Conservation laws provide important constraints on the solutions of partial differential equations (PDEs), therefore it is important to preserve them when discretizing such equations. In this paper, a new systematic method for discretizing a PDE, so as to preserve the local form of multiple conservation laws, is presented. The technique, which uses symbolic computation, is applied to the Korteweg–de Vries (KdV) equation to find novel explicit and implicit schemes that have finite difference analogues of its first and second conservation laws and its first and third conservation laws. The resulting schemes are numerically compared with a multisymplectic scheme.

Efficient methods for computing with matrices over finite fields often involve randomised algorithms, where matrices with a certain property are sought via repeated random selection. Complexity analyses for such algorithms require knowledge of the proportion of relevant matrices in the ambient group or algebra. We introduce a method for estimating proportions of families $N$ of elements in the algebra of all $d\times d$ matrices over a field of order $q$ , where membership of a matrix in $N$ depends only on its ‘invertible part’. The method is based on the availability of estimates for proportions of certain non-singular matrices depending on $N$ , so that existing estimation techniques for non-singular matrices can be used to deal with families containing singular matrices. As an application, we investigate primary cyclic matrices, which are used in the Holt–Rees MEATAXE algorithm for testing irreducibility of matrix algebras.

The Coleman integral is a $p$ -adic line integral that encapsulates various quantities of number theoretic interest. Building on the work of Harrison [J. Symbolic Comput. 47 (2012) no. 1, 89–101], we extend the Coleman integration algorithms in Balakrishnan et al. [Algorithmic number theory, Lecture Notes in Computer Science 6197 (Springer, 2010) 16–31] and Balakrishnan [ANTS-X: Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Series 1 (Mathematical Sciences Publishers, 2013) 41–61] to even-degree models of hyperelliptic curves. We illustrate our methods with numerical examples computed in Sage.

For an elliptic curve $E/\mathbb{Q}$ without complex multiplication we study the distribution of Atkin and Elkies primes $\ell$ , on average, over all good reductions of $E$ modulo primes $p$ . We show that, under the generalized Riemann hypothesis, for almost all primes $p$ there are enough small Elkies primes $\ell$ to ensure that the Schoof–Elkies–Atkin point-counting algorithm runs in $(\log p)^{4+o(1)}$ expected time.

We present a computer algebra package based onMagma for performing computations in rational Cherednik algebras with arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero, which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We can determine the decomposition matrices of the Verma modules, the graded $G$ -module structure of the simple modules, and the Calogero–Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we can also confirm Martino’s conjecture for several exceptional complex reflection groups.

Supplementary materials are available with this article.

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.

Our main purpose in this paper is to propose the piecewise Legendre spectral-collocation method to solve Volterra integro-differential equations. We provide convergence analysis to show that the numerical errors in our method decay in $h^{m}N^{-m}$ -version rate. These results are better than the piecewise polynomial collocation method and the global Legendre spectral-collocation method. The provided numerical examples confirm these theoretical results.

We construct explicit bases for spaces of overconvergent $p$ -adic modular forms when $p=2,3$ and study their interaction with the Atkin operator. This results in an extension of Lauder’s algorithms for overconvergent modular forms. We illustrate these algorithms with computations of slope sequences of some $2$ -adic eigencurves and the construction of Chow–Heegner points on elliptic curves via special values of Rankin triple product L-functions.

Polynomials for blending parametric curves in Lie groups are defined. Properties of these polynomials are proved. Blending parametric curves in Lie groups with these polynomials is considered. Then application of the proposed technique to construction of spline curves on smooth manifolds is presented. As an example, construction of spherical spline curves using the proposed approach is depicted.

Let $A$ be an abelian variety of dimension $g$ together with a principal polarization ${\it\phi}:A\rightarrow \hat{A}$ defined over a field $k$ . Let $\ell$ be an odd integer prime to the characteristic of $k$ and let $K$ be a subgroup of $A[\ell ]$ which is maximal isotropic for the Riemann form associated to ${\it\phi}$ . We suppose that $K$ is defined over $k$ and let $B=A/K$ be the quotient abelian variety together with a polarization compatible with ${\it\phi}$ . Then $B$ , as a polarized abelian variety, and the isogeny $f:A\rightarrow B$ are also defined over $k$ . In this paper, we describe an algorithm that takes as input a theta null point of $A$ and a polynomial system defining $K$ and outputs a theta null point of $B$ as well as formulas for the isogeny $f$ . We obtain a complexity of $\tilde{O} (\ell ^{(rg)/2})$ operations in $k$ where $r=2$ (respectively, $r=4$ ) if $\ell$ is a sum of two (respectively, four) squares which constitutes an improvement over the algorithm described in Cosset and Robert (Math. Comput. (2013) accepted for publication). We note that the algorithm is quasi-optimal if $\ell$ is a sum of two squares since its complexity is quasi-linear in the degree of $f$ .

We study the problem of efficiently constructing a curve $C$ of genus  $2$ over a finite field $\mathbb{F}$ for which either the curve $C$ itself or its Jacobian has a prescribed number $N$ of $\mathbb{F}$ -rational points.

In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus- $2$ curves necessarily takes time exponential in the size of its input.

On the other hand, we provide an algorithm for producing a genus- $2$ curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus- $2$ curve having exactly $10^{2014}+9703$ (prime) points, and two genus- $2$ curves each having exactly $10^{2013}$ points.

In an appendix we provide a complete parametrization, over an arbitrary base field $k$ of characteristic neither two nor three, of the family of genus- $2$ curves over $k$ that have $k$ -rational degree- $3$ maps to elliptic curves, including formulas for the genus- $2$ curves, the associated elliptic curves, and the degree- $3$ maps.

Supplementary materials are available with this article.

This paper revisits the solution of the word problem for ${\it\omega}$ -terms interpreted over finite aperiodic semigroups, obtained by J. McCammond. The original proof of correctness of McCammond’s algorithm, based on normal forms for such terms, uses McCammond’s solution of the word problem for certain Burnside semigroups. In this paper, we establish a new, simpler, correctness proof of McCammond’s algorithm, based on properties of certain regular languages associated with the normal forms. This method leads to new applications.

Two semigroups are distinct if they are neither isomorphic nor anti-isomorphic. Although there exist  $15\,973$  pairwise distinct semigroups of order six, only four are known to be non-finitely based. In the present article, the finite basis property of the other $15\,969$  distinct semigroups of order six is verified. Since all semigroups of order five or less are finitely based, the four known non-finitely based semigroups of order six are the only examples of minimal order.