Last updated: April 27, 2013.
The Bellairs Workshop in Number Theory

Arithmetic intersection theory on Shimura varieties and derivatives of L-functions
May 3-10, 2013

Speaker: Ben Howard (Boston College)

Organizers: Eyal Goren (McGill) and Steve Kudla (Toronto)
Coorganizer: Henri Darmon (McGill)

Arrival: Friday, May 3. Departute: Friday, May 10.

The workshop is dedicated to Arithmetic intersection theory on Shimura varieties and derivatives of L-functions. A series of lectures will be given by Ben Howard on this topic, supplemented by lectures from attending experts, for a total of 20 hours and many more hours of stimulating exchanges. The intention of the workshop is to provide an access point to the field, assuming at least a background comparable to advanced graduate students. Participation is by invitation only.

Program	The Bellairs Research Institute	Recreational Activities
Participants	Getting to Bellairs	Coming prepared
	The facilities	Pictures

Program
The general schedule is morning lectures by Howard and after-dinner lectures by participating experts. The program is designed to encourage discussion and new research initiatives by allowing enough free time and having all the participants in the same residence.

One of the appealing aspects of the geometry of Shimura varieties is the existence of large supplies of special cycles.  For example, all Shimura varieties come with special points in the sense of Deligne.  In the simple case of modular curves these special points include the Heegner points. In higher dimensions one has, for example, the Hirzebruch-Zagier divisors on Hilbert modular surfaces, and the Humbert surfaces on the Siegel threefold. In all these examples, the intersection multiplicities of the cycles in question are known to be related to modular forms.  For example, Hirzebruch and Zagier proved that the intersection multiplicities of their divisors appear as the Fourier coefficients of a modular form, and Gross and Zagier showed that the Neron-Tate heights of Heegner points appear as central derivatives of L-functions of modular forms.  Many more relations of this flavor are included in what has come to be called Kudla's program.

The focal point for the lectures will be one  case of such relations, in which special divisors on a Shimura variety of type GU(n-1,1) are intersected with special points.  The calculations of these intersection multiplicities leads to a Gross-Zagier type formula for the central derivative of the Rankin-Selberg convolution L-function attached to a cuspidal modular form of weight n and a  Siegel theta series of weight n-1, with the original Gross-Zagier theorem being (more or less) the case n=2.

Day	Morning  9:00 - 11:00	Evening  7:30 - 9:30
Friday	------ (arrival)
Saturday	Ben Howard: Unitary Shimura varieties, integral models, and special cycles	Steve Kudla: Special cycles on unitary Shimura varieties
Sunday	No scheduled scientific activity (although we may organize an excursion)	Nothing planned at present
Monday	Ben Howard: Green functions, regularized theta lifts, and the CM value formula	Michael Rapoport: The arithmetic fundamental lemma
Tuesday	Ben Howard: Intersection multiplicities and Gross's formula	Keerthi Madapusi-Pera: Orthogonal Shimura varieties, the Kuga-Satake construction, and K3 surfaces
Wednesday	Ben Howard: The adjunction formula	Stephan Ehlen: Borcherds products
Thursday	Ben Howard: Related topics	Dimitar Jetchev: An Euler system for unitary Shimura varieties
Friday	----- (departure)

 Here is some suggested reading material (the organizers had asked the speaker to supply it):

1) Heights of Kudla-Rapoport divisors and derivatives of L-functions.  Bruinier-Howard-Yang, preprint
2) Moduli spaces of CM elliptic curves and derivatives of Eisenstein series.  Howard.
3) Lectures on Arakelov geometry.  Soule, Abramovich, Burnol, and Kramer.
4) Special cycles on unitary Shimura varieties I and II.  Kudla and Rapoport.
5) Faltings heights of CM cycles and derivatives of L-functions.  Bruinier and Yang.
6) On canonical and quasi-canonical lifts.  Gross.

Ben's (co-authored and solo) papers are available from his webpage.

Ben had shared written notes from which much of the material will be selected. It is available here.

Participants (* = to be confirmed)

	Name	Affiliation	Arrival	Departure
1	Amir Khosravi, Zavosh	Toronto	May 3, AA1385, 12:55	May 10, AA2050, 14:55
2	Andreatta, Fabrizio	Milan	May 5, BA2155, 14:50	May 10, BA2154, 16:40
3	Attwell-Duval, Dylan	McGill	May 3, AC966, 14:00	May 10, AC967, 15:00
4	Bergamaschi, Francesca	Milano and McGill	May 3, AC966, 14:00	May 10, AC967 15:00
5	Bertolini, Massimo	Milano	May 3, BA2155, 14:50	May 10, BA2154, 16:40
6	Castella, Francesc	McGill	May 3, AA1995, 21:15	May 9, AA2050, 14:55
7	Cheung, Amy	McGill	May 3, WS2512, 14:45	May 10, WS598, 15:40
8	Darmon, Henri	McGill	May 3, AA1089, 13:45	May 11, AA1194, 07:15
9	De Quehen, Victoria	McGill	May 3, AA1995, 21:15	May 11, AA1384, 14:05
10	Disegni, Daniel	Columbia	May 3, VS29 14:10	May 9, BW447, 21:05
11	Ehlen, Stephan	Darmstadt	May 3,  BA2155, 14:50	May 10, BA2154, 16:40
12	Funke, Jens	Durham	May 3, BA2155, 14:50	May 10, BA2154, 16:40
13	Garcia, Luis	Columbia	May 3, AA1385, 12:55	May 10, AA1384, 14:05
14	Gomez, Clement	McGill	May 3, WS2512, 14:45	May 10, WS2513, 15:40
15	Goren, Eyal	McGill	May 3, AC966, 14:00	May 10, AC967, 15:00
16	Howard, Ben and "Mrs"	Boston College	May 3, AA1995, 21:15	May 10, AA2050, 14:55
17	Jetchev, Dimitar	EPFL	May 6, BA2155, 14:50	May 10, BA2154, 16:40
18	Kudla, Steve	Toronto	May 3, AC 966, 14:00	May 10, AC967, 15:00
19	Kuhn, Ulf	Hamburg	May 3, AB4531, 21:15	May 10, AB4514, 07:15
20	Madapusi-Pera, Keerthi	Harvard	May 3, AA1995, 21:15	May 10, AA602, 07:15
21	Prassana, Kartik	Ann-Arbor	May 5, AA1089, 13:45	May 10, AA602, 07:15
22	Rapoport, Michael and Mrs.	Bonn	May 3, BA2155, 14:50	May 10, BA2154, 16:40
23	Schnidman, Ari	Ann-Arbor	May 3, AA1385, 12:55	May 10, AA1384, 14:05
24	Shin, Sug Woo and Mrs.	MIT	May 3, AA1385, 12:55	May 10, AA2050, 14:55
25	Viray, Bianca	Brown	May 3, AA1089, 13:45	May 10, AA602, 07:15
26	Walls, Patrick	Toronto	May 3, WS2512, 14:45	May 10, WS2513, 15:40

