Related papers: Progress Towards Counting D_5 Quintic Fields

We prove an improvement on Schmidt's upper bound on the number of number fields of degree $n$ and absolute discriminant less than X for $6 \leq n \leq 94$. We carry this out by improving and applying a uniform bound on the number of monic…

Given a number field $k$, we show that, for many finite groups $G$, all the Galois extensions of $k$ with Galois group $G$ cannot be obtained by specializing any given finitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$…

In previous works, we described algorithms to compute the number field cut out by the mod ell representation attached to a modular form of level N=1. In this article, we explain how these algorithms can be generalised to forms of higher…

We investigate the Galois module structure of the Tate-Shafarevich group of elliptic curves. For a Dirichlet character $\chi$, we give an explicit conjecture relating the ideal factorization of $L(E,\chi,1)$ to the Galois module structure…

We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological…

We calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing $\zeta_5$. Applied to imaginary quadratic fields, this proportion is at least $2/5$. Applied to cyclotomic…

The complexity of computing the Galois group of a linear differential equation is of general interest. In a recent work, Feng gave the first degree bound on Hrushovski's algorithm for computing the Galois group of a linear differential…

Let $GP(q,d)$ be the $d$-Paley graph defined on the finite field $\mathbb{F}_q$. It is notoriously difficult to improve the trivial upper bound $\sqrt{q}$ on the clique number of $GP(q,d)$. In this paper, we investigate the connection…

We count abelian number fields ordered by arbitrary height function whose generator of tame inertia is restricted to lie in a given subset of the Galois group, and find an explicit formula for the leading constant. We interpret our results…

Let $\mathsf{r}_k$ be the unique positive root of $x^k - (x+1)^{k-1} = 0$. We prove the best known bounds on the number $n_{g,d}$ of $d$-dimensional generalized numerical semigroups, in particular that \[n_{g,d} > C_d^{g^{(d-1)/d}}…

We introduce a generalisation of norm relations in the group algebra Q[G], where G is a finite group. We give some properties of these relations, and use them to obtain relations between the S-unit groups of different subfields of the same…

We improve the best known constant $\frac{3-\sqrt 5}{2}$ for which the union-closed conjecture is known to be true, by using dependent samples as suggested by Sawin and the entropy approach on this problem initiated by Gilmer. Meanwhile, we…

We establish asymptotic formulae for the number of biquadratic number fields of bounded discriminant that can be embedded into a quaternionic or a dihedral extension. To prove these results, we express the solvability of these inverse…

We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper…

We prove that function fields of varieties of dimension at least two over an algebraic closure of a finite field are determined, modulo purely inseparable extensions, by the quotient by the second term in the lower central series of their…

We establish upper bounds for the smallest height of a generator of a number field $k$ over the rational field $\Q$. Our first bound applies to all number fields $k$ having at least one real embedding. We also give a second conditional…

This is the second in a pair of papers about residually reducible Galois deformation rings with non-optimal level. In the first paper, we proved a Galois-theoretic criterion for the deformation ring to be as small as possible. This paper…

Let $X\subseteq \mathbb{P}^3$ be a smooth projective surface of degree $d\ge 4$ defined over a number field $K$, and let $N_{X^{\prime}}(B)$ be the number of rational points of $X$ of height at most $B$ that do not lie on lines contained in…

We improve a result of H. L. Montgomery and J. P. Weinberger by establishing the existence of infinitely many fundamental discriminants $d>0$ for which the class number of the real quadratic field $\mathbb{Q}(\sqrt{d})$ exeeds…

The main result in this paper concerns a new five-variable expander. It is proven that for any finite set of real numbers $A$, $$|\{(a_1+a_2+a_3+a_4)^2+\log a_5 :a_1,a_2,a_3,a_4,a_5 \in A \}| \gg \frac{|A|^2}{\log |A|}.$$ This bound is…