We test R. van Luijk’s method for computing the Picard group of a K3 surface. The examples considered are the resolutions of Kummer quartics in ℙ3. Using the theory of abelian varieties, the Picard group may be computed directly in this case. Our experiments show that the upper bounds provided by van Luijk’s method are sharp when sufficiently many primes are used. In fact, there are a lot of primes that yield a value close to the exact one. However, for many but not all Kummer surfaces V of Picard rank 18, we have for a set of primes of density at least 1/2.

We find all quadratic post-critically finite (PCF) rational functions defined over $\mathbb{Q}$, up to conjugation by elements of $\mathop{\rm PGL}_2(\overline{\mathbb{Q}})$. We describe an algorithm to search for possibly PCF functions. Using the algorithm, we eliminate all but 12 rational functions, all of which are verified to be PCF. We also give a complete description of all possible rational preperiodic structures for quadratic PCF functions defined over $\mathbb{Q}$.

In order to assess the security of cryptosystems based on the discrete logarithm problem in non-prime finite fields, as are the torus-based or pairing-based ones, we investigate thoroughly the case in $\mathbb{F}_{p^{6}}$ with the number field sieve. We provide new insights, improvements, and comparisons between different methods to select polynomials intended for a sieve in dimension 3 using a special-$\mathfrak{q}$ strategy. We also take into account the Galois action to increase the relation productivity of the sieving phase. To validate our results, we ran several experiments and real computations for various polynomial selection methods and field sizes with our publicly available implementation of the sieve in dimension 3, with special-$\mathfrak{q}$ and various enumeration strategies.

This paper [1], which was published online on 1 June 2011, has been retracted by agreement between the authors, the journal’s Editor-in-Chief Derek Holt, the London Mathematical Society and Cambridge University Press. The retraction was agreed to prevent other authors from using incorrect mathematical results. (In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no. 1, 50–58; J. Number Theory 130 (2010) no. 4, 1109–1114. Furthermore, we formulate a criterion for the partial Riemann hypothesis and we provide some numerical evidence for it using new formulas for the Li coefficients.)

In this paper, the authors re-examine the reduction of Maurer and Wolf of the discrete logarithm problem to the Diffie-Hellman problem. They give a precise estimate for the number of operations required in the reduction, and then use this to estimate the exact security of the elliptic curve variant of the Diffie-Hellman protocol for various elliptic curves defined in standards.

Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of these genus 2 modular polynomials and give an improved way to explicitly compute them.

We show how to efficiently evaluate functions on Jacobian varieties and their quotients. We deduce an algorithm to compute $(l,l)$ isogenies between Jacobians of genus two curves in quasi-linear time in the degree $l^{2}$.

We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite-index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.

We give a natural geometric condition that ensures that sequences of interpolation polynomials (of fixed degree) of sufficiently differentiable functions with respect to the natural lattices introduced by Chung and Yao converge to a Taylor polynomial.

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.

The Verheul homomorphism is a group homomorphism from a finite subgroup of the multiplicative group of a field to an elliptic curve. The hardness of computation of the Verheul homomorphism was shown by Verheul to be closely related to the hardness of the computational Diffie-Hellman problem. Let p ≥ 5 be a prime, and let N be a prime satisfying √(12p) < N < 2p / √3, where N ≠ p. Let E be an ordinary elliptic curve over Fp, and let C ⊂ E be a cyclic subgroup of order N. Let H be the group of all Nth roots of unity (contained in the algebraic closure of Fp), and let phi be the Verheul isomorphism from H to C.

We consider a polynomial P such that P(z) is the X-coordinate of phi(z) for all z ∈ H – {1}. We show that, for at least approximately 58% of pairs (E, C), none of the coefficients of the non-constant terms of P vanishes.

The authors of this paper consider the anti-ferromagnetic Potts model on the the integer lattice Z2. The model has two parameters: q, the number of spins, and λ = exp(−β), where β 4 is ‘inverse temperature’. It is known that the model has strong spatial mixing if q > 7, or if q = 7 and λ = 0 or λ > 1/8, or if q = 6 and λ = 0 or λ > 1/4. The λ = 0 case corresponds to the model in which configurations are proper q-colourings of Z2. It is shown that the system has strong spatial mixing for q ≥ 6 and any λ. This implies that Glauber dynamics is rapidly mixing (so there is a fully-polynomial randomised approximation scheme for the partition function), and also that there is a unique infinite-volume Gibbs state. We also show that strong spatial mixing occurs for a larger range of λ than was previously known for q = 3, q = 4 and q = 5.

