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1. Unramified Heisenberg group extensions of number fields

   Frauke M. Bleher, Ted Chinburg (February 2022, Israel journal of mathematics)

   We construct ́etale generalized Heisenberg group covers of hyperelliptic curves over number fields. We use these to produce infinite families of quadratic extensions of cyclotomic fields that admit everywhere unramified generalized Heisenberg Galois extensions. 

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2. Geometric sieve over number fields for higher moments

   https://doi.org/10.1007/s40993-023-00466-6  

   Micheli, Giacomo; Schraven, Severin; Tinani, Simran; Weger, Violetta (August 2023, Research in Number Theory)

   Abstract The geometric sieve for densities is a very convenient tool proposed by Poonen and Stoll (and independently by Ekedahl) to compute the density of a given subset of the integers. In this paper we provide an effective criterion to find all higher moments of the density (e.g. the mean, the variance) of a subset of a finite dimensional free module over the ring of algebraic integers of a number field. More precisely, we provide a geometric sieve that allows the computation of all higher moments corresponding to the density, over a general number fieldK. This work advances the understanding of geometric sieve for density computations in two ways: on one hand, it extends a result of Bright, Browning and Loughran, where they provide the geometric sieve for densities over number fields; on the other hand, it extends the recent result on a geometric sieve for expected values over the integers to both the ring of algebraic integers and to moments higher than the expected value. To show how effective and applicable our method is, we compute the density, mean and variance of Eisenstein polynomials and shifted Eisenstein polynomials over number fields. This extends (and fully covers) results in the literature that were obtained with ad-hoc methods. 

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3. Number theoretical locally recoverable codes

   https://doi.org/10.1142/S0219498825500604  

   Ferraguti, Andrea; Goldfeld, Dorian; Micheli, Giacomo (October 2023, Journal of Algebra and Its Applications)

   In this paper, we give constructions for infinite sequences of finite nonlinear locally recoverable codes [Formula: see text] over a product of finite fields arising from basis expansions in algebraic number fields. The codes in our sequences have increasing length and size, constant rate, fixed locality, and minimum distance going to infinity. 

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4. Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence

   https://doi.org/10.1515/crelle-2022-0056  

   Frei, Sarah; Hassett, Brendan; Várilly-Alvarado, Anthony (September 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Given a smooth projective variety over a number field and an elementof its Brauer group, we consider the specialization of the Brauerclass at a place of good reduction for the variety and the class. Weare interested in the case of K3 surfaces.We show that a Brauer class on a very general polarized K3 surfaceover a number field becomes trivial after specialization at a set ofplaces of positive natural density. We deduce that there exist cubic fourfolds over number fields that are conjecturally irrational, with rational reduction at a positive proportion of places. We also deduce that there are twisted derivedequivalent K3 surfaces which become derived equivalent after reductionat a positive proportion of places. 

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5. 2-Adic point counting on K3 surfaces

   https://doi.org/10.1007/s40993-022-00378-x  

   Elsenhans, Andreas-Stephan; Jahnel, Jörg (December 2022, Research in Number Theory)

   Abstract This article reports on an approach to point counting on algebraic varieties over finite fields that is based on a detailed investigation of the 2-adic orthogonal group. Combining the new approach with a p -adic method, we count the number of points on some K 3 surfaces over the field $$\mathbb {F}_{\!p}$$ F p , for all primes $$p < 10^8$$ p < 10 8 . 

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