Sextic Twists Research Articles

Vanishing of quartic and sextic twists of L-functions

Let E be an elliptic curve over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Q}}}$$\\end{document}. We conjecture asymptotic estimates for the number of vanishings of L(E,1,χ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L(E,1,\\chi )$$\\end{document} as χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi $$\\end{document} varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis on E. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Compared to earlier work by David, Fearnley and Kisilevsky that formulated analogous conjectures for characters of any odd prime order, in the composite order case, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. To do this, we introduce the notion of totally order ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} characters to quantify how quickly the quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (resp., sextic) characters is much slower than in the sub-family of totally quartic (resp., sextic) characters. We provide a conceptual explanation for this phenomenon by observing that the full family of order ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} twisted elliptic curve L-functions, with ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} even and composite, is a mixed family with both unitary and orthogonal aspects.

Maximal curves and Tate-Shafarevich results for quartic and sextic twists

We study elliptic surfaces corresponding to an equation of the specific type y2=x3+f(t)x, defined over the finite field Fq for a prime power q≡3mod4. It is shown that if s4=f(t) defines a curve that is maximal over Fq2 then the rank of the group of sections defined over Fq on the elliptic surface is determined in terms of elementary properties of the rational function f(t). Similar results are shown for elliptic surfaces given by y2=x3+g(t) using prime powers q≡5mod6 and curves s6=g(t). Finally, for each of the forms used here, existence of curves with the property that they are maximal over Fq2 is discussed, as well as various examples.

Ready-made short basis for GLV+GLS on high degree twisted curves

<abstract><p>The crucial step in elliptic curve scalar multiplication based on scalar decompositions using efficient endomorphisms—such as GLV, GLS or GLV+GLS—is to produce a short basis of a lattice involving the eigenvalues of the endomorphisms, which usually is obtained by lattice basis reduction algorithms or even more specialized algorithms. Recently, lattice basis reduction is found to be unnecessary. Benjamin Smith (AMS 2015) was able to immediately write down a short basis of the lattice for the GLV, GLS, GLV+GLS of quadratic twists using elementary facts about quadratic rings. Certainly it is always more convenient to use a ready-made short basis than to compute a new one by some algorithm. In this paper, we extend Smith's method on GLV+GLS for quadratic twists to quartic and sextic twists, and give ready-made short bases for $ 4 $-dimensional decompositions on these high degree twisted curves. In particular, our method gives a unified short basis compared with Hu et al.'s method (DCC 2012) for $ 4 $-dimensional decompositions on sextic twisted curves.</p></abstract>

On conjectural rank parities of quartic and sextic twists of elliptic curves

We study the behavior under twisting of the Selmer rank parities of a self-dual prime-degree isogeny on a principally polarized abelian variety defined over a number field, subject to compatibility relations between the twists and the isogeny. In particular, we study isogenies on abelian varieties whose Selmer rank parities are related to the rank parities of elliptic curves with [Formula: see text]-invariant 0 or 1728, assuming the Shafarevich–Tate conjecture. Using these results, we show how to classify the conjectural rank parities of all quartic or sextic twists of an elliptic curve defined over a number field, after a finite calculation. This generalizes the previous results of Hadian and Weidner on the behavior of [Formula: see text]-Selmer ranks under [Formula: see text]-twists.

GOLDFELD’S CONJECTURE AND CONGRUENCES BETWEEN HEEGNER POINTS

Given an elliptic curve$E$over$\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (respectively 1). We show that this conjecture holds whenever$E$has a rational 3-isogeny. We also prove the analogous result for the sextic twists of$j$-invariant 0 curves. For a more general elliptic curve$E$, we show that the number of quadratic twists of$E$up to twisting discriminant$X$of analytic rank 0 (respectively 1) is$\gg X/\log ^{5/6}X$, improving the current best general bound toward Goldfeld’s conjecture due to Ono–Skinner (respectively Perelli–Pomykala). To prove these results, we establish a congruence formula between$p$-adic logarithms of Heegner points and apply it in the special cases$p=3$and$p=2$to construct the desired twists explicitly. As a by-product, we also prove the corresponding$p$-part of the Birch and Swinnerton–Dyer conjecture for these explicit twists.

