# L-functions and Number Theory

### Overview

Number theory is a subject as diverse as it is ancient, and this diversity is well represented in the mathematics departments of PIMS universities. These universities are home to academics with expertise in algebraic and analytic number theory, arithmetic algebraic geometry, computational number theory, number theoretic cryptography and information security, and representation theory, and have long-held reputations for producing cutting-edge research in these fields. Much has changed, though, in the arithmetic landscape of Western Canada since PIMS last funded a CRG in number theory during the period 2003-2005. An influx of new, young researchers (Bauer, Doran, Cunningham, Gordon, Greenberg, Kadiri, LalÃn, Ng, Silberman) into PIMS universities, particularly in Alberta, has added to the regionâ€™s already impressive list of accomplished number theorists. We feel the time is ripe to bring all of these mathematicians together, again under the umbrella of a PIMS CRG, to encourage the development of promising young faculty and their work while continuing to promote the number theoretic excellence for which Western Canada is acknowledged worldwide.

### Timeliness and importance

Canada is a world leader in number theoretic research. Recently, Western Canadian universities have invested in number theory by adding many strong young faculty members to a core of established researchers with world-class reputations. In parallel, important scientific developments have spurred exciting progress in the field and opened the doors to new avenues of research:

- Explosive growth in subjects like random matrix theory and mock theta functions is leading to impressive new results in analytic number theory.
- The field of Diophantine equations has recently undergone a significant change of perspective. Although the negative resolution of Hilbert's tenth problem renders experts generally pessimistic about the existence of an algorithm to decide solvability in general. Recent computational work has inspired important heuristics that indicate that an algorithm to decide the existence of rational points on curves may in fact exist and might already be known.
- The rapid development of Darmonâ€™s theory of modular points and advances Kudlaâ€™s program are increasing our understanding of the arithmetic nature of cycles on modular varieties and their connections with L-functions and representation theory.
- Recently, there have been developments in both local and global aspects of the Langlands program, e.g., the proof of the Fundamental Lemma by Ngo and the potential modularity results of Harris and Taylor. These developments have led to spectacular results, e.g., the proof by Taylor of the Sato-Tate conjecture, and suggest many promising avenues for further investigation of the nature of arithmetic statistics.

Our CRG looks forward to playing a major role in the advancement of the fundamental scientific theories listed above.

### Participants

**Coordinator**

Matthew Greenberg, University of Calgary

**Principal investigators**

Amir Akbary, University of Lethbridge

Nils Bruin, Simon Fraser University

Clifton Cunningham, University of Calgary

Charles Doran, University of Alberta

Matthew Greenberg, University of Calgary

Greg Martin, University of British Columbia

Lior Silberman, University of British Columbia

Vinayak Vatsal, University of British Columbia

**Participating faculty**

Mark Bauer, University of Calgary

Michael Bennett, University of British Columbia

Peter Borwein, Simon Fraser University

David Boyd, University of British Columbia

Paul Buckingham, University of Alberta

William Casselman, University of British Columbia

Imin Chen, Simon Fraser University

Stephen Choi, Simon Fraser University

Gerald Cliff, University of Alberta

Michael Jacobson, University of Calgary

Julia Gordon, University of British Columbia

Dragos Ghioca, University of Lethbridge

Habiba Kadiri, University of Lethbridge

Matilde LalÃn, University of Alberta

Kristen Lauter, Microsoft Research

James Lewis, University of Alberta

Nathan Ng, University of Lethbridge

Renate Scheidler, University of Calgary

Hugh Williams, University of Calgary/CSEC

**Scientific advisory committee**

William Casselman, University of British Columbia

Stephen Kudla, University of Toronto

Ram Murty, Queens University

Bjorn Poonen, MIT

Vinayak Vatsal, University of British Columbia

Scientific Events

- West End Number Theory Seminar, March 6, 2010, University of Calgary
- PIMS Calgary CRG Launch 2010, April 6, 2010, University of Calgary
- 2010 Alberta Number Theory Days, April 30-May 2, 2010, BIRS
- PIMS Number Theory CRG Distinguished Lecture: How many points can a curve have? Speaker: Noam Elkies (Harvard University), January 14, 2011, University of Calgary
- PIMS Number Theory CRG Distinguished Lecture: On the areas of rational triangles Speaker: Noam Elkies (Harvard University), January 17, 2011, University of Calgary
- Workshop on Analytic Aspects of L-functions and Applications to Number Theory, May 29-June 3, 2011, University of Calgary, Organizers: A. Akbary, M. Greenberg, G. Martin
- Alberta Number Theory Days, June 17-19, 2011, BIRS, Organizers: Amir Akbary, Brandon Fodden
- Analytic Aspects of L-functions and Applications to Number Theory, UC, May 29 - June 3, 2011
- Special session in number theory, CMS Summer Meeting, June 3-5, 2011, University of Alberta, Organizers: M. Greenberg, C. Cunningham
- L-packets, June 26-July 1, BIRS, Organizers: C. Cunningham, C. Moeglin, V. Vatsal
- Cycles on modular varieties, October 30-November 4, 2011, BIRS, Organizers: P. Charollois, S. Dasgupta, M. Greenberg, B. Gross
- WIN 2: Women in Numbers, November 6-11, 2011, BIRS, Organizers: C. David, M. Lalin, M. Manes
- Alberta number theory days, May or June, 2012 Organizers: TBA
- 12th Canadian Number Theory Association Meeting (CNTA XII), June 17-22, 2012, University of Lethbridge, Organizers: Amir Akbary, Michael Bennett, Nils Bruin, Charles Doran, Matthew Greenberg, Habiba Kadiri, Nathan Ng, Renate Scheidler, Vinyak Vatsal, Soroosh Yazdani
- Sage Days 36: p-adics in Sage, February 19-23, 2013, UC San Diego, Organizers: K. Kedlaya , D. Roe, W. Stein

### Seminars

- Seminar on nearby cycles, Wednesdays, 2:00pm-3:30 pm, in University Services Building Conference Room 0115.
- Nearby Cycles Seminar Series at the University of Calgary
- Nearby Cycles Seminar, February 9, 2011
- Apollonian Circle Packings, Stephan Erikson, Colorado College: April 21 2011 - 4:00pm, Bio Sciences 540B
- West End Number Theory Seminar (WENTS)
- UBC-SFU Number Theory Seminar

### Visitors

**Distinguished chairs: TBA**

- Michael Stoll (Bayreuth, Germany), Host: Nils Bruin (SFU), May 1-17, 2010
- Hugo Chapdelaine (Laval), December 14-19, 2010
- David Roe (Harvard), January 19-22, 2011 (seminar on January 21)
- Pierre Charollois (Jussieu), March 2011, (seminar on March 10)
- Cameron Franc (McGill), April 2011, (seminar on April 4)
- Paul Mezo (Ottawa), June 2011
- Hadi Salmasian (Ottawa) June 2011
- Pramod Achar (LSU) June 2011
- Samit Dasgupta (UCSC) TBA

### Post-doctoral Fellows

- Johnson Jia, Supervisor: Vinayak Vatsal (UBC), September 2010-August 2012
- Christopher Marks, Supervisor: Terry Gannon (UA), September 2011-August 2013
- Ce Bian, Supervisor: Matthew Greenberg (UC), September 2011-August 2013
- David Roe, Supervisor: Matthew Greenberg (UC), September 2011-August 2013

### Organizers

Clifton Cunningham and Mathew Greenberg