Title: On truncated pretentious distances of multiplicative functions, and applications Abstract: We will be interested in recent progress in the study of correlations of multiplicative functions and on Eliott’s conjecture. In particular, I will present an upper bound, due to Klurman, Mangerel, and Teräväinen, for correlations of multiplicative functions in terms of their truncated pretentious distance. I will then describe a construction,…

Title: Constructing Large Galois Groups via Small Perturbations Abstract: A classical theorem of van der Waerden shows that, in a density sense, most integer polynomials of a fixed degree have the full symmetric group as their Galois group. However, this result is non-constructive and does not explain how to explicitly produce such polynomials close to a given one. In this talk, I will present a constructive approach to this problem. Given…

Title: Counting number fields by their smallest defining polynomial Abstract: When do two irreducible polynomials with integer coefficients define the same number field? Improving on work of Bhargava, Shankar, and Wang, we show that in a certain statistical sense, this usually only happens if the polynomials lie in the same orbit of a particular action of GL_2 x GL_1. We use this to count number fields whose smallest defining polynomial has…

Title: Point Counting on Elliptic Curves over Finite Fields Abstract: Elliptic curves over finite fields play a central role in modern number theory and cryptography. A fundamental computational problem is to determine the number of rational points on a given elliptic curve defined over a finite field, a quantity closely related to the trace of the Frobenius endomorphism. In this talk, I will present an overview of algorithms for point…

Title: Enumerating Orders in Number Fields Abstract: For a given number field K we study the distribution of suborders of the maximal order O_K of K. The Chinese Remainder Theorem shows that it suffices to describe the orders whose index in O_K is a power of p for some prime p. Zassenhaus' well-known Round 2 algorithm yields an effective method to compute the p-maximal order of a given order in K. We show how to reverse this process to…

Title: On the inverse Galois problem for del Pezzo surfaces of degree 1 Abstract: Del Pezzo surfaces over a field are nice (= smooth projective geometrically integral) surfaces with an ample anticanonical sheaf. The self-intersection number of the canonical sheaf is called the degree of the del Pezzo surface and is always an integer between 1 and 9. There is a natural action of the absolute Galois group of the ground field on the geometric…

Title: Finding the optimal fraction for wildly ramified finite abelian $p$-extensions in characteristic $p$ Abstract: Let $p$ be a fixed prime, $G$ a fixed finite abelian group of order $p^n$ and let the type of $G$ be denoted by the corresponding partition $\lambda$ of $n$. In dependence of $\lambda$, we define the map $f_{\lambda}: D(\lambda) \rightarrow \mathbb{Q}, \ d = (m_1, \ldots, m_n) \mapsto \frac{1 + \sum\limits_{i=1}^{n} \left( m_i…

Title: The Grunwald Problem for solvable groups Abstract: Let $K$ be a number field. The Grunwald problem for a finite group (scheme) G/K asks what is the closure of the image of $H^1(K,G) \to \prod_{v \in M_K} H^1(K_v,G)$. For a general $G$, there is a Brauer—Manin obstruction to the problem, and this is conjectured to be the only one. In 2017, Harpaz and Wittenberg introduced a technique that managed to give a positive answer (BMO is the…

Title: Twisted Artin-Schreier theory Abstract: Over any field K of characteristic p, we parametrize the (Fₚ ⋊ Fₚˣ)-extensions of K by pairs of elements (g, a) ∈ K×Kˣ using a “twisted” variant of Artin–Schreier theory. Focusing on the case of the local function fields and of the rational function fields K over a finite field of characteristic p, we use this parametrization to study the number of (Fₚ ⋊ Fₚˣ)-extensions of K with prescribed…

Title: Binary quadratic forms with non-fundamental discriminants Abstract: Let F be a primitive positive definite binary quadratic form of discriminant D, and let r_F(n) denote the number of representations of n by F up to automorphisms. Blomer and Granville (Duke Math. J., 2006) established estimates for the β-th moments of r_F(n) uniformly for fundamental discriminants |D| = o(x) for fixed non-negative real β. In this talk, we discuss the…

