On the number of distinct elliptic curves in some families

We give explicit formulas for the number of distinct elliptic curves over a finite field (up to isomorphism over the algebraic closure of the ground field) in several families of curves of cryptographic interest such as Edwards curves and their generalization due to D. J. Bernstein and T. Lange as well as the curves introduced by C. Doche, T. Icart and D. R. Kohel.     

On some anabelian properties of arithmetic curves

In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicated to describe the position of decomposition groups of points at the boundary of the scheme \({{\rm Spec}\, \mathcal{O}_{K, S}}\) , where K is a number field and S a set of primes of K, intrinsically in terms of the fundamental group. We prove that it is equivalent to give the following pieces of information additionally to the fundamental group \({\pi_1({\rm Spec}\, \mathcal{O}_{K, S})}\) : the location of decomposition groups of boundary points inside it, the p-part of the cyclotomic character, the number of points on the boundary of all finite étale covers, etc. Under a certain finiteness hypothesis on Tate–Shafarevich groups with divisible coefficients, one can reconstruct all these quantities simply from the fundamental group.     

On universal elliptic curves over Igusa curves

Summary The purpose of this note is to introduce the arithmetic, study of the universal elliptic curve over Igusa curves. Specifically, its Hasse-WeilL-function is computed in terms of modular forms and is shown to have interesting zeros. Explicit examples are presented for which the Birch and Swinnerton-Dyer conjecture is verified.This paper summarizes part of the author's Ph.D. thesis. He wishes to thank the Sloan Foundation for financial support in the form of a Doctoral Dissertation Fellowship and his advisor, Dick Gross, for mathematical guidance and inspirational enthusiasm.To my parents in their 50th year     

On oriented supersingular elliptic curves

We revisit theoretical background on OSIDH (Oriented Supersingular Isogeny Diffie-Hellman protocol), which is an isogeny-based key-exchange protocol proposed by Colò and Kohel at NutMiC 2019. We give a proof of a fundamental theorem for OSIDH. The theorem was stated by Colò and Kohel without proof. Furthermore, we consider parameters of OSIDH, give a sufficient condition on the parameters for the protocol to work, and estimate the size of the parameters for a certain security level.     

Potential modularity for elliptic curves and some applications

In this paper we prove the simultaneous potential modularity for a finite number of elliptic curves defined over a totally real field. As an application we prove the meromorphic continuation of some L-functions associated to elliptic curves and Tate conjecture for a product of 2 or 4 elliptic curves defined over a totally real field.     

On construction of certain CM elliptic curves

Let E be a CM elliptic curve defined over an algebraic number field F. In the previous paper [N. Murabayashi, On the field of definition for modularity of CM elliptic curves, J. Number Theory 108 (2004) 268-286], we gave necessary and sufficient conditions for E to be modular over F, i.e. there exists a normalized newform f of weight two on Γ₁(N) for some N such that HomF(E,Jf)≠{0}. We also determined the multiplicity of E as F-simple factor of Jf when HomF(E,Jf)≠{0}. In this process we separated into the three cases. In this paper we construct certain CM elliptic curves which satisfy the conditions of each case. In other words, we show that all three cases certainly occur.     

On the supersingular reduction of elliptic curves

Let a∈Q and denote byE _a the curvey ² = (x ² + l)(x + a). We prove thatE _a(F_p) is cyclic for infinitely many primesp. This fact was known previously only under the assumption of the generalized Riemann hypothesis. Research partially supported by NSERC grant A9418.     

On standardized models of isogenous elliptic curves

Let be isogenous elliptic curves over given by standardized Weierstrass models. We show that (in the obvious notation)

and, moreover, that there are integers such that

where .

     

On points of finite order on elliptic curves

A determination is made of all elliptic curves which possess, over a field K, points of order 10.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 563–567, May, 1970.     

On the Tate-Shafarevich groups of certain elliptic curves

The Tate-Shafarevich groups of certain elliptic curves over Fq(t) are related, via étale cohomology, to the group of points of an elliptic curve with complex multiplication. The Cassels-Tate pairing is computed under this identification.     

Solutions concentrating at curves for some singularly perturbed elliptic problems

We study positive solutions of the equation ?ε²Δu+u=u^p, where p>1 and ε>0 is small, with Neumann boundary conditions in a three-dimensional domain $\text{Ω}$. We prove the existence of solutions concentrating along some closed curve on $\text{?Ω}$. To cite this article: A. Malchiodi, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=F_q(t) having conductor (p)·∞, where (p) is a prime ideal in F_q[t]. Let d∈F_q[t] be an irreducible polynomial of odd degree, and let . Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗_FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗_FK,1)≠0.     

On the elliptic curves modulo p

We first prove Sun's three conjectures [Z.H. Sun, On the number of incongruent residues of x⁴+ax²+bx modulo p, J. Number Theory 119 (2006) 210-241; Z.H. Sun, http://sfb.hytc.edu.cn/xsjl/szh/, 2000, June] on the number of rational points of some elliptic curves over finite fields Fp, which are related to the congruence cubic and quartic residue. And we provide some examples and comments concerning these conjectures.