We use an invariant-theoretic method to compute certain twists of the modular curves $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X(n)$ for $n=7,11$. Searching for rational points on these twists enables us to find non-trivial pairs of $n$-congruent elliptic curves over ${\mathbb{Q}}$, that is, pairs of non-isogenous elliptic curves over ${\mathbb{Q}}$ whose $n$-torsion subgroups are isomorphic as Galois modules. We also find a non-trivial pair of 11-congruent elliptic curves over ${\mathbb{Q}}(T)$, and hence give an explicit infinite family of non-trivial pairs of 11-congruent elliptic curves  over ${\mathbb{Q}}$.

Supplementary materials are available with this article.

Let $G(q)$ be a finite Chevalley group, where $q$ is a power of a good prime $p$, and let $U(q)$ be a Sylow $p$-subgroup of $G(q)$. Then a generalized version of a conjecture of Higman asserts that the number $k(U(q))$ of conjugacy classes in $U(q)$ is given by a polynomial in $q$ with integer coefficients. In [S. M. Goodwin and G. Röhrle, J. Algebra 321 (2009) 3321–3334], the first and the third authors of the present paper developed an algorithm to calculate the values of $k(U(q))$. By implementing it into a computer program using $\mathsf{GAP}$, they were able to calculate $k(U(q))$ for $G$ of rank at most five, thereby proving that for these cases $k(U(q))$ is given by a polynomial in $q$. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of $k(U(q))$ for finite Chevalley groups of rank six and seven, except $E_7$. We observe that $k(U(q))$ is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write $k(U(q))$ as a polynomial in $q-1$, then the coefficients are non-negative.

Under the assumption that $k(U(q))$ is a polynomial in $q-1$, we also give an explicit formula for the coefficients of $k(U(q))$ of degrees zero, one and two.

Let χ be the primitive Dirichlet character of conductor 49 defined by χ(3)=ζ for ζ a primitive 42nd root of unity. We explicitly compute the slopes of the U7 operator acting on the space of overconvergent modular forms on X1(49) with weight k and character χ7k−6 or χ8−7k, depending on the embedding of ℚ(ζ) into ℂ7. By applying results of Coleman and of Cohen and Oesterlé, we are then able to deduce the slopes of U7 acting on all classical Hecke newforms of the same weight and character.

Van Wamelen [Math. Comp. 68 (1999) no. 225, 307–320] lists 19 curves of genus two over $\mathbf{Q}$ with complex multiplication (CM). However, for each curve, the CM-field turns out to be cyclic Galois over $\mathbf{Q}$, and the generic case of a non-Galois quartic CM-field did not feature in this list. The reason is that the field of definition in that case always contains the real quadratic subfield of the reflex field.

We extend Van Wamelen’s list to include curves of genus two defined over this real quadratic field. Our list therefore contains the smallest ‘generic’ examples of CM curves of genus two.

We explain our methods for obtaining this list, including a new height-reduction algorithm for arbitrary hyperelliptic curves over totally real number fields. Unlike Van Wamelen, we also give a proof of our list, which is made possible by our implementation of denominator bounds of Lauter and Viray for Igusa class polynomials.

We present a new method to propagate $p$-adic precision in computations, which also applies to other ultrametric fields. We illustrate it with some examples and give a toy application to the stable computation of the SOMOS 4 sequence.

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of $n$-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over $\mathbb{Q}$, ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$, ordered by height, in which we compute ranks and $2$-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon that we observe in our database is that the average rank eventually decreases as height increases.

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.

Extensive numerical experiments have been conducted by the authors, aimed at finding the admissible range of the ratios of the first three eigenvalues of a planar Dirichlet Laplacian. The results improve the previously known theoretical estimates of M. Ashbaugh and R. Benguria. Some properties of a maximizer of the ratio λ3/λ1 are also proved in the paper.