منابع مشابه
On Invariants of Elliptic Curves on Average
Abstract. We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over Q and p be a prime of good reduction for E. Let eE (p) be the exponent of the group of rational points of the reduction modulo p of E over the finite field Fp. Let C be the family o...
متن کاملAverage Frobenius Distribution of Elliptic Curves
The Sato-Tate conjecture asserts that given an elliptic curve without complex multiplication, the primes whose Frobenius elements have their trace in a given interval (2α √ p, 2β √ p) have density given by 2 π R β α √ 1− t2 dt. We prove that this conjecture is true on average in a more general setting.
متن کاملAverage Twin Prime Conjecture for Elliptic Curves
Let E be an elliptic curve over Q. In 1988, N. Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over Fp is prime. This is an analogue of the Hardy–Littlewood twin prime conjecture in the case of elliptic curves. Koblitz’s Conjecture is still widely open. In this paper we prove that Koblitz’s Conjecture is true on average...
متن کاملAverage Size of 2-selmer Groups of Elliptic Curves, I
In this paper, we study a class of elliptic curves over Q with Qtorsion group Z2×Z2, and prove that the average order of the 2-Selmer groups is bounded.
متن کاملAverage Size of 2-selmer Groups of Elliptic Curves, Ii
In this paper, we consider the average order of the 2-Selmer groups of elliptic curves over Q given by the equation E : y2 = x(x + a)(x + b), where a and b are integers. We show that, with a being fixed, the average order of the 2-Selmer groups of such curves closely depends on a. More exactly, we show that the average order is bounded if |a| is not a square and unbounded if |a| is the square o...
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ژورنال
عنوان ژورنال: Bulletin of the Australian Mathematical Society
سال: 2002
ISSN: 0004-9727,1755-1633
DOI: 10.1017/s0004972700040211