On invariants of elliptic curves on average
نویسندگان
چکیده
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 کاملOn the asymptotics for invariants of elliptic curves modulo p
Let E be an elliptic curve defined over Q. Let E(Fp) denote the elliptic curve modulo p. It is known that there exist integers i p and f p such that E(Fp) ∼= Z/ i pZ × Z/ i p f pZ. We study questions related to i p and f p . In particular, for any α > 0 and k ∈ N, we prove there exist positive constants cα and ck such that for any A > 0 ∑ p≤x (log i p) α = cα li(x) + O ( x (log x)A ) and ∑ p≤x ...
متن کاملClassical Invariants and 2-descent on Elliptic Curves
The classical theory of invariants of binary quartics is applied to the problem of determining the group of rational points of an elliptic curve deened over a eld K by 2-descent. The results lead to some simpliications to the method rst presented in (Birch and Swinnerton-Dyer, 1963), and can be applied to give a more eecient algorithm for determining Mordell-Weil groups over Q, as well as being...
متن کاملOn Silverman's conjecture for a family of elliptic curves
Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...
متن کاملOn the rank of certain parametrized elliptic curves
In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2015
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa168-1-3