PIMS Workshop on Computational Arithmetic Geometry 2017

Sponsors: SFU KEY Tutte Institute PIMS
Dates: June 5 - 9, 2017.
Location: SFU's Big Data Hub, Applied Sciences Building (ASB) 10900.
Simon Fraser University,
Burnaby, BC V5A 1S6,

The map of the previous occupant may be useful.

URL: http://www.cecm.sfu.ca/~nbruin/WCAG2017/
Registration: Please contact the organizers if you are interested in participating.
Practical Information: Here
Dinner Thursday June 8, 7pm at Boat House Restaurant 2770 Esplanade St, Port Moody, BC V3H 0C8, Canada
Description: A workshop bringing together experts in arithmetic geometry, with an emphasis on computational aspects.

Motivation: In the last 20-30 years, great advances have been made in the study of elliptic curves, thanks to the development of excellent computational tools for doing explicit, computational experiments. Similarly, the existence of efficient computational tools for working with hyperelliptic curves has advanced mathematical knowledge greatly. These computational advances are also finding practical applications, for instance in cryptography and coding theory. By comparison, the support for efficient computation with non-hyperelliptic curves is much more limited (essentially the only platform available for it is the closed-source computer algebra system Magma). This workshop aims to bring together experts to improve the computational support for arithmetic geometric objects in general, and for divisor groups and function fields of algebraic curves in particular.

The program will consist of a variety of scientific lectures, as well as ample time for informal collaboration and code sprints for software development. One of the targets is to improve the computational support for arithmetic geometry in the open-source computer algebra system SageMath.
Preliminary Programme:
Monday, June 5
9:00 Coffee/Tea
9:30 Talk - David Roe
Title: Makdisi's algorithm for Jacobians of p-adic curves
Abstract: Makdisi's algorithm allows for computation with divisors on curves over an arbitrary field, using linear algebra and Riemann-Roch spaces. For p-adic fields, naive application of row reduction can lead to catastrophic precision losses. This talk will consist of two parts: an introduction to Makdisi's algorithm, and an introduction to differential precision methods for working with p-adic matrices.
10:30 Break
11:00 Public Lecture - Julian Rüth
Title: Introduction to Sage: a free mathematics software system
Abstract: The SageMath project started in 2004 to create a viable free open source alternative to Magma, Maple, Mathematica and Matlab. Since then, a community of hundreds of contributors has built Sage, a tool to perform computations ranging from undergraduate calculus to all kinds of research level mathematics.

This talk is an introduction to Sage. We will start by exploring its Python-powered interface in a tour of its main features. We will then use the CoCalc to run Sage in a web browser; since no local installation is required, this can be a great tool for research (or homework) collaboration or just to jointly work on LaTeX documents. After a quick discussion of ways to integrate Sage with other computer algebra systems, we will finally look at some internals of Sage to see how you could start contributing to Sage.

12:00 Lunch served in Atrium
1:00 Coding Session
5:30 Welcome reception at Club Ilia

Tuesday, June 6
9:00 Coffee/Tea
9:30 Talk - Stefan Wewers
Title: Computing stable reduction of superelliptic curves
Abstract: I will survey methods on how to compute the semistable reduction of superelliptic curve, and report on the implementation of these methods in Sage (joint with Julian Rüth)
10:30 Break
11:00 Talk - Peter Bruin
Title: Computing with modular curves over finite fields
Abstract: K. Khuri-Makdisi has developed an algorithmic framework for doing computations in Picard groups of general smooth projective curves by reducing them to linear algebra. This approach is very well suited for modular curves over finite fields. I will explain how this can be done in practice, with applications to computing modular Galois representations and to classifying elliptic curves over number fields with specific torsion groups.
12:00 Lunch break
1:00 Coding Session
5:00 End

Wednesday, June 7
9:00 Coffee/Tea
9:30 Talk - René Schoof
Title: Modular forms invariant under a non-split Cartan subgroup
Abstract: TBA
10:30 Break
11:00 Talk - David Harvey
Title: A quick survey of average polynomial time point counting for curves
Abstract: Let X be a curve over Q. In the last few years there have appeared a number of algorithms that efficiently compute the zeta function of the reduction of X modulo p, simultaneously for all good primes p less than a prescribed bound N. I will discuss the current status of theoretical and implementation work on such "average polynomial time" algorithms for various classes of curves.
12:00 Tutte Institute Presentation
12:30 Free afternoon / Coding session

Thursday, June 8
9:00 Coffee/Tea
9:30 Talk - Claus Fieker
Title: Large Class Groups in Nemo
Abstract: Nemo/Hecke/Antic/Oscar are names for the big new project in computer algebra undertaken here in Kaiserslautern. The BIG project aims to bring together Singular, Gap and Polymake, add number theory (formally Hecke/Antic) and make them work together seamlessly. This is a collaborative effort involving (part of) the Gap group, the Singular Team, Polymake and Antic.

