- Claus Fieker (Technische Universität Kaiserslautern, Germany)
- Florian Hess (Universität Oldenburg, Germany)

Number Theory and Arithmetic Geometry share many common problems and techniques: Initially independent areas of mathematics, the similarity between number fields and algebraic curves over finite fields lead to the (common) theory of global fields. This combined viewpoint proved to be very fruitful - theory, algorithms and software were transported and extended in both directions. Algebraic curves over global fields provide further challenges in theory and computational practice, while relying on tools for global fields. The computational instrumentarium used and developed here spans a rather broad field: Exact symbolic computations from computer algebra need to be complemented with numerical computations using complex numbers, power series rings or p-adic rings, and with geometric methods for lattices. Algorithms are often designed and analysed based on randomisation and statistical traits. This session aims at showcasing some state of the art implementations as well as bringing together researchers with different applications and using different tools.

14:00-14:30 | Elsenhans, Sijsling, Streng |

15:50-16:20 | Johansson, Molin, Pauli |

16:30-17:10 | Hofmann, Kirschmer, Kulkarni |

Stephan Elsenhans (Würzburg) :

** Computational investigations on algebraic surfaces **

The geometry and arithmetic of algebraic curves has been
studied intensively in the past. The classification of algebraic
surfaces was done by the old Italian school about a century ago.
However, some geometric and many arithmetic questions for surfaces are
still open. In this talk I will show how to use modern computer algebra
to generate interesting examples and to study open conjectures.

Fredrik Johansson (Bordeaux) :

** What's new in Flint **

Flint is a C library for number theory and one of the core
computational packages in SageMath as well as the new Oscar system. This
talk will present the latest version of Flint which represents nearly five
years of development over the previous release. New features include sparse
multivariate polynomials, multithreaded algorithms, improved integer
factorisation and primality testing, embeddings of finite fields, new and
improved linear algebra and power series functions, and numerous other
additions, optimisations and bug fixes.

Tommy Hofmann (Kaiserslautern) :

** Norm relations and computational problems in number fields **

In this talk, I will present joint work with Jean-François Biasse, Claus
Fieker
and Aurel Page on the application of norm relations to computational
problems
in number fields. For a finite group $G$, I will introduce a
generalization of
norm relations in the group algebra $\mathbf{Q}[G]$ and their
classification.
These norm relations can be used to obtain relations between arithmetic
invariants of the subfields of an algebraic number field with Galois
group $G$.
As an applications I present subfield based algorithms for computing
rings of
integers, $S$-unit groups and class groups, which are practical and
fast.
On the theoretical side, for the $S$-unit group computation this yields
a
quasi-polynomial-time reduction to the corresponding problem in
subfields.

Markus Kirschmer (Paderborn):

** Definite hermitian lattices with small class number **

The local-global principle states that two hermitian spaces over some
number field $K$ are isometric if and only if they are isometric over
every completion of $K$.
The genus of a lattice $L$ in a hermitian space consists of those lattices
which are isometric to $L$ locally everywhere. Every genus decomposes into
finitely many isometry classes. The number of isometry classes in a genus
is called its class number. Hence the genera with class number one are
precisely those lattices for which the local-global principle holds.
For indefinite lattices, the class number can be expressed a-priori in terms
of some local invariants. For definite lattices, such a description is not
possible. However, up to similarity, there are only finitely many genera
with a given class number.
I will present some recent computer assisted classification of all definite
hermitian lattices of class number one.

Avinash Kulkarni (Dartmouth):

** Super-linear convergence in the p-adic QR-algorithm **

The QR-algorithm is one of the most important algorithms in linear
algebra. Its several variants make feasible the computation of the
eigenvalues and eigenvectors of a numerical real or complex matrix,
even when the dimensions of the matrix are enormous. The first
adaptation of the QR-algorithm to local fields was given by the first
author in 2019. However, in this version the rate of convergence is
only linear and in some cases the decomposition into invariant
subspaces is incomplete. We present a refinement of this algorithm
with a superlinear convergence rate in nearly all cases, and treat
some of the pathological cases. When the eigenvalues are simple, the
convergence rate is quadratic. Joint work with Tristan Vaccon.

Pascal Molin (Paris 7):

** Tba. **

Sebastian Pauli (Greensboro):

** Evaluating fractional derivatives of the Riemann zeta function **

We present a method for evaluating the reverse Grünwald-Letnikov fractional
derivatives of the Riemann Zeta function and use it to explore the location
of zeros of integral and fractional derivatives on the left half plane.
This is joint work with Ricky E. Farr and Filip Saidak.

Jeroen Sijsling (Ulm):

** Jacobians and their endomorphisms in computer algebra **

Jacobians of algebraic curves form one of the most
tangible classes of abelian varieties. Their linear structure gives
rise to Galois representations that have played an important role both
in the Modularity Theorem due to Wiles et al. and more recently in the
systematic exploration of the Langlands philosophy, as initiated in
the L-functions and Modular Forms Database (LMFDB).
This talk describes the results of joint work with Costa, Mascot and
Voight that enable one to rigorously compute the endomorphism rings of
a Jacobian over a number field by reducing such computations to
manipulations on algebraic curves instead of on the more complicated
abelian variety itself. After this, we briefly discuss work in
progress with Booker, Sutherland, Voight and Yasaki that makes
modularity for curves of higher genus explicit, and conclude by
demonstrating the resulting matches in the LMFDB as well as the
relevant software.

Marco Streng (Leiden, tentative):

** Tba. **

© 2020. All rights reserved.