We construct two new families of basis for finite field extensions. Basis in the first family, the so-called elliptic basis, are not quite normal basis, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Basis in the second family, the so-called normal elliptic basis are normal basis and allow fast (quasi linear) arithmetic. We prove that all extensions admit models of this kind.
In this article, we construct algebraic equations for a curve C and a map f to an elliptic curve E, with pre-specified branching data. We do this by determining certain relations that the periods of C and E must satisfy and use these relations to approximate the coefficients to high precision. We then conjecture which algebraic numbers the coefficients are, and then we prove this conjecture to be correct.
We describe iterated integrals as unipotent periods on families of marked elliptic curves in terms of multiple zeta values and elliptic multiple zeta values.
In our PH.D. thesis we have showed that the Generalized Grothendieck's Conjecture of Periods applied to 1-motives, whose underlying semi-abelian variety is a product of elliptic curves and of tori, is equivalent to a transcendental conjecture involving elliptic integrals of the first and second kind, and logarithms of complex numbers. In this paper we investigate the Generalized Grothendieck's Conjecture of Periods in the case of 1-motives whose underlying semi-abelian variety is a non trivial extension of a product of elliptic curves by a torus. This will imply the introduction of elliptic in…
In this paper we prove a relation between the period of an elliptic curve and the period of its real and imaginary quadratic twists. This relation is often misstated in the literature.
We give an account of the complex Arithmetic-Geometric Mean (AGM), as first studied by Gauss, together with details of its relationship with the theory of elliptic curves over $\C$, their period lattices and complex parametrisation. As an application, we present efficient methods for computing bases for the period lattices and elliptic logarithms of points, for arbitrary elliptic curves defined over $\C$. Earlier authors have only treated the case of elliptic curves defined over the real numbers; here, the multi-valued nature of the complex AGM plays an important role. Our method, which we hav…
We compute the image of Enriquez' elliptic KZB associator in the (maximal) meta-abelian quotient of the fundamental Lie algebra of a once-punctured elliptic curve. Our main result is an explicit formula for this image in terms of Eichler integrals of Eisenstein series, and is analogous to Deligne's computation of the depth one quotient of the Drinfeld associator. We also show how to retrieve Zagier's extended period polynomials of Eisenstein series, as well as the values at zero of Beilinson--Levin's elliptic polylogarithms from the meta-abelian elliptic KZB associator.
We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.
We prove the generalised Tate conjecture for H^3 of products of elliptic curves over finite fields, by slightly modifying an argument of M. Spiess concerning the Tate conjecture. We prove it fully if the elliptic curves run among at most 3 isogeny classes. We also show how things become more intricate from H^4 onwards, for more that 3 isogeny classes.
It is proved that the rank of elliptic curves with complex multiplication introduced by B. H. Gross is one less an arithmetic complexity of the corresponding noncommutative tori with real multiplication.
Let E/K be an elliptic curve defined over a number field, and let p be a prime number such that E(K) has full p-torsion. We show that the order of the p-part of the Shafarevich-Tate group of E/L is unbounded as L varies over degree p extensions of K. The proof uses O'Neil's period-index obstruction. We deduce the result from the fact that, under the same hypotheses, there exist infinitely many elements of the Weil-Chatelet group of E/K of period p and index p^2.
We study elliptic functions in quaternionic analysis, and prove some analogues of classical theorems from the complex case. The main result is a relation between the periods of closed differential 1-forms and 3-forms on H/L where L is a lattice in H. By applying to forms that are regular in the sense of quaternionic analysis, we obtain quaternionic analogues of Riemann's period relations for an abelian surface. We also obtain a quaternionic analogue of Legendre's relation, as an equality between $2π^2$ and a linear combination of the four quasi-periods of the quaternionic Weierstrass $ζ$-funct…
Let $X_0^6(1)/W_6$ be the Atkin-Lehner quotient of the Shimura curve $X_0^6(1)$ associated to a maximal order in an indefinite quaternion algebra of discriminant $6$ over $\mathbb Q$. By realizing modular forms on $X_0^6(1)/W_6$ in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer's formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over $\overline\mathbb Q$ with complex multiplication.
