Descending Rational Points on Elliptic Curves to Smaller Fields
نویسنده
چکیده
In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = 1, 2), S4 × Cm (m = 1, 2). Next, we consider the case where E has complex multiplication by the ring of integers O of an imaginary quadratic field K contained in K. Suppose that the O-rank over a Galois extension F is 1 or 2. If K 6= Q( √ −1) and Q( √ −3) and hK (class number of K) is odd, we show that E acquires positive O-rank over a cyclic extension of K or over a field whose Galois group is one of SL2(Z/3Z), an extension of SL2(Z/3Z) by Z/2Z, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of L-functions. Table of
منابع مشابه
Efficient elliptic curve cryptosystems
Elliptic curve cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on elliptic curve is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation...
متن کاملOn the elliptic curves of the form $ y^2=x^3-3px $
By the Mordell-Weil theorem, the group of rational points on an elliptic curve over a number field is a finitely generated abelian group. There is no known algorithm for finding the rank of this group. This paper computes the rank of the family $ E_p:y^2=x^3-3px $ of elliptic curves, where p is a prime.
متن کاملThe Number of Rational Points on Elliptic Curves y2 = x3 + a3 on Finite Fields
In this work, we consider the rational points on elliptic curves over finite fields Fp. We give results concerning the number of points Np,a on the elliptic curve y ≡ x + a(mod p) according to whether a and x are quadratic residues or non-residues. We use two lemmas to prove the main results first of which gives the list of primes for which -1 is a quadratic residue, and the second is a result ...
متن کاملThe family of indefinite binary quadratic forms and elliptic curves over finite fields
In this paper, we consider some properties of the family of indefinite binary quadratic forms and elliptic curves. In the first section, we give some preliminaries from binary quadratic forms and elliptic curves. In the second section, we define a special family of indefinite forms Fi and then we obtain some properties of these forms. In the third section, we consider the number of rational poi...
متن کاملOn the Elliptic Curves of the Form $y^2 = x^3 − pqx$
By the Mordell- Weil theorem, the group of rational points on an elliptic curve over a number field is a finitely generated abelian group. This paper studies the rank of the family Epq:y2=x3-pqx of elliptic curves, where p and q are distinct primes. We give infinite families of elliptic curves of the form y2=x3-pqx with rank two, three and four, assuming a conjecture of Schinzel ...
متن کاملGenerating Elliptic Curves over Finite Fields Part I: Generating by Complex Multiplication
We study the theory of rational points on elliptic curves over nite elds and the theory of complex multiplication through which we construct elliptic curves over F p such that their orders of the group of rational points over F p are of the form mr where r is a prime and m is a small integer.
متن کامل