We study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.

We give an enumeration of all positive definite primitive $ \mathbb{Z} $-lattices in dimension $n\geq 3$ whose genus consists of a single isometry class. This is achieved by using bounds obtained from the Smith–Minkowski–Siegel mass formula to computationally construct the square-free determinant lattices with this property, and then repeatedly calculating pre-images under a mapping first introduced by G. L. Watson.

We hereby complete the classification of single-class genera in dimensions 4 and 5 and correct some mistakes in Watson’s classifications in other dimensions. A list of all single-class primitive $ \mathbb{Z} $-lattices has been compiled and incorporated into the Catalogue of Lattices.

We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in MAGMA. This is illustrated with the computation of an example with an irreducible cubic factor in the singular locus of the defining pencil of quadrics (in contrast to previous examples, which had at worst quadratic irreducible factors). We exploit the relationship with the Tate-Shafarevich group to give new types of examples of III [2], for families of curves of genus 2 of the form y2 = f(x), where f(x) is a quintic containing an irreducible cubic factor.

We study the p-form spectrum of the Laplace-Beltrami operator acting on lens spaces as considered by Ikeda [Geometry of manifolds (Academic Press, Boston, MA, 1989) 383–417]. Ikeda gave examples of such spaces that are non-isometric but isospectral for all p ≤ p0. In this paper we exhibit examples of such spaces that are not isometric, and are isospectral for various, but not for all. values of p. In particular, examples are given of non-isometric lens spaces that are isospectral for some values of p but not for the case p = 0.

This paper concerns the existence and algorithmic determination of minimal models for curves of genus 1, given by equations of the form y2 = Q(x), where Q(x) has degree 4. These models are used in the method of 2-descent for computing the rank of an elliptic curve. The results described here are complete for unramified extensions of Q2 and Q3, and for all p-adic fields for p greater than or equal to 5. The primary motivation for this work was to complete the results of Birch and Swinnerton-Dyer, which are incomplete in the case of Q2. The results in this case (when applied to 2-coverings of elliptic curves over Q) yield substantial improvements in the running times of the 2-descent algorithm implemented in the program mwrank. The paper ends with a section on implementation and examples, and an appendix gives constructive proofs in sufficient detail to be used for implementation.

We show that if a Barker sequence of length $n>13$ exists, then either n $=$ 3 979 201 339 721 749 133 016 171 583 224 100, or $n > 4\cdot 10^{33}$. This improves the lower bound on the length of a long Barker sequence by a factor of nearly $2000$. We also obtain eighteen additional integers $n<10^{50}$ that cannot be ruled out as the length of a Barker sequence, and find more than 237 000 additional candidates $n<10^{100}$. These results are obtained by completing extensive searches for Wieferich prime pairs and using them, together with a number of arithmetic restrictions on $n$, to construct qualifying integers below a given bound. We also report on some updated computations regarding open cases of the circulant Hadamard matrix problem.

We give a mathematically rigorous analysis which confirms the surprising results in a recent paper of Benilov, O‘Brien and Sazonov [J. Fluid Mech. 497 (2003) 201-224] about the spectrum of a highly singular non-self-adjoint operator that arises in a problem in fluid mechanics. We also show that the set of eigenvectors does not form a basis for the operator.

In this paper, we present the outcome of vast computer calculations, locating several of the very rare instances of level one cuspidal Bianchi modular forms that are not lifts of elliptic modular forms.

The authors construct faithful permutation representations of maximal 2-local subgroups and classify the radical chains of the Janko simple group J4; hence the Alperin weight conjecture and the Dade reductive conjecture for J4 are verified.

The algorithms described in this paper were developed to investigate three problems regarding polynomials with restricted coefficients: (i) determining whether there exist polynomials with {0, 1} coefficients and repeated noncyclotomic factors, (ii) searching for polynomials with {−1, 1} coefficients and small Mahler measure, and (iii) finding polynomials with {−1, 0, 1} coefficients with a root of high multiplicity off the unit circle. The results in the first problem presented here answer a question of Odlyzko and Poonen.

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 ${|\unicode[STIX]{x1D6E5}_{K}|}^{(1-\unicode[STIX]{x1D700})/n}$.

