C e n t r e d e r e c h e r c h e s m a t h é m a t i q u e s

The theme year in number theory and arithmetic geometry will emphasize several current directions: Algebraic cycles and Shimura varieties, Elliptic curves and modular forms, Representations of p-adic groups, Analytic theory of automorphic L-functions. The year will be organized around a certain number of workshops, seminar courses and mini-courses spread throughout the year. Following is a schedule of events:

- NATO Advanced Study Institute/CRM Summer School 1998:
**The Arithmetic and Geometry of Algebraic Cycles** - Workshop on Algebraic modular forms and modular forms mod p
- Workshop on Analytic number theory
- CMS Winter Meeting
- Workshop on Representations of reductive p-adic groups
- Workshop on Arithmetical algebraic geometry
- Moonshine Workshop
- 6
^{th}Conference of the Canadian Number Theory Association - Seminar courses
- Mini-courses
- Chaire Aisenstadt Lectures
- Organizing Committee
- Registration Form

The last decades have witnessed the emergence of a "Langlands philosophy mod p" which (among other things) relates Galois representations to modular forms mod p. This instructional workshop will survey this circle of ideas, with a special emphasis on (1) Gross's theory of "algebraic modular forms" and (2) the progress on Serre's conjectures arising from the work of Ribet, Taylor, and Wiles.

**B. Gross (Harvard):**

Algebraic modular forms**K. Ribet (UC Berkeley):**

Congruences between modular forms**S. Kudla (Maryland):**

The Siegel-Weil formula

**Organizers:**- H. Darmon (McGill and CICMA) and G. Savin (Utah)

The workshop will focus on recent developments in analytic number theory with special emphasis on non-vanishing theorems for L-functions attached to automorphic forms.

**Organizer:**- R. Murty (Queen's)
**Invited Speakers:**- K. Dilcher, J. Friedlander, S. Gonek, J. Hoffstein, H. Iwaniec, K. Murty, R. Murty, M. Nair, R. Raghunathan, C.S. Rajan, C. Stewart

Special session in number theory

13-15 December 1998

**Organizers:**- R. Murty (Queen's) and N. Yui (Queen's)
**Invited Speakers:**- H. Darmon, C. David, J. Fabrykowski, C. Greither, H. Kisilevsky, M. Kolster, A. Ledet, C.
Levesque, K. Murty, V. Platonov, D. Roy, G. Walsh, H. Williams, K. Williams

The main focus of the workshop will be construction of *K*-types
for admissible representations, and Hecke algebras and their representations.

**Organizer:**- F. Murnaghan (Toronto)
**Invited Speakers:**- J. Adler, S. DeBacker, D. Goldberg, A. Helminck, C. Jantzen, P. Kutzko, D. Manderscheid,
L. Morris, A. Moy, M. Reeder, B. Roberts, A. Roche, P. Sally Jr., G. Savin, J.-K. Yu

Arithmetic algebraic geometry covers a range of possible topics, and a number of graduate students will be attending the workshop, so participants have been asked to address their lectures to a wide audience and to attempt to deliver "survey talks".

**Organizers:**- M. Goresky (IAS) and K. Murty (Toronto)
**Invited Speakers:**- A. Baragar, D. Boyd, W. Casselman, H. Darmon, B. Gordon, M. Goresky, E. Kani, M. Kolster,
J. Kramer, J. Lewis, K. Murty, M. Nori, Frans Oort, J. Scherk, N. Wieslawa, N. Yui, S.-W. Zhang

"Moonshine" started in 1978 with the observation that representations of some sporadic groups are naturally parametrized by the Fourier coefficients of certain modular forms. The functions arising here are axiomatizable as replicable functions from their behaviour under a generalized Hecke operator.

There is a panoply of mathematics used in these investigations. Vertex algebras, graded Lie algebras, conformal field theory, automorphic functions, Fuchsian groups, computational algebra, Schwarz derivatives, Hecke operators, representation theory, and mirror maps all appear.

The workshop, one in a two yearly series, will focus on recent developments on the subject.

