Titel: Counting extensions of division algebras over number fields (joint work with Fabian Gundlach)
Abstract: We present and explain the proof of results concerning the asymptotical density of discriminants of extensions of a given division algebra over a number field.
This is an extension of the question of the distribution of number fields to the case of non-commutative fields.
We explain what happens both in the case of "inner Galois…
Titel: Orders in number fields
Abstract: Zassenhaus' well known Round 2 algorithm yields an effective method to compute the maximal order Z_K in a given number field K.
We show how to reverse this process to enumerate the orders of a given index in Z_K or containing some fixed order.
This is joint work with J. Klüners.
Titel: Asymptotics of nilpotent extensions of number fields
Abstract: We give an overview about the proof of the weak Malle conjecture for nilpotent groups. Given a group G and a number field k, Malle defines a counting function.
Z(k,G;x) which counts all (finitely many) number fields with Galois group G and norm of the discriminant bounded by x. Malle conjectures that this counting function is O(x^a * log(x)), where a(G) and b(k,G) are…
Titel: Estimates for the number of representations of binary quadratic forms
Abstract: Given a positive definite binary quadratic form g, we study the number of representations of an integer n by g, denoted rn(g). In particular, we generalize an estimate of Blomer and Granville for the quantity ∑ n≤xrg(n)β with β a positive integer, to the case where g has a non fundamental discriminant. To do this, we study the non-maximal orders of imaginary…
Titel: "Solving embedding problems in characteristic p"
Abstract: In this talk, we will describe some aspects of field theoretic embedding problems with a focus on fields of characteristic p.We will present a well-known approach to explicitly solve embedding problems with p-groups over fields with characteristic p which only requires the existence of an element with non-zero trace.
Titel: Covers and rigidity in inverse Galois theory
Abstract: Celebrated bridges between analytic geometry and algebraic geometry lead to an equivalence of categories between finite
extensions of ℂ(T) and finite ramified covers of the Riemann sphere (i.e., the complex projective line). These covers are
well-understood, and this correspondence directly implies a positive answer to the inverse Galois problem over ℂ(T), as
well as a…
Titel: Counting abelian extensions of number fields
Abstract: We will count abelian extensions of number fields with bounded discriminant or product of ramified primes. This was first done by David Wright [1], but we will stay a bit closer to a rephrased and simplified proof by Melanie Matchett Wood [2].
[1] https://doi.org/10.1112/plms/s3-58.1.17
[2] https://doi.org/10.1112/S0010437X0900431X
