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IT-Beauftragte
Anschrift
Geometry, Number Theory, Algorithms and Applications in Cryptography
February 20-21
This is a joint, local workshop of groups at the Universities of Groningen and Oldenburg.
Organisation: Florian Hess (Oldenburg), Jan Steffen Müller (Groningen).
The workshop is funded by the Universitätsgesellschaft Oldenburg and by the University of Groningen.

Location
The workshop will be held at the Bernoulli Institute of Mathematics, University of Groningen. The talks take place at the rooms FBG 5616.0125 and FBG 5616.0128 of the Feringa building on Thursday and BB 5161.0267 in Bernoulliborg on Friday.
Programme
Thursday, February 20
11:00 FBG 5616.0125
Martin Lüdtke (Groningen, MPI Bonn, Ben Gurion):
Refined Chabauty–Kim for the thrice-punctured line over \({\mathbb Z}[1/6]\)
If \(X\) is a curve of genus at least 2 defined over the rational numbers, we know by Faltings's Theorem that the set \(X({\mathbb Q})\) of rational points is finite but we don't know how to systematically compute this set. In 2005, Minhyong Kim proposed a new framework for studying rational (or \(S\)-integral) points on curves, called the Chabauty–Kim method. It aims to produce p-adic analytic functions on \(X({\mathbb Q}_p)\) containing the rational points \(X({\mathbb Q})\) in their zero locus. We apply this method to solve the S-unit equation for \(S=\{2,3\}\) and computationally verify Kim's Conjecture for many choices of the auxiliary prime p.
11:30 FBG 5616.0125
Begüm Çaktı (Groningen):
An effective asymptotic generalized Fermat over imaginary quadratic fields of class number one
The asymptotic generalized Fermat conjecture (AGFC) predicts that for a number field \(K\), and non-zero elements \(A\), \(B\), \(C\) of \(\mathcal{O}_K\) such that \(A\omega_1+B\omega_2+C\omega_3\neq0\) for any roots of unity \(\omega_1,\omega_2,\omega_3\) in \(K,\) there is a constant \(\mathcal{B}(K,A,B,C)\) such that for all primes \(p>\mathcal{B}(K,A,B,C)\), the equation \[Ax^p+By^p+Cz^p=0\] has only trivial solutions in \(K\). Even though there has been promising progress in proving AGFC over general number fields, it remains unclear whether the bound \(\mathcal{B}(K,A,B,C)\) is effectively computable. In this talk, we will first provide recent results regarding AGFC and briefly discuss the modular approach. We will then study the equation \[q^rx^p+y^p+z^p=0\] over imaginary quadratic fields of class number one when \(q\) is an odd unramified prime. We will observe that the AGFC holds for this equation over imaginary quadratic fields of class number one with an effectively computable bound.
12:00 Lunch
Lunch is held at the restaurant in the Feringa building.
13:00 FBG 5616.0128
Joris Dannemann (Oldenburg):
Using Automorphisms on Curves for Speeding Up Computations
I give a brief overview over my plans for my master thesis. The starting point is a recent paper by Couveignes and Gasnier. They consider an unramified abelian galois cover \(Y \to X\) of two curves \(X\) and \(Y\) over a finite field with galois group \(G\), and study the effect of a \(G\)-action on two typical linear spaces on \(Y\) associated with pullbacks of divisors on \(X\). As one application they use this \(G\)-action to obtain better complexities for encoding and decoding algebraic-geometric codes. In my talk, I first give a summary of some of their results. Thereafter, I formulate my preliminary goals and one early result.
13:30 FBG 5616.0128
Ludwig Fürst (Groningen, Bayreuth):
Explicit methods for hyperelliptic curves: Height bounds and the Kummer variety
If \(F\in \mathbb{Z}[x]\) is a square-free polynomial, then \(y^2 = F(x)\) defines a hyperelliptic curve. If its genus is at least \(2\), then Faltings' theorem tells us that there are only finitely many rational points on this curve. Bugeaud, Mignotte et. al. introduced in 2008 a method to find all integral points on this curve if we know its Mordell-Weil group. One key ingredient for this method and for computing the Mordell-Weil group itself is a height bound on the curve. In this talk we will take a look at these methods and how to make them explicit. In particular we will give an explicit height bound for genus \(4\).
14:00 Coffee
14:30 FBG 5616.0128
Marcel Salmon (Oldenburg):
Working on a classification of simple non-isolated CMC2 singularities
It is well known that in the classes of hypersurface singularities and complete intersection singularities simple non-isolated singularities do not exist. On the other hand, rigid, and therefore simple, non-isolated singularities do exist. In this talk we look at the class of Cohen-Macaulay codimension 2 (CMC2) singularities, where simple non-isolated singularities do occur. After the introduction to the classification problem, I will present the methodology which is mainly used while working on this problem.
15:00 FBG 5616.0128
Tianci Kang (Groningen):
Explicit Neron functions over hyperelliptic curves
In this talk, I will compute Neron function on hyperelliptic curves associated with theta divisor. I will recall the definition of Neron function, and admissible pairing which is raised by Shou-Wu Zhang. Moreover I will introduce the determinant of cohomology and universal line bundle on \(\operatorname{Pic}^n\). They are important tools in the computation. Then I will compute the pullback of theta bundle from Jacobian variety of curves. I will compute the value of Neron function on hyperelliptic curves with genus \(g\).
