In this paper, we generalise to the family of Fermat quartics $X^4 + Y^4 = 2^m, m \in \mathbb {Z}$, a result of Aigner [‘Über die Möglichkeit von $x^4 + y^4 = z^4$ in quadratischen Körpern’, Jahresber. Deutsch. Math.-Ver. 43 (1934), 226–228], which proves that there is only one quadratic field, namely $\mathbb {Q}(\sqrt {-7})$, that contains solutions to the Fermat quartic $X^4 + Y^4 = 1$. The $m \equiv 0 \pmod 4$ case is due to Aigner. The $m \equiv 2 \pmod 4$ case follows from a result of Emory [‘The Diophantine equation $X^4 + Y^4 = D^2Z^4$ in quadratic fields’, Integers 12 (2012), Article no. A65, 8 pages]. This paper focuses on the two cases $m \equiv 1, 3 \pmod 4$, classifying for $m \equiv 1 \pmod 4$ the infinitely many quadratic number fields that contain solutions, and proving for $m \equiv 3 \pmod 4$ that $\mathbb {Q}(\sqrt {2})$ and $\mathbb {Q}(\sqrt {-2})$ are the only quadratic number fields that contain solutions.

Let $n$ be an integer congruent to $0$ or $3$ modulo $4$. Under the assumption of the ABC conjecture, we prove that, given any integer $m$ fulfilling only a certain coprimeness condition, there exist infinitely many imaginary quadratic fields having an everywhere unramified Galois extension of group $A_n \times C_m$. The same result is obtained unconditionally in special cases.

Let $f(X) \in {\mathbb Z}[X]$ be a polynomial of degree $d \ge 2$ without multiple roots and let ${\mathcal F}(N)$ be the set of Farey fractions of order N. We use bounds for some new character sums and the square-sieve to obtain upper bounds, pointwise and on average, on the number of fields ${\mathbb Q}(\sqrt {f(r)})$ for $r\in {\mathcal F}(N)$, with a given discriminant.

Using a recent breakthrough of Smith [18], we improve the results of Fouvry and Klüners [4, 5] on the solubility of the negative Pell equation. Let $\mathcal {D}$ denote the set of positive squarefree integers having no prime factors congruent to $3$ modulo $4$. Stevenhagen [19] conjectured that the density of d in $\mathcal {D}$ such that the negative Pell equation $x^2-dy^2=-1$ is solvable with $x, y \in \mathbb {Z}$ is $58.1\%$, to the nearest tenth of a percent. By studying the distribution of the $8$-rank of narrow class groups $\operatorname {\mathrm {Cl}}^+(d)$ of $\mathbb {Q}(\sqrt {d})$, we prove that the infimum of this density is at least $53.8\%$.

We give a family of real quadratic fields such that the 2-class field towers over their cyclotomic $\mathbb Z_2$-extensions have metabelian Galois groups of abelian invariants $[2,2,2]$. We also consider the boundedness of the Galois groups in relation to Greenberg’s conjecture, and calculate their class-2 quotients with an explicit example.

For any odd prime p, we construct an infinite family of imaginary quadratic fields whose class numbers are divisible by p. We give a corollary that settles Iizuka’s conjecture for the case n=1 and p>2.

We discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei [16] in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are ‘governed’ by Frobenius conditions on the primes dividing the discriminant, a fact used in the recent work of A. Smith [18, 19]. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation $x^2-dy^2=-1$.

Aigner showed in 1934 that nontrivial quadratic solutions to $x^4 + y^4 = 1$ exist only in $\mathbb Q(\sqrt {-7})$. Following a method of Mordell, we show that nontrivial quadratic solutions to $x^4 + 2^ny^4 = 1$ arise from integer solutions to the equations $X^4 \pm 2^nY^4 = Z^2$ investigated in 1853 by V. A. Lebesgue.

Let $p$ be an odd prime. For a number field $K$, we let $K_{\infty }$ be the maximal unramified pro-$p$ extension of $K$; we call the group $\text{Gal}(K_{\infty }/K)$ the $p$-class tower group of $K$. In a previous work, as a non-abelian generalization of the work of Cohen and Lenstra on ideal class groups, we studied how likely it is that a given finite $p$-group occurs as the $p$-class tower group of an imaginary quadratic field. Here we do the same for an arbitrary real quadratic field $K$ as base. As before, the action of $\text{Gal}(K/\mathbb{Q})$ on the $p$-class tower group of $K$ plays a crucial role; however, the presence of units of infinite order in the ground field significantly complicates the possibilities for the groups that can occur. We also sharpen our results in the imaginary quadratic field case by removing a certain hypothesis, using ideas of Boston and Wood. In the appendix, we show how the probabilities introduced for finite $p$-groups can be extended in a consistent way to the infinite pro-$p$ groups which can arise in both the real and imaginary quadratic settings.

Let $n$ be a positive integer and $a$ an integer prime to $n$. Multiplication by $a$ induces a permutation over $\mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$. Lerch’s theorem gives the sign of this permutation. We explore some applications of Lerch’s result to permutation problems involving quadratic residues modulo $p$ and confirm some conjectures posed by Sun [‘Quadratic residues and related permutations and identities’, Preprint, 2018, arXiv:1809.07766]. We also study permutations involving arbitrary $k$th power residues modulo $p$ and primitive roots modulo a power of $p$.