The Bellairs Research Institute
Here is a link to the Institute's website. It is located in St. James, which is just to the north of Holetown. The exact location is
13.192104, -59.640130 (I think... you can plug these coordinates in  http://maps.google.ca)

Getting to Bellairs
Here you can find maps of Barabados, as well as much information about tourism. The local airport is in Bridgetown, Barbados (code BGI), while the Bellairs institue is more towards the north of the island, on its west coast. The easiest is to share a taxi from the airport to the institute. Most taxi drivers will know where is the Bellairs reserach institue; if not, tell them is in St. James, north to Holetown. When you pass the St. James Parish and the Folkstone marine park look for the Bellairs research institute on the left (sea side) of the road. Hook up at the airport with other participants based on arrival information to reduce the cost.

View Larger Map

The facilities

• The institute is well equipped for marine research. One can rent air tanks there very cheaply. Regardless, it is an excellent place for scuba diving and swimming, but you should bring your own gear (good quality mask and snorkel can be purchased for a reasonable price at a local sport store).
• Most people take their breakfast at the institute, but this is optional. Dinner at the institute is mandatory and is charged separately at $25US. This is not the cheapest, but the food is great and local and in generous portions and it is a nice way for us to get together. For lunch everyone is on his/her own. Usually people get together and go to one of the restaurants in Holetown. There is a kitchen there where, if needed, you can store food and cook for yourself if you cannot eat the food served.
• There is a flat rate of $15US for use of internet for the whole week. There is wireless in the Institute and also a few terminals and a printer (not for heavy jobs!). Laptop is the easiest way to be connected and to call home (using Skype, say). There is a public phone at the institute.
• Though Barbados in the large is a safe place, there have been cases of thefts from rooms in the institute. Some prudence is recommended: do not leave valuables unattended. Lock your room at night and do not leave valuables near the window (allowing someone from the outside to get to them). The institute has a safe where you can deposit your valuables.
• There is a modest size blackboard, and there is a data projector (to use with a laptop).
• Please wash your body from sand very carefully using the taps outside (in the main yard, for example). Sand can cause very difficult problems of blockage at the showers.
• The institute accepts Visa, Mastercard, cash and checques, which gives flexibility in settling your bill.
• The prices of room are as per the posted rate on the Institute website. Postdocs and “up” are considered research faculty and students, well… as students. The price depends on the specific room. Students, as a rule, will be staying at the more affordable rooms.
• Breakfast at a cost of $7 will be served on Monday morning (it includes toast, cereal, fruit, tea, coffee, juice and something hot like pancakes, eggs or bacon). After that we’ll ask people to choose either to have breakfast at the institute for the rest of the week, or not (at the same cost). One has the option of buying groceries and preparing breakfast independently, however what they offer seem to me a rather fair deal and the only option people may wish to take is to have their breakfast outside the institute altogether (there are no obvious choices, though).

Recreational Activities

• Here is a link to a tourist information site.
• The institute is located 1 minute walk (literally) from the beach. It is a beautiful beach with a coral reef. The Folkstone marine park is next door.
• Some world-class cricket (an acquired taste...).
• Bridgetown is a fun city to visit.
• Crane beach on the east-coast is great fun (waves and white sands).

Coming prepared

• The local currency is Bajan dollars. US dollars are accepted everywhere, at a fair rate, and it is easy to withdraw money from any ATM. You can change some money at the airport before exiting, though the taxi will accept US dollars.
• Barbados is hot and sunny. Some information about the weather can be found here and claims almost constant temperature during night and day at the range of 25-30 Celsius. In particular, sunscreen is a must and one should plan on summer-like weather with the possibility of showers and occasional chilly evenings. A musquito repellent is also highly recommended.
• The institute is in St. James, which is a short walking distance to Holetown, where there is an excellent supermarket, banks, pharmacies and souvenir shops and so on, as well as cheap to expensive (and excellent) restaurants, cafes and bars. If you have a special diet (kosher, halal, vegan, ...) it is worth doing some research before hand and making sure you have what you need. Dinners in the institute will not have special dishes prepared for people with special diet. Life, in general, gravitates between the institute and Holetown. For an occasional trip to Bridgetown the public buses are a fun experience and cheap, and shared taxis are a reasonable expense. Normally, one would not rent a car for the stay.
• Stationary is not provided.
• Coffee lovers should be mindful that the local coffee is instant coffee and bring with them ground coffee and filters.
  
• Here is some screenshot of the area with indication where you can get basic stuff like water, sunscreen, etc..