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.

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.

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.

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 \(4\).
 

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.

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.

Dinner takes place in Osteria Da Vinci, Turfsingel 33-1, Groningen. The food is as in Menu Misto here.