We construct a random model to study the distribution of class numbers in special families of real quadratic fields ${\open Q}(\sqrt d )$ arising from continued fractions. These families are obtained by considering continued fraction expansions of the form $\sqrt {D(n)} = [f(n),\overline {u_1,u_2, \ldots ,u_{s-1} ,2f(n)]} $ with fixed coefficients u1, …, us−1 and generalize well-known families such as Chowla's 4n2 + 1, for which analogous results were recently proved by Dahl and Lamzouri [‘The distribution of class numbers in a special family of real quadratic fields’, Trans. Amer. Math. Soc. (2018), 6331–6356].

Boston, Bush and Hajir have developed heuristics, extending the Cohen–Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro- $p$ extensions of imaginary quadratic number fields for $p$ an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of $\mathbb{F}_{q}(t)$ , the Galois groups of the maximal unramified pro- $p$ extensions, as $q\rightarrow \infty$ , have the moments predicted by the Boston, Bush and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.

In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding three-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid three-term geometric progressions. When unique factorization fails, or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets ‘greedily’, a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.

Since its introduction in 2010 by Lyubashevsky, Peikert and Regev, the ring learning with errors problem (ring-LWE) has become a popular building block for cryptographic primitives, due to its great versatility and its hardness proof consisting of a (quantum) reduction from ideal lattice problems. But, for a given modulus $q$ and degree $n$ number field $K$ , generating ring-LWE samples can be perceived as cumbersome, because the secret keys have to be taken from the reduction mod $q$ of a certain fractional ideal ${\mathcal{O}}_{K}^{\vee }\subset K$ called the codifferent or ‘dual’, rather than from the ring of integers ${\mathcal{O}}_{K}$ itself. This has led to various non-dual variants of ring-LWE, in which one compensates for the non-duality by scaling up the errors. We give a comparison of these versions, and revisit some unfortunate choices that have been made in the recent literature, one of which is scaling up by ${|\unicode[STIX]{x1D6E5}_{K}|}^{1/2n}$ with $\unicode[STIX]{x1D6E5}_{K}$ the discriminant of $K$ . As a main result, we provide, for any $\unicode[STIX]{x1D700}>0$ , a family of number fields $K$ for which this variant of ring-LWE can be broken easily as soon as the errors are scaled up by ${|\unicode[STIX]{x1D6E5}_{K}|}^{(1-\unicode[STIX]{x1D700})/n}$ .

In this paper, we present novel algorithms for finding small relations and ideal factorizations in the ideal class group of an order in an imaginary quadratic field, where both the norms of the prime ideals and the size of the coefficients involved are bounded. We show how our methods can be used to improve the computation of large-degree isogenies and endomorphism rings of elliptic curves defined over finite fields. For these problems, we obtain improved heuristic complexity results in almost all cases and significantly improved performance in practice. The speed-up is especially high in situations where the ideal class group can be computed in advance.

We provide a criterion for the central norm to be any value in the simple continued fraction expansion of $\sqrt{D}$ for any non-square integer $D\,>\,1$ . We also provide a simple criterion for the solvability of the Pell equation ${{x}^{2}}\,-\,D{{y}^{2}}\,=\,-1$ in terms of congruence conditions modulo $D$ .

Let hK denote the class number of the imaginary quadratic field , where m and n are positive integers, k is an odd integer with k > 1 and 22m < kn. In this paper we prove that if either 3 ∣ n and 22m − kn ≡ 5(mod 8) or n = 3 and k = (22m+2 −1)/3, then ∣ hK. Otherwise, we have n ∣ hK.

The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields.

Soient ${{p}_{1}},\,{{p}_{2}},\,{{p}_{3}}$ et $q$ des nombres premiers distincts tels que ${{p}_{1}}\,\equiv \,{{p}_{2}}\,\equiv \,{{p}_{3}}\,\equiv \,-q\,\equiv \,1\,(\bmod \,4)$ , $k=\mathbf{Q}(\sqrt{{{p}_{1}}},\sqrt{{{p}_{2}}},\sqrt{{{p}_{3}}},\sqrt{q})$ et $\text{C}{{\text{l}}_{2}}(k)$ le 2-groupe de classes de $k$ . A. Fröhlich a démontré que $\text{C}{{\text{l}}_{2}}(k)$ n’est jamais trivial. Dans cet article, nous donnons une extension de ce résultat, en démontrant que le rang de $\text{C}{{\text{l}}_{2}}(k)$ est toujours supérieur ou égal à 2. Nous démontrons aussi, que la valeur 2 est optimale pour une famille infinie de corps $k$ .

Starting with a paper of Jacobson from the 1960s, many authors became interested in characterizing all algebraic number fields in which each integer is the sum of pairwise distinct units. Although there exist many partial results for number fields of low degree, a full characterization of these number fields is still not available. Narkiewicz and Jarden posed an analogous question for sums of units that are not necessarily distinct. In this paper we propose a generalization of these problems. In particular, for a given rational integer n we consider the following problem. Characterize all number fields for which every integer is a linear combination of finitely many units εi in a way that the coefficients ai∈ℕ are bounded by n. The paper gives several partial results on this problem. In our proofs we exploit the fact that these representations are related to symmetric beta expansions with respect to Pisot bases.