# Workshop - Women in Numbers Europe 3 (3rd edition of the European WIN Workshop)

## List of projects

### Group 1: Local fields and Galois groups. Ramla Abdellatif, Université de Picardie, (France) & Agnès David, Université de Franche-Comté (France) & Lara Thomas, Université de Franche-Comté (France)

Understanding what are the Galois groups of number fields extensions is still a huge challenge and many number theorists are working, in a way or the other, on this topic. Via decomposition subgroups, this is directly related to the study of Galois groups of local fields extensions, which is at the core of this research project. Our working group aims to progress as far as possible in the two following directions :

- the first one has an inverse Galois problem flavour, as it aims to find necessary and/or sufficient conditions on $p$-adic groups to allow them to appear as Galois groups of local fields extensions;

-the second one is more related to Langlands program, as it aims to understand the interplay between the deformation theory of $p$-modular representations of some reductive $p$-adic groups and its counterpart on the Galois side.

### Group 2: Reduction types of plane quartics in special strata of their moduli space. Irene Bouw, University of Ulm (Germany) & Elisa Lorenzo García, Université de Rennes (France).

In this project we will work with some special families of plane quartic curves having non-trivial automorphism group. We will consider their invariants and we will try to characterize in term of them the reduction type of these plane quartics at the different primes. If possible, we also want to compute a semi-stable model for each different prime and determine the isomorphism classes of the different components of its special fiber.

### Group 3: q-Analogues in Combinatorics Eimear Byrne, University College Dublin (Ireland) & Relinde Jurrius, The Netherlands Defense Academy (The Netherlands).

A q-analogue arises in combinatorics when an object that is defined in terms of sets has an analogous description in terms of vector spaces. Matroids, codes and designs all have q-analogues and have been the subject of substantial recent activity. In this project we seek to address open problems on these q-combinatorial objects and on the connections between them.

### Group 4: Fields of definition of elliptic fibrations on K3 surfaces. Alice Garbagnati, Università Statale di Milano (Italy) & Cecilia Salgado, Federal University of Rio de Janeiro (Brazil).

The aim of this project is to use algebraic-geometric properties of certain K3 surfaces to determine arithmetic features of its elliptic fibrations. More precisely, we want to study the fields of definition of the Mordell-Weil groups of elliptic fibrations. We plan to use previous work of the group leaders on the classification of elliptic fibrations on K3 surfaces, and algeraic-geometric tools (as intersection properties of divisors) to determine the fields of definition of the Mordell-Weil groups of elliptic fibrations on certain K3 surfaces.

### Group 5: Universal Groebner bases. Elisa Gorla, University of Neuchâtel (Switzerland) & Emanuela De Negri, University of Genova (Italy).

In a joint work with Aldo Conca, we introduced a family of multigraded ideals, which we named after Cartwright and Sturmfels. These ideals, which for brevity we call CS ideals, are characterized by the property that their multigraded generic initial ideals are radical. In previous works we identified some families of CS ideals and studied their universal Groebner bases and multigraded generic initial ideals. Our goal for this project is identifying other families of CS ideals and studying their universal Groebner bases.

### Group 6: Arithmetic Dynamics. Holly Krieger, University of Cambridge (United Kingdom) & Nicole Looper, University of Cambridge (United Kingdom).

Given a global field $K$ and a degree $d\ge 2$ rational function $f:\mathbb{P}^1(K)\to\mathbb{P}^1(K)$, one can consider the dynamical systems arising from reducing $f$ mod $\mathfrak{p}$ at the various prime ideals $\mathfrak{p}$ of $K$. Expressing a rational function on $\mathbb{P}^1(K)$ as $f=[F,G]$ , where $F,G\in\mathcal{O}_K[X,Y]$ and at least one coefficient of $F$ or $G$ is a $\mathfrak{p}$-adic unit, the reduction $\tilde{f}$ of $f$ modulo $\mathfrak{p}$ is well-defined. By analogy with arithmetic geometry, a rational function $f$ is said to have good reduction at $\mathfrak{p}$ if $\tilde{f}$ has the same degree as $f$. This project will explore two aspects of reduction of dynamical systems: the reduction of dynatomic varieties, and the dynamical Shafarevich question.

### Group 7: Isogeny graphs. Kristin Lauter, Microsoft Research, (United States) & coleader TBC.

In this project we will study some properties of isogeny graphs, with a view towards applications to cryptography.

### Group 8: On the representation of tuples of integers by systems of quadratic forms. Lilian Matthiessen, KTH (Sweden) & Damaris Schindler, Utrecht University (The Netherlands).

The goal of this project is to study the failure of the representation of pairs of integers by systems of two quadratic equations in four variables. We will explore in special families how often the Brauer-Manin obstruction can be responsible for a failure of the local-global principle for this representation problem. For the counting process we will use techniques from analytic number theory.

### Group 9: Failures of the Hasse norm principle in function fields. Rachel Newton, University of Reading (United Kingdom) & Ekin Ozman, Bogazici University (Turkey).

In this project we will study failures of the Hasse norm principle in function fields.

### Group 10: Construction of non-holomorphic Poincarè series by generating kernels. Kathrin Maurischat, University of Heidelberg (Germany) & Lejla Smajlovic, University of Sarajevo (Bosnia and Herzegovina).

In this project we study solutions of differential equations for the Laplacian by resolvent integral kernels. Integrating kernels against special test functions will produce series, which hopefully will have new interesting properties, in particular in higher dimensional settings. Further, we study the arising representation theoretic implications.
We will start with the baby case of the weighted Laplacian on the upper
half plane, and work through more challenging cases later on.