Narrow Results

In this paper, we extend an inequality that compares the pull-back of the total dimension divisor of an étale sheaf and the total dimension divisor of the pull-back of the sheaf due to Saito. Using this formula, we generalize Deligne and Laumon’s lower semi-continuity property for Swan conductors of étale sheaves on relative curves to higher relative dimensions in a geometric situation.

In this work, we present some arithmetic properties of families of abelian $p$-extensions of global function fields, among which are their generators and their type of ramification and decomposition.

The main goal of this work is to develop a robust framework for an exhaustive description of essential properties of a fractal object. For this purpose, the inherent features of fractal sets are scrutinized. The topological, metrological, morphological, and topographical attributes of fractal systems are delineated. The criteria of the fractal connectedness are established. The characteristics of the fractal connectivity and ramification are ascertained. The index of the fractal loopiness is introduced. The quantifications of the fractal heterogeneity, lacunarity, and anisotropy are briefly sketched out. A set of key attributes which enable a proper characterization of fractal system are suggested.

Serre's conjecture predicts the nonexistence of certain nonsolvable Galois extensions of ℚ which are unramified outside one small prime. These nonexistence theorems have been proven by the techniques of discriminant bounding. In this paper, we will apply these techniques to nonsolvable extensions of small degree number fields.

We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form ±p^A for p ≤ 7 a prime. Our main polynomial has degree 25, Galois group of the form PSL₂(5)⁵.10, and field discriminant 5⁶⁹. A closely related polynomial has degree 120, Galois group of the form SL₂(5)⁵.20, and field discriminant 5³¹¹. We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125 · 5^-1/12500 ≈ 124.984 and the class number of the latter field is divisible by 5⁴.

For a prime p, let ζ_p denote a fixed primitive pth root of unity. Let E be an elliptic curve over a number field K with a p-torsion point. Then the p-torsion subgroup of E gives a Kummer extension over K(ζ_p), and in this paper, we study the ramification of such Kummer extensions using the Kummer generators directly computed by Verdure in 2006. For quadratic fields K, we also give unramified Kummer extensions over K(ζ_p) generated from elliptic curves over K having a p-torsion point with bad reduction at certain primes. Many of these unramified Kummer extensions have not appeared in the previous work using fundamental units of quadratic fields.

The generalized Rikuna polynomials are an iterative generalization of Rikuna's generic cyclic polynomials, which themselves generalize Shanks' cubic polynomials. In this paper we study Galois properties of the generalized Rikuna polynomials under specialization.

We tabulate polynomials in ℚ[t] with a given factorization partition, bad reduction entirely within a given set of primes, and satisfying auxiliary conditions associated to 0, 1, and ∞. We explain how these polynomials are of particular interest because of their role in the construction of nonsolvable number fields of arbitrarily large degree and bounded ramification.

For a prime p, let ζ_p denote a fixed primitive pth root of unity. Let E be an elliptic curve over a number field k with a p-torsion point. Then the p-torsion subgroup of E gives a Kummer extension over k(ζ_p). In this paper, for p = 5 and 7, we study the ramification of such Kummer extensions using explicit Kummer generators directly computed by Verdure in 2006.

There are several variants of the inverse Galois problem which involve restrictions on ramification. In this paper, we give sufficient conditions that a given finite group $G$ occurs infinitely often as a Galois group over the rationals $\mathbb{Q}$ with all nontrivial inertia groups of order $2$. Notably any such realization of $G$ can be translated up to a quadratic field over which the corresponding realization of $G$ is unramified. The sufficient conditions are imposed on a parametric polynomial with Galois group $G$ — if such a polynomial is available — and the infinitely many realizations come from infinitely many specializations of the parameter in the polynomial. This will be applied to the three finite simple groups $A_5$, $PS{L}_2(7)$ and $PS{L}_3(3)$. Finally, the applications to $A_5$ and $PS{L}_3(3)$ are used to prove the existence of infinitely many optimally intersective realizations of these groups over the rational numbers (proved for $PS{L}_2(7)$ by the first author in [J. König, On intersective polynomials with nonsolvable Galois group, Comm. Alg.46(6) (2018) 2405–2416.

This paper describes cubic function fields $L/K$ with prescribed ramification, where $K$ is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely cubic closure $K^{\prime }/K$ of $L/K$ is of genus zero, and a description of the twists of $L/K$ up to isomorphism over $K$. For cubic function fields of genus at most one, we also describe the twists and isomorphism classes obtained when one allows Möbius transformations on $K$. The paper concludes by studying the more general case of covers of elliptic and hyperelliptic curves that are ramified above exactly one point.

Let K be a number field. The Pólya field concept is used to know when the module of integer-valued polynomials over the ring of integers ${{𝒪}}_K$ of K has a regular basis. In [C. W.-W. Tougma, Some questions on biquadratic Pólya fields, J. Number Theory 229 (2021) 386–398], the author used cohomological results from [C. B. Setzer, Units over totally real $C_2\times C_2$ fields, J. Number Theory 12 (1980) 160–175] to answer questions raised in [A. Leriche, Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, J. Théor. Nr. Bordx. 23 (2011) 235–249; A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory  133 (2013) 59–71] on biquadratic Pólya fields. Here we first prove that number fields were omitted from the list of exceptional fields cited in [A. Leriche, Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, J. Théor. Nr. Bordx. 23 (2011) 235–249; A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory  133 (2013) 59–71]. We therefore identify new biquadratic Pólya fields, where the prime number 2 is totally ramified. This result corrects and completes some others on the literature. On the other hand, we show that the main results of [C. W.-W. Tougma, Some questions on biquadratic Pólya fields, J. Number Theory  229 (2021) 386–398] and this paper can be proved with a single method using Kuroda/Kubota’s unit formula without cohomological results of [C. B. Setzer, Units over totally real $C_2\times C_2$ fields, J. Number Theory  12 (1980) 160–175].