Conference: A day of arithmetic geometry

Speaker: Dimitrios Noulas (Athens)
Time: 9:30-10:30
Room: Cramer at SU
Title: From Topology to Arithmetic: around curves over the algebraic numbers
Abstract: Can tools from topology unlock the arithmetic of curves? In this talk, we will firstly review classical facts about Galois groups as fundamental groups, around the celebrated theorem of Belyi and the outer Galois representation associated to the thrice-punctured projective line. Motivated by the striking analogies between knots and primes, we will see how tools from topology, such as Fox calculus combined with tangential basepoints, can be potentially used to obtain arithmetic information regarding the periods and Jacobians of curves, with an emphasis on the complex multiplication property. We will showcase an adaptation of these ideas by Ihara to the p-power tower of Fermat curves. Then, motivated by questions of Ihara and Oort, we will discuss recent progress of this setting concerning Heisenberg curves, which arise as cyclic covers of Fermat curves.

Speaker: Simon Cooper (King's College London)
Time: 11:00-12:00
Room: Cramer at SU
Title: TBA
Abstract: TBA

Speaker: Christian Dahlhausen (Heidelberg)
Time: 13:30-14:30
Room: Kovalevsky at SU
Title: Continuous K-theory and motivic homotopy theory
Abstract: Algebraic K-theory satisfies Nisnevich descent and, on regular schemes, is \(\mathbb{A}^1\)-invariant. These properties lead to its representability in motivic homotopy theory. We aim to establish analogues of these statements for rigid analytic spaces. In this talk, I will discuss continuous K-theory and its representability in a motivic homotopy category of rigid analytic spaces. After reviewing the relevant background on algebraic K-theory and rigid analytic spaces, I will explain the local construction of continuous K-theory for certain topological rings. I will then introduce a motivic homotopy category in which the analytic affine line is contractible. In the end, I will state a representability theorem assuming a suitable form of resolution of singularities. This is based on joint work with Can Yaylali and Yicheng Zhou.

Speaker: Richard Pink (ETH Zurich)
Time: 15:00-16:00
Room: Kovalevsky at SU
Title: Schur \(\sigma\)-groups of Zassenhaus type \((3,3)\).
Abstract: For any odd prime p, the Galois group \(G_K\) of the maximal unramified pro-p-extension of an imaginary quadratic field \(K\) is a Schur \(\sigma\)-group. Its maximal abelian quotient is naturally isomorphic to the p-part of the ideal class group of \(\mathcal{O}_K\), and the full group \(G_K\) contains additional information about the field \(K\). It is therefore an interesting problem to classify all possibilities for \(G_K\).

In certain cases when \(G_K\) is non-abelian of Zassenhaus type \((3,3)\), it is actually a p-adic analytic group of dimension \(\le3\), and a full classification seems feasible. I will report on the current status of joint work with Eric Ahlqvist in that direction.