and the city is very pretty but it seems like it's mostly pretty expensive too.

After finally getting around to planning the rest of my journey, supposedly to Turkey, I discovered that Turkey was very expensive and difficult to get to. Also, my Australian friends Brennen and Richard that I've been traveling with are meeting some people we met in Rome in Amsterdam, then going to Paris then Barcelona; which is essentially my perfect replacement route since it hits two cities I'd like to see and leaves me in Barcelona at the right time.

Before leaving Florence, we managed to correct our egregious culinary errors by going out to a nice restaurant on the Piazza della Signoria, where I had Rebollita Toscana e Petto de pollo ai funghi which was amazing; I highly recommend the rebollita; it's a traditional bready soup. I don't really know how else to describe it.

In Pisa for lunch the next day we had Pizze Romantica; pizza with ham and pine nuts; and I had gnocchi quattro formaggi.

I have also had so many flavors of Gelato. Frutti di bosco, caramel, vanilla, chocolate, coconut, stratticelli, traditional flavors I can't remember the names of... Unfortunately we never managed to track down what I'd heard were the best gelaterias in Florence, but c'est la vie.

Firenze

Firenze is far too hot. The place I've been staying at has these terrible tents which really retain the heat in the morning in a terrible terrible way.

My first meal in Florence was at a McDonalds and I have not yet gotten a chance to really go out for dinner.

And my four friends from Wales that I'd been traveling with left our company last night, and me and the two Australian guys I'm also with walked them to the train station and didn't get back from saying our goodbyes until 5:30 in the morning. That was pretty exhausting, but we ended up passing the David right at sunrise, which is one of the most beautiful things I've ever seen.

On the way from Siena we passed through a medieval village and climbed a castle, and tonight we have a restaurant and gelateria lined up to get a true Tuscan dining experience.

Tomorrow we'll head out for Venice, and after that the Aussies are going to Amsterdam then Paris then arriving in Barcelona about when I intended to. Doing some looking into it, I'm realizing that going to Turkey is probably a lot more expensive and inconvenient than I'd thought, so I'm considering just staying along with them for a while longer since I'd end up where I needed to be anyway. I need to talk to my Turkish friends though... I feel a little bad because I definitely am interested in Istanbul, but I suppose the real truth here is that it's my vacation so I can do what I want :P

The internet where we're staying in Florence is very cheap (1 euro for 12 hours) but there are only a half dozen plugs across the entire place which makes it pretty hard to keep everything charged. I've just finished charging everything though so hopefully that'll keep me set for a while.

The gelato is great. Meeting friends has been great, and I've had some great conversations. Enough so that I've used up all my big words for now (!!!!) and I'll have to come back when I've got some mathematics to talk about or something.

Siena

I ended up traveling with several friends from Rome up into Siena, which was an absolutely gorgeous trip, although Siena was not particularly exciting as a town in my opinion. Some pretty towers but not as exciting as the ride there. There is, however, a supermarket near our campground which is EXCELLENT because it means we get lots of cheap food! =D

However I've had a good time having people to travel with, and we'll soon be heading to Florence, and I also finally have a (relatively) stable internet connection with which I am uploading a big load of photos to facebook (over 100) from Alghero and Rome.

I'm traveling fairly slowly so I'm not sure that I'll be able to get to Georgia on my trip, which makes me a bit sad but we'll see how things work out and I have been having a good time.

I have not been doing any math, which sometimes makes me sad but I'm often too busy to really think about it.

Rome, The Next Generation

So Rome has turned into a great time! I've met a load of friends at the hostel and seen a bunch of sights. The other day I went to see the pantheon, the colloseum, and trevi fountain, and today I went to the Vatican with several friends. It was rather hot and my phone ran out of batteries so I had to stop taking pics halfway through. But I did get pictures of the mummy, and of some of the statues looking like they're wearing my pink hat.

I had some lasagna in town for lunch after the Vatican, and it was quite good. Then I came back and hopped in the pool.

I'll be hopping in for dinner in a few minutes and tonight will be my last night in Rome, as tomorrow morning I'll be heading to Siena with some friends I met here.

I have not been doing any math. ):

Rome

Yesterday I flew out of Alghero to Rome; getting into the flight was actually not bad at all. Going through security in Italy is much nicer than in the US.

After getting into Rome, I had a hell of a time trying to figure out where I should go to get to the hostel I'm staying at, in large part because the automated ticket machines were not very robust.

I managed to eventually get a three day bus/train/metro pass for 25 euros which seems like a good deal to me in that I will no longer have to care at all about paying for public transit for the rest of my stay.

I haven't had much chance to meet up with anyone in Rome despite knowing at least four people from the conference are in town, because I haven't had internet until just now.

