Title: Nilpotent Artin-Schreier theory Abstract: In this talk, we review techniques used for parametrizing extensions of fields of characteristic p, and we show how these techniques specialize to known theories (Artin-Schreier-Witt theory, ϕ-modules, ...). We then review the Lazard correspondence, which relates p-groups of nilpotency class smaller than p with Lie algebras. By combining these two…

Title: Higher ramification groups and counting Galois extensions Abstract: This is a follow-up to last week's talk on nilpotent Artin-Schreier theory. We recall some basic facts about higher ramification groups of Galois extensions and where they come up in class field theory. Then, we study how they fit in with Abrashkin's nilpotent Artin-Schreier theory, and we sketch how we can (for some…

Title: Classification of imaginary quadratic number fields with class number 1, part I Abstract: We give an overview of the problem of determining how many imaginary quadratic number fields have class number 1. We then present Heilbronn and Linfoot's result, that there are at most 10 such number fields.

Title: Erdos sieves and repeated patterns Abstract: Recently Gundlach and Klüners introduced the concept of sieves on the ring of integers of a number field, and studied the associated dynamical systems. In this talk, we give an overview of what is referred in the literature as Sarnak's Program for these dynamical systems. We then show how the study of such dynamical systems gives insights into…

Title: Classification of imaginary quadratic number fields of class number 1, part II Abstract: In this talk, we show how Baker's result on linear forms in logarithms of algebraic numbers provides a solution to the problem of classifying imaginary quadratic number fields of class number one, following the approach of Bundschuh and Hock.

Title: On Galois extensions with cyclic decomposition groups Abstract: Call a Galois extension of number fields "locally cyclic" if all its decomposition groups are cyclic. Such extensions are interesting by themselves, but are also of significance for other problems in and around inverse Galois theory. I will give a survey of several recent results by myself and others, in part on realizing…

Title: Algebraic geometry of equilibria in cooperative games Abstract: The classical notion of Nash equilibria imposes the somewhat unnatural assumption of independent non-cooperative acting on the players of a game. In 2005, the philosopher Wolfgang Spohn introduced a new concept, called dependency equilibria, that also takes into consideration cooperation of the players. Dependency…

Title: Algebraic Patching for Beginners Abstract: Using the language and the tools of rigid analytic geometry, Harbater (1987) has defined a "patching operation" which can be used to solve the inverse Galois problem over fields like Qₚ(T) or Fₚ((X))(T). Later, Haran and Völklein (1996) rephrased this construction in a purely algebraic language, replacing all geometric arguments with (almost…

Title: The distribution of quadratic non-residues: A stroll through the garden Abstract: We shall have a stroll through the garden of classical results about the distribution quadratic (non-)residues modulo a prime with an emphasis on the methods involved. We shall also stress the depressing state of the affairs pertaining to the distribution of quadratic non-residues in non-initial segments,…

Title: Symmetries of the set of squarefree integers in a number field Abstract: Let K be a number field. We answer the following question and several generalizations: What are the Z-linear maps O_K -> O_K that send every squarefree algebraic integer to a squarefree algebraic integer? In the talk at 11:15am, Michael Baake will introduce dynamical systems associated to k-free algebraic integers. …

Title: Dynamical and spectral properties of some shift spaces of number-theoretic origin Abstract: While B-free systems in one dimension have been studied for a long time, considerably less is known on their higher-dimensional analogues. Starting from the visible points of the integer lattice, a large class of such systems emerge via k-free integers in algebraic number fields. We discuss typical…

Di, 16. April 2024 in Raum A 3.339 14:15 Uhr - 15:45 Uhr Titel: Fermat's Last Theorem for regular primes Abstract: Definition. For a number field K, let h(K) denote its class number. A prime number p is said to be regular if p ∤ h(Q(ζp)), otherwise irregular. Theorem. (Fermat’s Last Theorem for regular primes) For a regular prime p ≥ 3, the equation xp + yp = zp does not have a solution in…

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: Counting abelian extensions with Witt vectors

Titel: Describing abelian extensions with Witt vectors

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