منابع مشابه
Eta-quotients and Elliptic Curves
In this paper we list all the weight 2 newforms f(τ) that are products and quotients of the Dedekind eta-function η(τ) := q ∞ Y n=1 (1− q), where q := e2πiτ . There are twelve such f(τ), and we give a model for the strong Weil curve E whose Hasse-Weil L−function is the Mellin transform for each of them. Five of the f(τ) have complex multiplication, and we give elementary formulae for their Four...
متن کاملConstructing elliptic curves over finite fields using double eta-quotients
We examine a class of modular functions for Γ(N) whose values generate ring class fields of imaginary quadratic orders. This fact leads to a new algorithm for constructing elliptic curves with complex multiplication. The difficulties arising when the genus of X0(N) is not zero are overcome by computing certain modular polynomials. Being a product of four η-functions, the proposed modular functi...
متن کاملQuotients of Elliptic Curves over Finite Fields
Fix a prime `, and let Fq be a finite field with q ≡ 1 (mod `) elements. If ` > 2 and q ` 1, we show that asymptotically (`− 1)/2` of the elliptic curves E/Fq with complete rational `-torsion are such that E/〈P 〉 does not have complete rational `-torsion for any point P ∈ E(Fq) of order `. For ` = 2 the asymptotic density is 0 or 1/4, depending whether q ≡ 1 (mod 4) or 3 (mod 4). We also show t...
متن کاملElliptic Nets and Elliptic Curves
Elliptic divisibility sequences are integer recurrence sequences, each of which is associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higher-dimensional analogue over arbitrary base fields. Suppose E is an elliptic curve over a field K, and P1, . . . , Pn are points on E defined over K. To this information we associate an ...
متن کاملGeneralized Jacobian and Discrete Logarithm Problem on Elliptic Curves
Let E be an elliptic curve over the finite field F_{q}, P a point in E(F_{q}) of order n, and Q a point in the group generated by P. The discrete logarithm problem on E is to find the number k such that Q = kP. In this paper we reduce the discrete logarithm problem on E[n] to the discrete logarithm on the group F*_{q} , the multiplicative group of nonzero elements of Fq, in the case where n | q...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1997
ISSN: 0002-9939,1088-6826
DOI: 10.1090/s0002-9939-97-03928-2