Because I've been wrapped up in travelling I actually have not had much chance to eat since leaving Sardinia. I did have a Cannoli at the airport which was pretty tasty though.

On the last night in Sardinia, several of us from the conference went out and had a late night on the town wherein I purchased a fabulous sparkly pink hat which has now earned me the nickname shiny hat math guy; which seems like not a bad way to be known.

I seem to have left my soap and shampoo in Sardinia, which is a bit frustrating but it hasn't been too bad.

Anyway I need to eat some food, just wanted to put an update here so that anyone who is following doesn't think I've fallen off the face of the earth.

Chris' Third Talk

Winding Numbers

Recall the function lambda which takes a rational number, then sends it to the sum of a_n/n times e^(2pi i n r)

Now define the modular symbol [r]+ := Re(lambda(r))/Omega+, which is rational.

Theorem: lambda(0) = L(E/Q,1)

proven by Mazur-Tate-Teitelbaum in their work on the p-adic BSD conjecture.

Corollary: [0] = L(E/Q,1)/Omega+ is rational

Also we can phrase BSD over Q in terms of modular symbols.

Theorem: (Kato, Urban-Skinner, et. al.)

L(E/Q,1) nonzero implies E(Q) and Sha finite; L(E/Q,1)=0 implies one is infinite.

In fact the rational point 0 viewed as a cusp is in X0(N)(Q), so phi(0) is a torsion point of E, so the denominator of [0] is a divisor of c*#E(Q) (i.e. bounded, and usually quite small)

Using the modular symbols one can find the size of Sha when finite.

Abelian Fields

let K be an abelian extension of Q; call m, the minimal integer such that Q(zeta_m) contains K, the conductor of K. Then there is a surjection from Z/mZ* onto the galois group G of K/Q.

Now we can view any character from G to C* as a Dirichlet character from Z to C by associating elements of G with their preimages in the above map which are clearly indexed by integers mod m*.

We now introduce three hypotheses.

I: no additive place ramifies in K/Q. This is quite necessary, or the results become false.

II: K is totally real. This is to avoid needing to introduce and use complex modular symbols.

III: d=[K:Q] is coprime to m. This is rarely used and simply excludes Iwasawa theory.

Now we will define and discuss Stickelberger elements of formal groups over all abelian extensions of Q, which nearly form an Euler system.

Define theta_E/K as the sum over a mod m* of [a/m]sigma_a in Q[G] where G = Gal(K/Q).

Lemma: let l be a prime not dividing m (here we use III)

define the Norm map N_L/K: Q[Gal(L/Q)] -> Q[Gal(K/Q)] by reducing the actions of Z/lmZ* to an action of Z/mZ*.

N_L/K(theta_E/L) is sigma l inverse times the sum (sigma l minus a_l plus delta_N(l) times sigma l inverse) times theta_E/K; where delta is 1 if l does not divide N and 0 if it does (N here is the conductor of E)

This essentially defines an Euler system, except from the factored sigma l inverse. So when that map is trivial, we do get an Euler system.

The proof of this lemma essentially involves pushing around the formulas for modular symbols and the function lambda. This is particularly exciting because Euler systems have deep structure that allows one to prove many things, but also because there are around 4 actual existing examples of Euler systems.

Vlad #3

To finish up Tuesday, as we now approach the end of Friday, I present Vlad Dokchitser's third lecture.

Compatibility of BSD w/ isogenies

Lemma: E/K, phi:E->E' an isogeny over K

then

(i) L(E/K,s)=L(E'/K,s) (in that the conjectural formulas are the same and where defined they are equal)

(ii) rk E/K = rk E'/K

Proof: the l-adic representations are equal, and isogenies must preserve rank due to having finite kernel and cokernel (since the dual isogeny gives you a multiplication by degree map).

Write the big conjectural formula from BSD as BSD(E/K). Then there is a theorem due to Cassels (extended by Tate to algebraic varieties): if phi is an isogeny from E to E' over K and Sha(E/K) finite, then Sha(E'/K) finite and BSD(E/K)=BSD(E'/K).

Corollary (due to Birch): if phi is an isogeny of prime degree p, then rk(E/K) is ord_p of the product of Cv and Omega_v of (E,w_E) over the product of Cv and Omega_v of (E',w_E').

Proving this essentially uses the definition of the regulator on the equality of the BSD parts.

Now suppose we have two families of elliptic curves over families of fields, Ei/Ki and E'j/K'j, such that the product of the L functions of each family is equal.

Then the sums of the ranks of each family is equal, the finiteness of Sha for one family implies it for the other family, and the products of the BSD(E/K) over each family are equal.

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!