Title: Generalizing Erdős B-free systems: Erdős Sieves Abstract: Given a set B = {b_1,b_2,...} of pairwise coprime numbers, we say it forms a B free system, if sum_i 1/b_i is finite. If for each i we instead consider a set of congruence classes mod b_i, we get Erdős sieves. For a sieve R we investigate the dynamical systems associated to the R-free numbers, those integers not contained in any…

Title: Local-global principle for p-extensions in characteristic p Abstract: One of the main themes of number theory is the description of extensions of a fixed (local or global) field. For abelian extensions, this is accomplished by class field theory, which has the distinctive property that local and global extensions are tightly connected. When restricted to abelian p-extensions in…

Title: Frobenius and Octopus Abstract: We count semisimple nxn-matrices over a finite field F_q that commute with their (p-th power) Frobenius conjugate (for fixed n and for q going to infinity). This involves some linear algebra, algebraic geometry, and graph theory.

Title: Number Fields with Absolutely Abelian Hilbert Class Fields

Title: Stably free cancellation for orders Abstract: A ring has the stably free cancellation property if every stably free module is free. We will discuss this property for orders in semisimple algebras and show how the question of whether an order has this property can be answered algorithmically. As an application we obtain new cancellation results for integral groups rings. This is joint work…

Title: Density of R-free sets Abstract: We will discuss a number of new results about the density of certain translates in R-free numbers. We present a solution to Problem 486 in the Erdős' Problems website, and a partial resolution of Problem 25. We will also discuss a new result from van Doorn and Tao.

Title: Counting $D_4$ extensions by multiple invariants Abstract: Malle’s conjecture is a central question in arithmetic statistics, predicting an asymptotic formula for the number of number fields with a prescribed Galois group and bounded discriminant. Recently, such counting problems have received considerable attention. In 2022 Gundlach introduced the notion of ordering number fields by…

Title: A Chebotarev Density Theorem over local fields Abstract: I will present an analog of the Chebotarev Density Theorem which works over local fields. As an application, I will use it to prove a conjecture of Bhargava, Cremona, Fisher, and Gajovi\'c. This is joint work with Asvin G and Yifan Wei.

Title: Linear Disjointness, Relativised Cyclotomic Polynomials and Inflated G-extensions for Number Fields Abstract: There had been many attempts to generalize cyclotomic polynomials by many authors: combinatorialapproach using MOebius Inversion, Number-Theoretic approacj=h by focussing on unitary divisors etc. Our approach takes the factorization of X^n-1 as the model and defines cyclotomic…

Title: Additive combinatorics and descent Abstract: In this talk I will outline a method introduced in joint work with Peter Koymans that allowed us to settle Hilbert 10th problem for all finitely generated rings and to show that every number field has an elliptic curve of rank 1. I will also present joint work with Alexandra Shlapentokh outlining some further consequences of these results and…

Title: On the stabiliser of k-free algebraic integers Abstract: tba

Title: Orbit separation: A complexity measure for primitive inflation tilings

Title: Admmissible sets for Erdos sieves Abstract: When considering a set of pairwise coprime numbers b_1,b_2,... the admissible sets, those which don't contain every congruence class mod b_i for any i, correspond to the closure of the shift {F+n} with n an integer and F those numbers not divisible by any b_i. When considering a generalization for Erdos sieves, this does not hold anymore. In this…

Title: Arithmetic on hyperelliptic Jacobians and singular curves Abstract: Mumford representation encodes each divisor class in the Jacobian of a nonsingular hyperelliptic curve as a unique pair of polynomials. And Cantor’s Algorithm provides an implementation of the addition of two divisor classes in this setting. In my thesis, I extended these ideas to study the arithmetic on Jacobians of…

Title: Baker's work on linear forms in logarithms, II Abstract: In part 1 of this two-part series, Marc Technau gave an overview of a method developed by Baker. In this talk, we will give a more detailed account of the proof of Baker's theorem [1]. [1] A. Baker. Linear forms in the logarithms of algebraic numbers. Mathematika 13:204–216, 1966.

Title: Baker's work on linear forms in logarithms, I Abstract: This talk is part of a series on the celebrated Baker–Heegner–Stark theorem concerning the determination of all imaginary-quadratic number fields with class number one, following the approach via Baker’s theorem on linear forms in logarithms of algebraic numbers. This talk and the subsequent talk by Fabian Gundlach shall give a…

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: 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: 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 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: 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: 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: 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…

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