MODULAR FORMS AND THEIR APPLICATIONS

Program: The summer school is particularly aimed at students in number theory, algebra and algebraic geometry.
The course will consist of three intertwined lecture series,

"Elliptic modular forms and their applications" (Don Zagier)
"Hilbert modular forms and their applications" (Jan Bruinier)
"Siegel modular forms and their applications" (Gerard van der Geer)

The first of these will treat the classical one-variable theory and some of its many applications in number theory (representations of numbers by quadratic forms, irrationality and transcendence results, moments of periodic functions, ...), algebraic geometry (counting of coverings of curves), and mathematical physics (appearance of modular forms in percolation theory, string theory, etc.)

The second, which has a more geometric flavor, will give an introduction to the theory of Hilbert modular forms in two variables (i.e., over real quadratic fields), the geometry of Hilbert modular surfaces, and to Borcherds products and the Borcherds lifting.

The third will give an introduction to Siegel modular forms (both scalar- and vector-valued) and present a beautiful application to the theory of curves of finite fields (Harder's conjecture).

In each case the theory will be built up from scratch, not presupposing any previous knowledge of modular forms. A reasonable general background in analysis (functions of one complex variable), algebra (e.g. finite fields), and topology (basic properties of manifolds and varieties and their homology and cohomology groups) will be supposed.

Location: The school will be held at Sophus Lie Conference Center in Nordfjordeid. Nordfjordeid, the birthplace of Sophus Lie, is a small town in the fjords of the west coast of Norway, halfway between Bergen and Trondheim. The Conference Center is located at Fjordane Folkehøgskule, situated just above Nordfjordeid, where the valley meets the fjord. Local guides will advice the participants on hikes in the nearby mountains.

Arrival: Sunday 15. or Monday morning 16. August
The first lecture is scheduled to start at 13.15 Monday 16. August.
The last lecture is scheduled to end at 14.00 Friday 20. August.
Departure: Friday 20. or Saturday morning 21.

How to arrive: For detailed travel information consult the home page of the conference center at http://www.math.uio.no/nordfjordeid/nordfjord.html

The following are the main possibilities:

a) By Plane: The natural place to fly to is Sandane via Oslo or Bergen. This airport is only 5 km from Nordfjordeid, and there is bus/taxi service from there.
b) By Bus: There is relatively good bus service between Oslo and Nordfjordeid, and Bergen and Nordfjordeid, but it takes 9 and 6 hours respectively.
c) By Boat and Bus: There is also a good express boat service between Bergen and Måløy, with bus connection between Måløy and Nordfjordeid, it takes about 5 hours altogether.

Weather: The weather is fairly unstable at Nordfjordeid. From cold, rain and wind to sunny and warm. The temperature is seldom below 10 and go up to 25 if we are lucky. It is advisable to bring a sweater and a jacket.

Registration fee: 200 NOK. The registration fee is payable at arrival (no credit cards).

Accomodation: The participants will be accommodated at the Sophus Lie Conference Center in single and double rooms. The center has nine pavilions, each pavilion has 4 double rooms and 2 single rooms and one shared bathroom. The cost for room and full board (three meals) pr person/night in a double room is NOK 320, for single room this cost is NOK 370.

Financial support: Partial financial support is available for participants from the nordic countries (for details check the web site http://www.mi.uib.no/nordag/ or contact johnsen@mi.uib.no)

Registration deadline is 1. June 2004

For registration and further information please contact K. Ranestad, Dep. of Math., Boks 1053 Blindern, N-0316 Oslo, Norway,
e-mail:ranestad@math.uio.no