We exhibit seven linear codes exceeding the current best known minimum distance $d$ for their dimension $k$ and block length $n$. Each code is defined over ${ \mathbb{F} }_{8} $, and their invariants $[n, k, d] $ are given by $[49, 13, 27] $, $[49, 14, 26] $, $[49, 16, 24] $, $[49, 17, 23] $, $[49, 19, 21] $, $[49, 25, 16] $ and $[49, 26, 15] $. Our method includes an exhaustive search of all monomial evaluation codes generated by points in the $[0, 5] \times [0, 5] $ lattice square.

Let EΓ be a family of hyperelliptic curves over F2alg cl with general Weierstrass equation given over a very small field F. The author of this paper describes an algorithm for computing the zeta function of Eγ, with γ in a degree n extension field of F, which has time complexity O(n3 + ε) bit operations and memory requirements O(n2) bits. Using a slightly different algorithm, one can get time O(n2.667) and memory O(n2.5), and the computation for n curves of the family can be done in time O(n3.376). All of these algorithms are polynomial-time in the genus.

The authors present an algorithm to construct conjugacy class representatives of the solvable primitive subgroups of Sd for a given degree d. Using this method, they determine the solvable primitive permutation groups of degree at most 6560 (that is, 38 – 1), up to conjugacy.

This paper describes an algorithm of 2-descent for computing the rank of an elliptic curve without 2-torsion, defined over a general number field. This allows one, in practice, to deal with fields of degree from 1 to 5.

Let $G$ be a simple simply connected exceptional algebraic group of type $G_{2}$, $F_{4}$, $E_{6}$ or $E_{7}$ over an algebraically closed field $k$ of characteristic $p>0$ with $\mathfrak{g}=\text{Lie}(G)$. For each nilpotent orbit $G\cdot e$ of $\mathfrak{g}$, we list the Jordan blocks of the action of $e$ on the minimal induced module $V_{\text{min}}$ of $\mathfrak{g}$. We also establish when the centralizers $G_{v}$ of vectors $v\in V_{\text{min}}$ and stabilizers $\text{Stab}_{G}\langle v\rangle$ of $1$-spaces $\langle v\rangle \subset V_{\text{min}}$ are smooth; that is, when $\dim G_{v}=\dim \mathfrak{g}_{v}$ or $\dim \text{Stab}_{G}\langle v\rangle =\dim \text{Stab}_{\mathfrak{g}}\langle v\rangle$.

We consider the 33 conjugacy classes of genus zero, torsion-free modular subgroups, computing ramification data and Grothendieck’s dessins d’enfants. In the particular case of the index 36 subgroups, the corresponding Calabi–Yau threefolds are identified, in analogy with the index 24 cases being associated to K3 surfaces. In a parallel vein, we study the 112 semi-stable elliptic fibrations over ${ \mathbb{P} }^{1} $ as extremal K3 surfaces with six singular fibres. In each case, a representative of the corresponding class of subgroups is identified by specifying a generating set for that representative.

Here we determine up to conjugacy all the maximal subgroups of the finite exceptional group of Lie-type $E_{7}(2)$.

Supplementary materials are available with this article.

We formulate a variational problem on a Riemannian manifold M whose solutions are piecewise smooth geodesies that best fit a given data set of time labelled points in M. By a limiting process, these solutions converge to a single point in M. which we prove to be the Riemannian mean of the given points for some particular Riemannian manifolds such as Euclidean spaces, connected and compact Lie groups, and spheres.

We introduce the computer algebra package PyCox, written entirely in the Python language. It implements a set of algorithms, in a spirit similar to the older CHEVIE system, for working with Coxeter groups and Hecke algebras. This includes a new variation of the traditional algorithm for computing Kazhdan–Lusztig cells and W-graphs, which works efficiently for all finite groups of rank ≤8 (except E8). We also discuss the computation of Lusztig’s leading coefficients of character values and distinguished involutions (which works for E8 as well). Our experiments suggest a re-definition of Lusztig’s ‘special’ representations which, conjecturally, should also apply to the unequal parameter case. Supplementary materials are available with this article.