**Organizer:**- J. McKay (Concordia)
**Invited Speakers:**- A.O.L. Atkin (Illinois, Chicago),
A. Baker (Glasgow),
S.J. Bloch (Chicago),
M. Brightwell (Glasgow),
J. Conway (Princeton),
I.V. Dolgachev (Michigan),
C.-Y. Dong (UC Santa Cruz),
B. Dubrovin (SISSA, Trieste),
P. Goddard FRS (Cambridge),
G.I. Glauberman (Chicago),
R.L. Griess Jr. (Michigan),
K. Harada (Ohio State),
M.J. Hopkins (M.I.T.),
G. Mason (UC Santa Cruz),
S.P. Norton (Cambridge),
A.J. Ryba (Michigan),
K. Saito (Kyoto),
H. Tamanoi (UC Santa Cruz),
M. Tuite (National University of Ireland, Galway),
S.-T. Yau (Harvard),
N. Yui (Queen's)

Co-sponsored by the CRM and The Fields Institute

20-24 June 1999

**Organizers:**- J. Borwein (Simon Fraser), D. Boyd (UBC), C. David (Concordia), R. Murty (Queen's), P. N. Shivakumar (Manitoba),
C. Stewart (Waterloo), H. Williams (Manitoba)
**Invited Speakers:**- M. Bennett (IAS), F. Beukers (Utrecht), A. Bremner (Arizona State), D. Bressoud (Macalester College, MN), H. Darmon
(McGill), J. Friedlander (Toronto), J. Grantham (Georgia), H. Kisilevsky (Concordia), M. Kolster (McMaster), H. W.
Jenstra, Jr. (Berkeley), L. Merel (Paris), A. Odlyzko (AT&T Labs, NJ), K. Ono (Penn State), B. Poonen (Berkeley), D Roy
(Ottawa), P. Sarnak (Princeton), W. Schmidt (Colorado), K. Soundararajan (Princeton) G. Stevens (Boston U), S. Vanstone
(Waterloo), T. Wooley (Michigan, Ann Arbor)

Seminar courses will last from one to three months.

- Lecturer: Henri Darmon (McGill and CICMA)
- The goal of this seminar is to survey the recent progress on the Birch and Swinnerton Dyer conjecture which
follows from the work of Kolyvagin and Wiles. In particular we will try to present a complete proof of the
following statement: Let
**E**be an elliptic curve over**Q**whose**L**-function is nonzero at**s=1**. Then the Mordell-Weil group**E(Q)**is finite. - Dates & Times: Every Thursday, from September 17, 1998 to April 1, 1999, 8:00-10:00
- Location: CRM, Univ. de Montréal, Pavillon André-Aisenstadt, Room 5340

- Speaker: Eyal Goren (CICMA, Concordia & McGill)
- Dates & Times: Every Tuesday from September 8, 1998 to April 6, 1999, 14:00-16:00
- Location: CRM, Univ. de Montréal, Pavillon André-Aisenstadt, Room 5340

- Speaker: Kumar Murty (Toronto)
- The Chebotarev Density Theorem is a fundamental tool in number theory and arithmetic geometry. We shall discuss effective versions of this theorem and some of its applications.
- Dates: October 9, 14, 16, 19, 21, and 30, 1998
- Complete Schedule

- Lecturer: Ram Murty (Queen's)
- This short course will survey sieve methods and some of its applications. After looking at the sieve of Eratosthenes, we will discuss the sieve methods of Brun, Selberg and Linnik. We will then examine applications of these methods to such questions as Artin's primitive root conjecture, squarefree values of polynomials, and structure of the group of points mod p of a global elliptic curve. The course will consist of a total of six lectures.
- Dates & Times: November 5, 10, 12, 17, 19 & 24, 1998, 10:30-12:00
- Complete Schedule

- Speaker: Henri Darmon (McGill & CICMA)
- Dates & Times: Every Thursday from January 7 to March 25, 1999, 8:30-10:00
- Location: CRM, Univ. de Montréal, Pavillon André-Aisenstadt, Room 5340

- Speaker: Eyal Goren (CICMA, Concordia & McGill)
- Dates & Times: Every Tuesday from January 12 to March 30, 1999, 15:30-17:00
- Location: CRM, Univ. de Montréal, Pavillon André-Aisenstadt, Room 5340

There will be several mini-courses of two weeks each, spread throughout the year.

