Efficient computation of universal elliptic Gau{\ss} sums

In a former paper it has been shown that the elliptic Gau{\ss} sums, whose use has been proposed in the context of counting points on elliptic curves and primality tests, can be computed by using modular functions. In this work we give detailed algorithms for the necessary computations, all of which have been implemented in C. We analyse the relatively straightforward algorithms derived from the theory and provide several improvements speeding up computations considerably. In addition, slightly generalizing former results we describe how (elliptic) Jacobi sums may be determined in a very similar way and show how this can be used. We conclude by an analysis of space and run-time requirements of the algorithms.Comment: 19 pages, follow-up to arXiv:1707.0807

Universal elliptic Gau{\ss} sums and applications

We present new ideas for computing elliptic Gau{\ss} sums, which constitute an analogue of the classical cyclotomic Gau{\ss} sums and whose use has been proposed in the context of counting points on elliptic curves and primality tests. By means of certain well-known modular functions we define the universal elliptic Gau{\ss} sums and prove they admit an efficiently computable representation in terms of the $j$-invariant and another modular function. After that, we show how this representation can be used for obtaining the elliptic Gau{\ss} sum associated to an elliptic curve over a finite field $\mathbb{F}_p$, which may then be employed for counting points or primality proving.Comment: 16 page

Universal elliptic Gau{\ss} sums for Atkin primes in Schoof's algorithm

This work builds on earlier results. We define universal elliptic Gau{\ss} sums for Atkin primes in Schoof's algorithm for counting points on elliptic curves. Subsequently, we show these quantities admit an efficiently computable representation in terms of the $j$-invariant and two other modular functions. We analyse the necessary computations in detail and derive an alternative approach for determining the trace of the Frobenius homomorphism for Atkin primes using these pre-computations. A rough run-time analysis shows, however, that this new method is not competitive with existing ones.Comment: 13 pages, follow-up to arXiv:1707.08075.pdf and https://arxiv.org/pdf/1707.08610.pd

Fast Evaluation of Zolotarev Coefficients

We review the theory of elliptic functions leading to Zolotarev's formula for the sign function over the range (\epsilon \leq|x| \leq1). We show how Gauss' arithmetico-geometric mean allows us to evaluate elliptic functions cheaply, and thus to compute Zolotarev coefficients ``on the fly'' as a function of (\epsilon). This in turn allows us to calculate the matrix functions (\sgn H), (\sqrt H), and (1/\sqrt H) both quickly and accurately for any Hermitian matrix (H) whose spectrum lies in the specified range.Comment: 21 pages, 2 figures. Made various minor correction

Real and complex multiplication on K3 surfaces via period integration

We report on a new approach, as well as some related experiments, to construct families of K3 surfaces having real or complex multiplication. Fundamental ideas include considering the period space of marked K3 surfaces, determining the periods by numerical integration, as well as tracing the modular curve by a numerical continuation method

WHFast: A fast and unbiased implementation of a symplectic Wisdom-Holman integrator for long term gravitational simulations

We present WHFast, a fast and accurate implementation of a Wisdom-Holman symplectic integrator for long-term orbit integrations of planetary systems. WHFast is significantly faster and conserves energy better than all other Wisdom-Holman integrators tested. We achieve this by significantly improving the Kepler-solver and ensuring numerical stability of coordinate transformations to and from Jacobi coordinates. These refinements allow us to remove the linear secular trend in the energy error that is present in other implementations. For small enough timesteps we achieve Brouwer's law, i.e. the energy error is dominated by an unbiased random walk due to floating-point round-off errors. We implement symplectic correctors up to order eleven that significantly reduce the energy error. We also implement a symplectic tangent map for the variational equations. This allows us to efficiently calculate two widely used chaos indicators the Lyapunov characteristic number (LCN) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). WHFast is freely available as a flexible C package, as a shared library, and as an easy-to-use python module.Comment: Accepted by MNRAS, 13 pages, 4 figures, source code and tutorials available at http://github.com/hannorein/reboun

On p-adic L-functions for GL(n)×GL(n−1){\rm GL}(n)\times{\rm GL}(n-1) over totally real fields

We refine and extend previous constructions of $p$-adic $L$-functions for Rankin-Selberg convolutions on \GL(n)\times\GL(n-1) for regular algebraic representations over totally real fields. We also prove an intrinsic functional equation for this $p$-adic $L$-function, which will be of interest in further study of its arithmetic properties

On pairs of p-adic L-functions for weight two modular forms

The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions L_p^\sharp(f,T) and L_p^\flat(f,T) for a weight two modular form \sum a_n q^n and a good prime p. This generalizes work of Pollack who worked in the supersingular case and also assumed a_p=0. The Iwasawa functions work in tandem to shed some light on the Birch and Swinnerton-Dyer conjectures in the cyclotomic direction: We bound the rank and estimate the growth of the Tate-Shafarevich group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.Comment: This paper replaces and corrects most of the arxiv submission arXiv:1211.1352, which has been split into two articles. The major corrections are the non-uniqueness of the sharp/flat p-adic L-functions in the ordinary case, and the correct bounds on the p-adic analytic cyclotomic ran

A note on the security of CSIDH

We propose an algorithm for computing an isogeny between two elliptic curves $E_1,E_2$ defined over a finite field such that there is an imaginary quadratic order $\mathcal{O}$ satisfying $\mathcal{O}\simeq \operatorname{End}(E_i)$ for $i = 1,2$. This concerns ordinary curves and supersingular curves defined over $\mathbb{F}_p$ (the latter used in the recent CSIDH proposal). Our algorithm has heuristic asymptotic run time $e^{O\left(\sqrt{\log(|\Delta|)}\right)}$ and requires polynomial quantum memory and $e^{O\left(\sqrt{\log(|\Delta|)}\right)}$ classical memory, where $\Delta$ is the discriminant of $\mathcal{O}$. This asymptotic complexity outperforms all other available method for computing isogenies. We also show that a variant of our method has asymptotic run time $e^{\tilde{O}\left(\sqrt{\log(|\Delta|)}\right)}$ while requesting only polynomial memory (both quantum and classical)

The Eight Epochs of Math as regards past and future Matrix Computation

This paper gives a personal assessment of Epoch making advances in Matrix Computations from antiquity and with an eye towards tomorrow. We trace the development of number systems and elementary algebra, and the uses of Gaussian Elimination methods from around 2000 BC on to current real-time Neural Network computations to solve time-varying linear equations. We include relevant advances from China from the 3rd century AD on, and from India and Persia in the 9th century and discuss the conceptual genesis of vectors and matrices in central Europe and Japan in the 14th through 17th centuries AD. Followed by the 150 year cul-de-sac of polynomial root finder research for matrix eigenvalues, as well as the superbly useful matrix iterative methods and Francis' eigenvalue Algorithm from last century. Then we explain the recent use of initial value problem solvers to master time-varying linear and nonlinear matrix equations via Neural Networks. We end with a short outlook upon new hardware schemes with multilevel processors that go beyond the 0-1 base 2 framework which all of our past and current electronic computers have been using.Comment: 3 figures with subpart