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…
