Description: 
A workshop bringing together experts in arithmetic geometry, with an
emphasis on computational aspects.
Motivation: In the last 2030 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 nonhyperelliptic curves is much
more limited (essentially the only platform available for it is the
closedsource 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 opensource 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 padic curves
Abstract: Makdisi's algorithm allows for computation with divisors on curves over an arbitrary field, using linear algebra and RiemannRoch spaces. For padic 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 padic 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
Pythonpowered 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. KhuriMakdisi 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 nonsplit 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 padic 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 LowGenus Hyperelliptic Curves
Abstract: The most efficient algorithms for arithmetic in the Picard group of a lowgenus 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 ongoing work on identifying families of genus 2 curves with fast ltupling 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 wellknown conjectures formulated for actual real
quadratic orders also hold in fake real quadratic orders. Two
such conjectures include the widely believed CohenLenstra
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 AnkenyArtinChowla
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


