# AIM SQUARE -- Explicit Reduction Theories (Feb 21-25, 2011)

## Purpose

This is a proposal to gather researchers with distinct perspectives on the topic of explicit computational methods for reduction theories and their application to study geometry, cohomology, and modular forms. The purpose of our week-long collaboration is to:

- Discuss/share current methods for using explicit reduction domains, and interesting open problems.
- Lay the groundwork for creating a unified computational framework for explicit reduction theory computations in SAGE, and discuss how to combine existing specialized projects to this end.
- Set concrete goals for future collaborations and software development.

(See the AIM Squares Proposal for more information.)

## Attendees

- Jonathan Hanke (Univ. of Georgia, Athens, USA)
- Achill Schurmann (Univ. of Rostock, Germany)
- Robert Miller (MSRI, Berkeley, CA)
- Marty Weissman (Univ of Califarnia, Santa Cruz, USA)
- Mathieu Dutour (Institut Rudjer Boskovic, Zagreb, Croatia)
- Paul Gunnells (Univ. of Massachussetts, Amherst, USA)
- Dan Yasaki (Univ. of North Carolina, Greensboro, USA)
- Herbert Gangl (Durham Univ, UK)

(See the AIM Squares Proposal for a short bio of each researcher.)

## Tentative Schedule

### Day 1 -- Research Overviews and Goals

#### Morning 20-25 minute Talks: (9:30am-12:30pm)

- J. Hanke -- Quadratic Forms and Theta series
- A. Schurmann -- Voronoi Reduction?
- M. Dutour -- Enumerating Perfect Forms?
- P. Gunnells -- Automorphic Forms and Cohomology of Algebraic Groups?
- D. Yasaki -- Perfect forms over number fields
- H. Gangl -- Connections with K-Theory?
- M. Weissman -- Something Cool?
- R. Miller -- Graph Isomorphism code in SAGE?

#### Lunch

#### Afternoon Goal Setting and Problem Discussion/Organization (2pm-5pm)

### Day 2 -- Research Projects

- Choose a two main themes for each day (one day in advance).

### Day 3 -- Research Projects

- Choose a two main themes for each day (one day in advance).

### Day 4 -- Research Projects

- Choose a two main themes for each day (one day in advance).

### Day 5 -- Wrap-up and Make Future Plans/Goals

## Questions to Discuss/Ponder:

### Computational Questions

- What can we compute? What do we want to compute?
- What is the state of the current (open-source) software projects?
- Can we make it easier to use and package it in Sage?

### Research Applications and Questions

- Computing cohomology of algebraic groups -- Gunnells/Yasaki
- Computing automorphic forms -- Gunnells/Yasaki
- Enumeration of perfect forms -- Schurmann/Dutour
- Tables of quadratic/hermitian reduced forms (fast isomorphism testing) -- Hanke
- Explicit reduction domains for group actions. -- Hanke
- How many facets do we need for large discriminant and fixed number of variables? -- Hanke/Schurmann
- How can we see the reduction conditions in relation to the genera? -- Hanke

- Linear relations among theta series? (Presently done by only by linear programming techniques.) -- Hanke
- Enumeration of class groups and other objects via Bhargava orbit parametrizations? -- Hanke
- Computing K-groups -- Gangl
- Compactifications of moduli spaces of abelian varieties -- Alexeev
- Computing cohomology of arithmetic groups using polyhedral decompositions
- GL(n) over number fields
- Rank 1 orthogonal groups
- The exceptional cone (???)

- Explicit reduction theory for the symplectic group
- Sp(6) ... is it feasible?
- Sharbly complex for the symplectic group
- Work of Mellit

- Hecke operators on cuspidal cohomology
- GL(5) and higher
- GL(4) over imaginary quadratics and higher
- Sp(4)

(See the AIM Squares Proposal for more topics/questions.)

## Relevant Reading

- Reduction Theory of Quadratic Forms:
- Achill's Thesis
- Enumeration of perfect forms

- Tables of Quadratic Forms:
- Ternary
- Quaternary

- Uniqueness of Ternary Theta series (over rationals)
- Compactification of Moduli Spaces:
- Alexeev Paper
- Namikawa

- Computing cohomology of arithmetic groups and Hecke operators
Ash, Gunnells, McConnell, papers I-IV (especially I, which discusses our techniques very explicitly, and IV, which talks about torsion classes)

- Gunnells, "Computing Hecke eigenvalues below the cohomological dimension"
- Gunnells, Yasaki, "Hecke operators and Hilbert modular forms"
- Gunnells, Hajir, Yasaki, "Modular forms and elliptic curves over the field of fifth roots of unity"