- Lecturer: Massimo Bertolini (Universita di Pavia)
- Dates & Times: September 17, 22, 24 & 29, 1998, 10:30-12:00
- Complete Schedule

- Lecturer: Chris Skinner (Institute for Advanced Study)
- Dates: October 15, 20 & 22, 1998
- Complete Schedule

- Lecturer: C.S. Rajan (Tata Institute)
- Dates & Times: November 5, 10, 12 & 17, 1998, 14:15-15:45
- Complete Schedule

- Lecturer: Imin Chen (CICMA, Concordia & McGill)
- Dates: January 7, 12, 14 and 19, 1999
- Complete Schedule

- Lecturer: Andreas Schweizer (CICMA, Concordia & McGill)
- January 21, 1999, McGill Univ., Burnside Hall, Room 920, 10:30-12:00
- January 26, 28, February 2, 1999, Univ. de Montréal, Pav. André-Aisenstadt, Room 5340, 10:30-12:00

- Lecturer: A. Iovita (CICMA, Concordia & McGill)
- February 4, 1999, McGill Univ., Burnside Hall, Room 920, 10:30-12:00
- February 9, 11, 16, 1999, Univ. de Montréal, Pav. André-Aisenstadt, Room 5340, 10:30-12:00

- Lecturer: Eyal Goren (CICMA, Concordia & McGill)
- Timetable:
- February 18, 1999, McGill Univ., Burnside Hall, Room 920, 10:30-12:00
- February 18, 1999, Concordia Univ., Library Building, Room LB540, 14:15-15:45
- March 2, 1999, Univ. de Montréal, Pav. André-Aisenstadt, Room 5340, 10:30-12:00
- March 2, 1999, Univ. de Montréal, Pav. André-Aisenstadt, Room 5340, 14:15-15:45

The distribution and extreme values of L-functions

- Lecturer: Andrew Granville (Georgia)
- March 4, 1999, McGill Univ., Burnside Hall, Room 920, 10:30-12:00
- March 9, 11, 16, 1999, Univ. de Montréal, Pav. André-Aisenstadt, Room 5340, 10:30-12:00

- Lecturer: Jean-François Mestre (Paris VII, Jussieu)
- March 18, 1999, McGill Univ., Burnside Hall, Room 920, 10:30-12:00
- March 23, 25, 30, 1999, Univ. de Montréal, Pav. André-Aisenstadt, Room 5340, 10:30-12:00

- Lecturer: Fiona Murnaghan (Toronto)
- April 20, 22, 27 & 29, 1999, Univ. de Montréal, Pav. André-Aisenstadt, Room 5340, 14:00-15:30

**NOTE:**
The lectures of Professeur Andrew Wiles (Princeton), scheduled for September 1998, have been cancelled.

Professor Frans Oort

Universiteit Utrecht

May 1999

a proof of a conjecture by Grothendieck, Montréal 1970

- May 14 & 18, 1999, 4 p.m.
- May 20, 25, 27 & 28, 1999, 2 p.m.
- Further Information

- Henri Darmon (McGill and CICMA)
- Mark Goresky (IAS)
- Hershy Kisilevsky (Concordia and CICMA)
- Fiona Murnaghan (Toronto)
- Kumar Murty (Toronto)
- Ram Murty (Queen's), Chair

Those wishing to participate in the above activities are invited to contact:

Louis Pelletier CRM, Université de Montréal C.P. 6128, Succ. centre-ville Montréal (Québec) CANADA H3C 3J7 E-mail: ACTIVITES@CRM.UMontreal.CA FAX: (514) 343-2254 Telephone: (514) 343-2197

**18 May 1999, webmaster@CRM.UMontreal.CA**