Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree d and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over Z) in a homogeneously expanding star body in R d+1 . The volume of this star body was computed by Chern and Vaaler, who also computed the volume of the codimension-one "slice" corresponding to monic polynomials -this led to results of Barroero on counting algebraic integers. We show how to estimate the volume of higher-codimension slices, which allows us to count units, algebraic integers of given norm, trace, norm and trace, and more. We also refine the lattice point-counting arguments of Chern-Vaaler to obtain explicit error terms with better power savings, which lead to explicit versions of some results of Masser-Vaaler and Barroero.

Let k be a number field, and denote by k^[d] the compositum of all degree d extensions of k in a fixed algebraic closure. We first consider the question of whether all algebraic extensions of k of degree less than d lie in k^[d]. We show that this occurs if and only if d < 5. Secondly, we consider the question of whether there exists a constant c such that if K/k is a finite subextension of k^[d], then K is generated over k by elements of degree at most c. This was previously considered by Checcoli. We show that such a constant exists if and only if d < 3. This question becomes more interesting when one restricts attention to Galois extensions K/k. In this setting, we derive certain divisibility conditions on d under which such a constant does not exist. If d is prime, we prove that all finite Galois subextensions of k^[d] are generated over k by elements of degree at most d.Comment: 14 pages, 2 figure

Abstract. A subfield K ⊆ Q has the Bogomolov property if there exists a positive ε such that no non-torsion point of K × has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of L × \ K × . We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K and K/Q is Galois.

Let F be an algebraic extension of the rational numbers and E an elliptic curve defined over some number field contained in F . The absolute logarithmic Weil height, respectively the Néron-Tate height, induces a norm on F * modulo torsion, respectively on E(F ) modulo torsion. The groups F * and E(F ) are free abelian modulo torsion if the height function does not attain arbitrarily small positive values. In this paper we prove the failure of the converse to this statement by explicitly constructing counterexamples.2010 Mathematics Subject Classification. 11G50, 11G05 (primary); 20K20 (secondary).

Bruinier and Ono recently developed the theory of generalized Borcherds products, which uses coefficients of certain Maass forms as exponents in infinite product expansions of meromorphic modular forms. Using this, one can use classical results on congruences of modular forms to obtain congruences for Maass forms. In this note we work out the example of Ramanujan's mock theta functions f and ω in detail.2010 Mathematics Subject Classification. 11F03, 11F33.