Let $f(x)$ be a square free quartic polynomial defined over a quadratic field $K$ such that its leading coefficient is a square. If the continued fraction expansion of $\displaystyle \sqrt{f(x)}$ is periodic, then its period $n$ lies in the set \[\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,22,26,30,34\}.\] We write explicitly all such polynomials for which the period $n$ occurs over $K$ but not over $\Q$ and $\displaystyle n\not\in\{ 13,15,17\}$. Moreover we give necessary and sufficient conditions for the existence of such continued fraction expansions with period $13, 15$ or $17$ over $K$.
For any elliptic curve $E$ over $k\subset \Bbb R$ with $E({\Bbb C})={\Bbb C}^\times/q^{\Bbb Z}$, $q=e^{2πiz}, \Im(z)>0$, we study the $q$-average $D_{0,q}$, defined on $E({\Bbb C})$, of the function $D_0(z) = \Im(z/(1-z))$. Let $Ω^+(E)$ denote the real period of $E$. We show that there is a rational function $R \in {\Bbb Q}(X_1(N))$ such that for any non-cuspidal real point $s\in X_1(N)$ (which defines an elliptic curve $E(s)$ over $\Bbb R$ together with a point $P(s)$ of order $N$), $πD_{0,q}(P(s))$ equals $Ω^+(E(s))R(s)$. In particular, if $s$ is $\Bbb Q$-rational point of $X_1(N)$, a rare o…
For an elliptic curve $E$ defined over a field $k\subset \mathbb C$, we study iterated path integrals of logarithmic differential forms on $E^\dagger$, the universal vectorial extension of $E$. These are generalizations of the classical periods and quasi-periods of $E$, and are closely related to multiple elliptic polylogarithms and elliptic multiple zeta values. Moreover, if $k$ is a finite extension of $\mathbb Q$, then these iterated integrals along paths between $k$-rational points are periods in the sense of Kontsevich--Zagier.
I prove lower bounds of some parameters of elliptic curve over finite field. There parameters are closely interrelated with cryptographic stability of elliptic curve.
We show that every elliptic curve over a finite field of odd characteristic whose number of rational points is divisible by 4 is isogenous to an elliptic curve in Legendre form, with the sole exception of a minimal respectively maximal elliptic curve. We also collect some results concerning the supersingular Legendre parameters.
The Langlands Programme predicts that a weight 2 newform f over a number field K with integer Hecke eigenvalues generally should have an associated elliptic curve E_f over K. In our previous paper, we associated, building on works of Darmon and Greenberg, a p-adic lattice to f, under certain hypothesis, and implicitly conjectured that this lattice is commensurable with the p-adic Tate lattice of E_f . In this paper, we present this conjecture in detail and discuss how it can be used to compute, directly from f, an explicit Weierstrass equation for the conjectural E_f . We develop algorithms to…
In this paper, we continue the recent work of Fukshansky and Maharaj on lattices from elliptic curves over finite fields. We show that there exist bases formed by minimal vectors for these lattices except only one case. We also compute their determinants, and obtain sharp bounds for the covering radius.
We introduce an analogue of the Mertens conjecture for elliptic curves over finite fields. Using a result of Waterhouse, we classify the isogeny classes of elliptic curves for which this conjecture holds in terms the size of the finite field and the trace of the Frobenius endomorphism acting on the curve.
Initially motivated by the relations between Anabelian Geometry and Artin's L-functions of the associated Galois-representations, here we study the list of zeta-functions of genus two abelian coverings of elliptic curves over finite fields. Our goal is to provide a complete description of such a list.
Let $E$ be an elliptic curve over the finite field $\mathbb F_q$. We prove that, when $n$ is a sufficiently large positive integer, $\#E(\mathbb F_{q^n})$ has a prime factor exceeding $n\exp(c\log n/\log\log n)$.
We prove Deligne's conjecture for symmetric fifth $L$-functions of elliptic newforms of weight greater than $5$. As a consequence, we establish period relations between motivic periods associated to an elliptic newform and the Betti-Whittaker periods of its symmetric cube functorial lift to ${\rm GL}_4$.
