How Many Rational Points Does a Random Curve Have?
نویسنده
چکیده
A large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties. For an elliptic curve over Q, the set of rational points forms a finitely generated abelian group. The ranks of these groups, when ranging over all elliptic curves, are conjectured to be evenly distributed between rank 0 and rank 1, with higher ranks being negligible. We will describe these conjectures and discuss some results on bounds for average rank, highlighting recent work of Bhargava and Shankar.
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 کاملSieving for rational points on hyperelliptic curves
We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even when our curve has points over all localizations Qp. We illustrate the practicality of the method with some ...
متن کاملA Conjecture on Rational Approximations to Rational Points
In this paper, we examine how well a rational point P on an algebraic variety X can be approximated by other rational points. We conjecture that if P lies on a rational curve, then the best approximations to P on X can be chosen to lie along a rational curve. We prove this conjecture for a wide range of examples, and for a great many more examples we deduce our conjecture from Vojta’s Main Conj...
متن کاملList of Problems
(a) Find Galois points and the Galois groups for singular plane curves. – for smooth curves, the number of Galois points is at most three (resp. four) if they are outer (resp. inner). The Galois groups are cyclic. [46, 62] – (i) How is the structure of Galois group and how many Galois points do there exist? Is it true that the maximal number of outer (resp. inner) Galois points is three (resp. ...
متن کاملGeneration Methods of Elliptic Curves
Let q be a prime power, and let E be an elliptic curve over the field F q of q elements. As usual we associate to E a finite set called the set of rational points of E over F q. We denote this set by E(F q). We will explain these terms in Chapter 2. Once we know that E(F q) actually is a finite Abelian group, we may define the discrete logarithm problem in E(F q) as usual. However, since the us...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012