In this presentation, I will talk a litte about what we have and what we want to do - and how we intend to get there. In particular, over the last (couple of) year(s), I and my group have been focussing on class groups in number fields. In the process we have been investigating closely all (well: a lot of them) fundamental algorithms and data structures typically used and investigated in particular with a view towards large fields. To a large degree: this reports on works in progress: very promising work, but not yet finished.

10:30 Break
11:00 Talk - Sebastian Pauli
Title: Enumeration of Extensions of Local Fields with Given Invariants
Abstract: We give an algorithm that constructs a minimal set of polynomials defining all extension of a p-adic field with given, inertia degree, ramification index, discriminant, ramification polygon, and residual polynomials of the segments of the ramification polygon.
12:00 Lunch break
1:00 Coding Session
7:00 Dinner at Boat House Restaurant 2770 Esplanade St, Port Moody, BC V3H 0C8, Canada

Friday, June 9
9:00 Coffee/Tea
9:30 Talk - Sebastian Lindner
Title: Improved Divisor Arithmetic for Low-Genus Hyperelliptic Curves
Abstract: The most efficient algorithms for arithmetic in the Picard group of a low-genus hyperelliptic curve are given by explicit formulas, where the group operations are expressed as an optimized sequence of field operations as opposed to arithmetic with polynomials. In this talk, we present improved explicit formulas for genus 2 imaginary hyperelliptic curves defined over finite fields. Our improvements are realized by identifying the best features of previous formulas and combining them in a novel way. In addition, we discuss on-going work on identifying families of genus 2 curves with fast l-tupling for l=2,3,5,7.
10:30 Break
11:00 Talk - Renate Scheidler
Title: Numerical Tests of Two Conjectures in Fake Real Quadratic Orders
Abstract: In an unpublished note from 2014, Henri Cohen coined the term “fake real quadratic order” for the ring obtained by adjoining to an imaginary quadratic order the inverse of a prime ideal above a split rational prime. The name was motivated by the surprising fact that allowing powers of a fixed prime as denominators in an imaginary quadratic order causes it to behave like a real quadratic order. In particular, just like real quadratic orders, fake real quadratic orders have unit rank 1 and tend to have a very large fundamental unit and a small class number. This invites the question of whether certain well-known conjectures formulated for actual real quadratic orders also hold in fake real quadratic orders. Two such conjectures include the widely believed Cohen-Lenstra heuristic which asserts that approximately 75 percent of all real quadratic fields are expected to have a class number whose odd part is one, and the more controversial Ankeny-Artin-Chowla conjecture which claims that if q is a prime congruent to 1 (mod 4) and $(t + u \sqrt{q})/2$ the fundamental unit of the quadratic field $Q(\sqrt{q})$, then q does not divide u. Both these conjectures have undergone extensive computational tests. In this talk, we present numerical data that speak to the validity of these two conjectures in the setting of fake real quadratic orders. This is joint work with Mike Jacobson and our jointly supervised recent Masters graduate Hongyan Wang.
12:00 Lunch break
1:00 Coding Session

Projects: See also the Sage Wiki Page
  • Basic arithmetic in function fields
  • Fast point adding on Jacobians of genus 2 imaginary hyperelliptic curves
  • Endomorphism ring computations via Riemann period matrices
  • P-adic computation in divisor class groups
Preliminary List of Participants: Jens Bauch (SFU)
Mark Bauer (University of Calgary)
Dean Bisogno (CSU)
Peter Bruin (Universiteit Leiden)
Nils Bruin (SFU)
Alyson Deines (CCR La Jolla)
Claus Fieker (TU Kaiserslautern)
David Harvey (University of New South Wales)
Nathan Ilten (SFU)
Michael Jacobson (University of Calgary)
Avi Kulkarni (SFU)
Sebastian Lindner (University of Calgary)
Su Min Leem (University of Calgary)
Pietro Mercuri (Sapienza University)
Brett Nasserden (University of Waterloo)
Sebastian Pauli (University of North Carolina Greensboro)
David Roe (University of Pittsburgh)
Edouard Rousseau (University of Waterloo)
Julian Rüth
Renate Scheidler (University of Calgary)
René Schoof (Università di Roma Tor Vergata)
Vijaykumar Singh (SFU)
Hanson Smith (University of Colorado Boulder)
Ha Tran (University of Calgary)
Charles Turo (SFU)
Shan Wang (SFU)
Colin Weir (Tutte Institute)
Stefan Wewers (Universität Ulm)
Sepehr Yadegarzadeh (SFU)
Alexandre Zotine (SFU)
Organizers: Nils Bruin (nbruin@sfu.ca)
Jens Bauch (jbauch@sfu.ca)