Fast Software Implementations of Bilinear Pairings

Advancement in pairing-based protocols has had a major impact on the applicability of cryptography to the solution of more complex real-world problems. However, the computation of pairings in software still needs to be optimized for different platforms including emerging embedded systems and high-performance PCs. Few works in the literature have considered implementations of pairings on the former applications despite their growing importance in a post-PC world. In this paper, we investigate the efficient computation of the Optimal-Ate pairing over special class of pairing friendly Barreto-Naehrig curves in software at different security levels. We target both applications and perform our implementations on ARM-powered processors (with and without NEON instructions) and PC processors. We exploit state-of-the-art techniques and propose new optimizations to speed up the computation in the different levels including tower field and curve arithmetic. In particular, we extend the concept of lazy reduction to inversion in extension fields, analyze an efficient alternative for the sparse multiplication used inside the Miller’s algorithm and reduce further the cost of point/line evaluation formulas in affine and projective homogeneous coordinates. In addition, we study the efficiency of using M-type and D-type sextic twists in the pairing computation and carry out a detailed comparison between affine, Jacobian, and homogeneous coordinate systems. Our implementations on various mass-market emerging embedded devices significantly improve the state-of-the-art of pairing computation on ARM-powered devices and x86-64 PC platforms. For ARM implementations we achieved considerably faster computations in comparison to the counterparts.

VARIATIONS ON HIGHER TWISTS OF PAIRS OF ELLIPTIC CURVES

In this note we show that for any pair of elliptic curves E1, E2 over ℚ with j-invariant equal to 0, we can find a polynomial D ∈ ℤ[u, v, w, t] such that the sextic twists of the curves E1, E2 by D(u, v, w, t) have rank ≥ 2 over the field ℚ(u, v, w, t). A similar result is proved for simultaneous quartic twists of pairs of elliptic curves with j-invariant 1728.

A NOTE ON HIGHER TWISTS OF ELLIPTIC CURVES

AbstractWe show that for any pair of elliptic curves E1, E2 over ℚ with j-invariant equal to 0, we can find a polynomial D ∈ ℤ[u, v] such that the cubic twists of the curves E1, E2 by D(u, v) have positive rank over ℚ(u, v). We also prove that for any quadruple of pairwise distinct elliptic curves Ei, i = 1, 2, 3, 4, with j-invariant j = 0, there exists a polynomial D ∈ ℤ[u] such that the sextic twists of Ei, i = 1, 2, 3, 4, by D(u) have positive rank. A similar result is proved for quadruplets of elliptic curves with j-invariant j = 1, 728.

On ranks of twists of elliptic curves and power-free values of binary forms

Let E be an elliptic curve over Q. By the rank of E we shall mean the rank of the group of rational points of E. Mestre [31], improving on the work of Neron [34] (cf. [13], [39] and [46]), has shown that there is an infinite family of elliptic curves over Q with rank at least 12. However, computational work (see [3], [4], [7] and [47]) suggests that a typical elliptic curve will have much smaller rank, with curves of rank 0 or 1 being predominant. Indeed, Brumer [6] has proved, subject to the Birch, Swinnerton-Dyer conjecture, the Shimura, Taniyama, Weil conjecture and the generalized Riemann hypothesis, that the average rank of an elliptic curve, ordered according to its Faltings height, is at most 2.3. In this article we shall study the behaviour of the rank as we run over twists of a given elliptic curve over Q. That is, we shall restrict our attention to families of elliptic curves defined over Q which are isomorphic over C. There are families of quadratic, cubic, quartic and sextic twists (see, for example, Proposition 5.4 of Chapter X of [42]). Let E be an elliptic curve over Q with Weierstrass equation y2 = X + ax + b and for any non-zero integer d let Ed 2 3 denote a quadratic twist of E given by the equation dy2 = X + ax + b. Let r(d) denote the rank of Ed . Note that if d1 and d2 are non-zero integers, then Ed is isomorphic to Ed over Q if and only if d1 /d2 is the square of a rational number. Subject to the conjectures of Birch and Swinnerton-Dyer and of Shimura, Taniyama, and Weil, Goldfeld [14] conjectured in 1979 that