15:30 Coffee
16:00 FBG 5616.0128
Tobias Geisler-Knickmann (Oldenburg):
Imaginary biquadratic number fields with small class numbers
In the paper to be presented, we determine complete lists of imaginary bicyclic biquadratic number fields for each class number up to and including \(50\) by using the lists of all imaginary quadratic number fields with class number less than or equal to \(100\) provided by Watkins in 2004. Thus far, these lists for imaginary biquadratic number fields had been determined only for class numbers \(1\), \(2\) and \(3\). More generally, we revisit known results in order to describe a general method for determining the unit group of an imaginary biquadratic number field which can then be used to compute the class number from the class numbers of the quadratic subfields.
16:30 FBG 5616.0128
Niek Veltman (Groningen):
An Extension of Minkowski's Statements to \(S\)-Integers
Minimal elements and reduced Arakelov divisors play a major role in the description of the infrastructure (an algorithm to compute the regulator of a real quadratic number field). The existence of a minimal element in a fractional ideal of the ring of integers relies on the statement that these fractional ideals form a lattice in a certain Euclidean space (sometimes referred to as Minkowski Theory, e.g., by Neukirch). The proof that there can only exist finitely many reduced Arakelov divisors depends on Minkowski's Convex Body Theorem. We were interested in extending the infrastructure to so-called fake real quadratic orders, a specific type of \(S\)-integers. Therefore, the first step was to extend the notions of minimal elements and reduced Arakelov divisors to S-integers. To recover the existence of minimal elements and the finiteness of reduced Arakelov divisors in this situation, we needed to extend Minkowski's statements. In this presentation we will sketch how we derived these statements by discussing lattices in the \(S\)-Minkowski space, fundamental regions, and covolumes.
18:30 Dinner
Dinner takes place in Osteria Da Vinci, Turfsingel 33-1, Groningen. The food is as in Menu Misto here.
Friday, February 21
09:30 BB 5161.0267
Konstantin Meiwald (Oldenburg):
Constructive aspects of Picard groups of non-reduced projective curves
Let \(X\) be a connected, projective curve over a perfect field \(\mathfrak{k}\), and \(X_{\operatorname{red}}\) be the corresponding reduced curve. Then the natural closed immersion \(X_{\operatorname{red}} \to X\) induces an epimorphism \[\phi : \operatorname{Pic}(X) \to \operatorname{Pic}(X_{\operatorname{red}})\]between the Picard groups. We use this epimorphism to develop explicit methods for \(\operatorname{Pic}(X)\) if \(\operatorname{Pic(X_{\operatorname{red}}})\) is known. Having explored in my talk last year how to calculate preimages under \(\phi\), in this talk we discuss how the kernel of \(\phi\) looks like. In particular, we explain how for \(\operatorname{char}(\mathfrak{k}) = 0\) the \(\operatorname{ker}(\phi)\) can be equipped with the structure of a \(\mathfrak{k}\)-vector space.
10:00 BB 5161.0267
Ruben van Dijk (Groningen):
On Howard’s Kolyvagin systems for residue characteristic 2
In his paper on the Heegner point Kolyvagin system, Howard reformulates Kolyvagin's proof of the bound on the \(p\)-Selmer group of an elliptic curve in a modern style, strengthening Kolyvagin's bound on the annihilator to a bound on the length; however, Howard omits the case where \(p=2\). In my Master's thesis we discuss Howard's proof in detail, and in an attempt to generalize his results to all primes \(p\), study where his proofs break down when \(p=2\). We find that under the assumption of two technical conjectures, a similar but weaker bound applies to the length of the \(2\)-Selmer group
10:30 Coffee
11:00 BB 5161.0267
Alejandro Martinez (Groningen):
A tropical adventure: Geometry from a combinatorial viewpoint
Combinatorial algebraic geometry is an emerging field which has exploded in the last 20 years. Classical geometric objects cast combinatorial 'shadows' which allow them to be worked with combinatorial methods and recent breakthroughs have come via tropical geometry, matroid theory and \({\mathbb F}_1\)-geometry, among other topics. This talk is a short overview of the field along with current work being developed on hyperfields and generalized scheme theory which is applicable to not only the above, but link to other areas such as non-archimedean geometry.
11:30 BB 5161.0267
Philipp Iber (Oldenburg):
Combinatorial studies of Fano varieties via toric embeddings
In the context of toric Fano varieties (i.e. toric varieties with an ample anticanonical divisor), a well established combinatorial tool is the Fano polytope that allows the combinatorial description of certain geometric properties, e.g. the singularity types and the Gorenstein index. In this talk we consider ways to apply similar combinatorial methods to certain other classes of varieties, where the Fano polytope is replaced with the so-called anticanonical complex.
12:00 Lunch
Lunch is held at the Food Court Zernike.
13:00 BB 5161.0267
Ander Arriola (Groningen):
A generalization of Manin-Mumford for Abelian schemes with isogenies
The Manin-Mumford conjecture, first solved by Raynaud in 1980 for Abelian varieties, has since then seen multiple proofs of varying nature. Building on Hindry's 1988 proof, Dill extended the result in 2022 to Abelian schemes using isogenies. In this talk, I will present this generalization and discuss some partially effective and explicit results we proved in my master’s thesis for products and powers of the Legendre family.